Showing papers in "Zeitschrift für Angewandte Mathematik und Physik in 1954"

Der Quotienten-Differenzen-Algorithmus

Abstract: 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.

Deformations possible in every isotropic, incompressible, perfectly elastic body

Abstract: Nous avons derive trois solutions des equations de l'elasticite non lineaire. Nous avons determine les deformations qui sont possibles dans un solide isotrope et incompressible quelconque.

Hautkondensation an feingewellten Oberflächen bei Berücksichtigung der Oberflächenspannungen

Abstract: Following a synopsis on film- and drop-condensation, one deals with condensation on an uneven surface. The surface tensions give rise to very great pressure gradients because of the different curvatures of the surface of the condensate. The action of the latter can amount to a multiple (up to a few powers with basis ten) of the value corresponding to the gravitation. Large gradients of pressure produce thin films of condensate, and according to the theory ofNusselt they lead to large coefficients of heat transmission. Heat transfer coefficients are given determined with calculus of finite differences for a condenser with vertical tubes, grooves parallel to the tube axis and given geometrical form. The latter are confirmed by tests. Adimensional numbers determinate theNusselt number.

The role of coordinate systems in boundary-layer theory

Abstract: The boundary layer approximation to a given flow problem is not invariant if different coordinate systems are used in the approximation process. However, a correlation theorem (Theorem 1) is given, which states that the boundary layer solution with respect to any given system can be found, by a simple substitution, from that with respect to any other system. On the basis of this theorem, the dependence of the solution on the choice of coordinates is investigated in detail. The skin friction is invariant, but the flow field is not invariant. At large distances from the wall, the flow field given by boundary layer theory depends almost entirely on the choice of coordinates, rather than on the physical problem. This dependence may be used to obtain a complete matching between the boundary layer solution and the external flow, in the following sense: Theorem 2 states how a coordinate system can be found such that the boundary layer solution with respect to this system is valid as an approximation for the entire flow field. It contains the external flow and the flow due to displacement thickness. The discussion is restricted to steady, two-dimensional, incompressible flow without separation. These restrictions, however, are not essential for many of the results.

Note on the boundary layer on a rotating sphere

Abstract: Die gleichmassige Stromung in der Grenzschicht bei einer gleichmassig rotierenden Kugel wird behandelt. Die Grenzschicht entsteht an den Polen und entwickelt sich in beiden Hemispharen nach dem Aquator zu. An den Polen verhalt sich die Kugel wie eine rotierende Scheibe, und Flussigkeit stromt hier ein. Am Aquator verhalt sich die Kugel wie ein rotierender Zylinder, und die Flussigkeit stromt in dessen Bereich aus.

On the dynamics of plastic circular plates

Abstract: 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.

Zur iterativen Auflösung algebraischer Gleichungen

Abstract: The slow convergence ofNewton's method for the iterative approximation of the roots of an algebraic equation may be improved by using a formula due toLaguerre. It is shown that this formula together with certain generalizations are useful for the numerical calculation of both real and complex zeros. A method is also given for the calculation of a further root of an equation after several roots have already been found, such that errors on the determination of the previous roots will not affect the accuracy with which the present one may be approximated.

The problem of the turbulent boundary layer

Abstract: Existing measurements of low-speed turbulent surface friction on a flat plate, in the absence of pressure gradient and roughness, are shown to be consistent with a simple analysis based on functional similarity in the velocity profile. In particular, the fully developed turbulent boundary layer is found to be unique within the accuracy of the experimental data, with uniqueness defined as the existence of a definite correspondence between local friction coefficient and momentum thickness Reynolds number. The relationships known as the law of the wall and the velocity defect law are found to describe the turbulent velocity profiles accurately for a considerable range of Reynolds numbers, and an effort is made to clarify the physical significance of these formulae. Finally, the proper definition of a length Reynolds number is discussed in terms of the asymptotic local properties of the ideal boundary layer, and numerical values for ideal mean and local friction coefficients are tabulated against Reynolds numbers based on momentum thickness and on distance from the leading edge.

Zur Theorie der Mahlvorgänge

Abstract: A mathematical theory of milling processes is developed by deriving and solving a partial integro-differential equation which describes the time-dependent particle size distribution of the milling charge. This basic equation contains a characteristic function of both the mill and the material milled, which is conveniently determined from measurements conducted in accordance with an approximation derived for this purpose from the general theory. This approximation, which can as well be used independently is equally convenient for obtaining experimental evidence of the validity of the general theory. It has been established in a previous paper that this approximation is in agreement with results obtained by experiments.

Anwendungen des Quotienten-Differenzen-Algorithmus

Abstract: Ist eine im Unendlichen regulare Funktion $$ % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qadaaeWbWdaeaapeWaaSaaa8aabaWdbiaadohapaWaaSbaaSqaa8qa % caWG4baapaqabaaakeaapeGaamOEa8aadaahaaWcbeqaa8qacaWG4b % Gaey4kaSIaaGymaaaaaaGccaaMc8Uaeyypa0JaaGPaVlaadAgacaGG % OaGaamOEaiaacMcaaSWdaeaapeGaaGimaaWdaeaapeGaeyOhIukani % abggHiLdaaaa!48BD! \sum\limits_0^\infty {\frac{{s_x }} {{z^{x + 1} }}\, = \,f(z)} $$ (1) gegeben, so kann man aus dem QD-Schema der Koeffizienten nach (I, 9) unmittelbar die fur die Aufstellung des S-Kettenbruches notwendigen Grossen ablesen; es ist namlich $$ % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX % garmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wy % Ubqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj-hEeeu0xXdbb % a9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXd % bPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaaa % baaaaaaaaapeGaamOzaiaacIcacaWG6bGaaiykaiaaykW7cqGH9aqp % caaMc8+aaSaaa8aabaWdbmaaeiaapaqaa8qacaWGZbWdamaaBaaale % aapeGaaGimaaWdaeqaaaGcpeGaayjcSdaapaqaa8qadaabbaWdaeaa % peGaamOEaaGaay5bSdaaaiaaykW7cqGHsislcaaMc8+aaSaaa8aaba % Wdbmaaeiaapaqaa8qacaWGXbWdamaaDaaaleaapeGaaGymaaWdaeaa % peGaaiikaiaaicdacaGGPaaaaaGccaGLiWoaa8aabaWdbmaaeeaapa % qaa8qacaaIXaaacaGLhWoaaaGaaGPaVlabgkHiTiaaykW7daWcaaWd % aeaapeWaaqGaa8aabaWdbiaadwgapaWaa0baaSqaa8qacaaIXaaapa % qaa8qacaGGOaGaaGimaiaacMcaaaaakiaawIa7aaWdaeaapeWaaqqa % a8aabaWdbiaadQhaaiaawEa7aaaacaaMc8UaeyOeI0IaaGPaVpaala % aapaqaa8qadaabcaWdaeaapeGaamyCa8aadaqhaaWcbaWdbiaaikda % a8aabaWdbiaacIcacaaIWaGaaiykaaaaaOGaayjcSdaapaqaa8qada % abbaWdaeaapeGaaGymaaGaay5bSdaaaiaaykW7cqGHsislcaaMc8+a % aSaaa8aabaWdbmaaeiaapaqaa8qacaWGLbWdamaaDaaaleaapeGaaG % OmaaWdaeaapeGaaiikaiaaicdacaGGPaaaaaGccaGLiWoaa8aabaWd % bmaaeeaapaqaa8qacaWG6baacaGLhWoaaaGaaGPaVlabgkHiTiaayk % W7cqGHMacVcaaMc8UaaiOlaaaa!813E! f(z)\, = \,\frac{{\left. {s_0 } \right|}} {{\left| z \right.}}\, - \,\frac{{\left. {q_1^{(0)} } \right|}} {{\left| 1 \right.}}\, - \,\frac{{\left. {e_1^{(0)} } \right|}} {{\left| z \right.}}\, - \,\frac{{\left. {q_2^{(0)} } \right|}} {{\left| 1 \right.}}\, - \,\frac{{\left. {e_2^{(0)} } \right|}} {{\left| z \right.}}\, - \, \ldots \,. $$ (2)

Theorie und Anwendungen des echten Kugelgetriebes

Abstract: A definition is given for a certain class of gears known as ‘genuine spherical gears’. The theoretical behaviour of these gears is analysed and formulae for calculating their efficiency are deduced. A large number of existing and possible applications for computing machines, servo systems and other purposes are described.

Boundary layer on rotating spheroids

Abstract: Die Grenzschichtgleichungen fur dreidimensionale Stromung an der Aussenseite eines rotierenden Spharoids werden in konfokalen Koordinaten abgeleitet und numerisch mittels der Methode vonPohlhausen fur verlangerte und verkurzte Spharoide gelost. Die Losungen zeigen einen Zustrom an den Polen und einen Abfluss an den Aquator. Der Ausfluss hangt vom Verhaltnisr/h ab, wobeir der Radius des Aquators undh die Hohe des Korpers in der Rotationsachse ist.

Rohrströmung mit schwachem Drall

Abstract: The steady laminar flow at high Reynolds' numbers in a straight tube of circular cross-section is investigated under the condition that swirl is present. Given any axially symmetrical distribution of the rotational velocities, the flow downstream is calculated in the special case that the rotational velocities are small compared to the axial velocity. The problem is reduced to an eigen-value problem of an ordinary linear differential equation of second order, the theory of which is developed in detail. Numerical values for the five lowest eigen-values and the corresponding eigen-functions are given. As an example, the decay of an initially rigid rotation is considered.

An arithmetic unit for automatic digital computers

Abstract: Fur eine programmgesteuerte Rechenmaschine mit einer magnetischen Trommel als Speicher wird ein fur den allgemeinen Gebrauch geeignetes Verzeichnis der arithmetischen Befehle gegeben und der logische Aufbau des Rechenwerks beschrieben. Vorgesehen ist das Rechnen sowohl mit gleitendem Komma (Zahlen der Forma·10 b ) als mit festem Komma, was das automatische Berechnen von Rechenplanen sowie das Rechnen mit verdoppelter Stellenzahl gestattet.

Über eine Funktionalgleichung aus der Wärmeleitung

Abstract: The functional equation $$2{\mathbf{ }}f\left( {\begin{array}{*{20}c} {x - y} \\ {\ln x/y} \\ \end{array} } \right) = f(x) \div f(y)$$ occuring in a heat conduction problem is solved analytically. The continuous solutions are discussed.

The theory of an oscillating cylinder viscometer

Abstract: The theoretical details of periodic state of motion of a liquid (Newtonian) in a narrow annular gap between long vertical coaxial cylinders is given in a new way.

Note on two-dimensional indentation problems of a non-isotropic semi-infinite elastic medium

Abstract: Es wird ein Losungsverfahren besprochen fur das Problem der elastischen, langs ihres Randes durch einen starren Stempel belasteten Halbebene. Die Methode wird fur einen einfachen Fall von Anisotropie am Beispiel eines Stempels mit annahernd kreisbogenformigem Rand illustriert.

Contribution à l'étude des compresseurs axiaux supersoniques

Abstract: In an attempt to study the three-dimensional flow in a supersonic axial compressor, the normal-shock-in-stator case has been considered. It is shown that the kinematic and dynamic conditions which are necessary to produce a normal shock wave and a radially stable transsonic flow are very restrictive: Furthermore, it is necessary to take in account the radial variations of the polytropic stage efficiencyη p in order to induce the prescribed energy distribution through one or several shock-free rotor-blade rows. A numerical example shows the importance of the radial effects. Finally, some critical comments are made on the shock-in-rotor case.

Stress distributions in semi-infinite solids of revolution

Abstract: The problem of obtaining the stress distribution within semi-infinite solids bounded by two concentric cones with a common apex is formulated in terms of a fourth order ordinary differential equation This is accomplished by using a recently developed theory for obtaining similarity solutions of partial differential equations Such a formulation is valid when the external loadings are proportional to an arbitrary real number power of the distance from the apex Closed solutions are given for these equations

On rotational viscous flow through a tube

Abstract: Fur die Eigenwerte und Eigenfunktionen eines Eigenwertproblems, auf dasL. Collatz undH. Gortler im Zusammenhang mit einer kurzlichen Untersuchung [3] uber Rohrstromung mit schwachem Drall gefuhrt worden sind, werden asymptotische Naherungsformeln angegeben, die insbesondere die hohern Eigenwerte mit grosserer Genauigkeit als in [3] zu berechnen gestatten. Auch zur Ermittlung der niedrigen Eigenwerte wird ein einfaches Verfahren vorgeschlagen. Die Ergebnisse werden numerisch mit denen von [3] verglichen.

The effect of cross-viscosity on the performance of full journal bearing without side-leakage

Abstract: Sommerfelds Theorie ergibt fur den Verlagerungswinkel eines vollen Gleitlagers den Betrag von 90°. In Wirklichkeit entsteht ein grosserer Verlagerungswinkel. Es wird hier nachgewiesen, dass die Berucksichtigung der Querviskositat die Unstimmigkeit zwischen Theorie und praktischer Beobachtung beseitigt.

Ein Impulsgenerator mit Sekundäremissionsröhren

Abstract: A pulse generator employing secondary emission valves is described. The generator produces positive and negative output pulses with adjustable pulse duration from 10−7 to 10−5 s and a rise time of about 1,3·10−8s. In addition to these test-pulses, signals for the beam brightening and the sweep of a cathode-ray oscillograph are generated, permitting the study of high-speed transients with an ordinary oscilloscope.

Erratum zu: Asymptotische Entwicklungen für Fresnelsche Integrale und verwandte Funktionen und ihre Anwendungsmöglichkeiten bei der Berechnung spezieller Raketenbahnen

Abstract: Calculating the influence of the wind on the trajectories of rockets (see foot-note 2) we find functions of the form $$f(x) = \int\limits_0^x {e^{ - t(k - i)} t^{ - 1/2} dt(i = \sqrt { - 1} )} $$ where, in general, 0≦k≦1 andx≧10. The present article shows that integrals of such a kind can be represented by asymptotic series convenient for the numerical evaluation with regard to the large values ofx. The results are given by formulae (6) to (8). Puttingk=0 we get the analogous formulae (9) and (10) forFresnel's integrals. To facilitate the computation, two auxiliary tables are added to this paper.

Ein Oszillationssatz für algebraische Eigenwertprobleme

Abstract: Conditions on a linear algebraic eigenvalue problem are given, under which there are exactlyk−1 changes of sign in the sequence of components of thek-th eigenvector. This is analogous to the oscillation theorems of differential equations. A class of difference equations which satisfies these conditions is defined. Finally a modification of a method ofCollatz is given, by means of which upper and lower bounds for thek-th eigenvalue may be derived from a trial vector havingk−1 sign changes in the sequence of its components. This paper is merely a summary of results; no proofs are given.

Prüfgerät für Photomultiplier

Abstract: Equipment for testing photo-multipliers with diffused light and with a special electrometer amplifier with dynamic condenser is described.

Die exakte Lösung der Integralgleichungen gewisser Schwingungsprobleme

Abstract: The vibration problem of certain dynamic systems with polynomial mass and stiffness distributions can be expressed as a Fredholm integral equation with a degenerated, symmetric kernel.

Aerodynamische Kräfte auf schlanke Rotationskörper

Abstract: Comparison between measurements of the normal force of fiveslender bodies of revolution with different base areas in supersonic flow and the simple airshiphull theory ofMunk or the more accurate linearised theory ofpotential flow show considerable discrepancy not only at large, but also at small angles of attack.

Die Konstruktion der mittleren Linie bei Oberflächenprofilen und einige Zusammenhänge zwischen den zur Kennzeichnung von Oberflächen benutzten Grössen

Abstract: Attention is directed first to the analogy between the problem of evaluating collectives and that of finding a numerical expression for an existing surface.