Showing papers in "Journal of The Franklin Institute-engineering and Applied Mathematics in 1948"

Partially plastic thick-walled tubes☆

Abstract: A theory is presented for the partial plastic yielding of thick-walled cylindrical tubes acted upon by any combination of internal pressure, external pressure and end load when the material follows an arbitrary stress-strain law. The solution combines (a) the distortion energy theory of plastic flow and (b) the effects of elastic compressibility of the plastic material. Numerical values for stresses and strains are given for certain special cases of internal pressure in thick-walled open-ended tubes for an idealized stress-strain law, and the results are compared with earlier approximate theories.

An integral-equation approach to problems of vibrating beams☆

Abstract: In this paper integral equations are applied for the calculation of the normal modes of vibrating beams. Both exact and approximate methods of solving the integral equation are considered. The Green's function, or kernel, of the integral equation is constructed for both uniform and nonuniform beams. Solutions for the normal modes of a uniform cantilever are given. A nonuniform, naturally-twisted turbine blade is studied in detail and the first and second normal modes are calculated by the integral-equation method.

Adiabatic flow of hydrogen gas through a rocket nozzle with and without composition change

Abstract: A procedure is described for determining the characteristics of adiabatic flow through a rocket nozzle with and without composition change. The method of calculation is illustrated for the expansion of pure hydrogen gas from a chamber temperature of 306° K. and a pressure of 20.42 atm. to atmospheric pressure. The study indicates that the exhaust velocity and temperature are highest for flow where complete equilibrium is reached at each temperature with respect to the reaction H 2 ⇋2H Flow with composition change requires a nozzle exit to nozzle throat area ratio somewhat greater than that determined for adiabatic flow without composition change for the same ratio of chamber pressure to exit pressure. The residence time in a given temperature range is computed as a function of gas temperature for the two types of flow. The results of this calculation may be used to determine the minimum required reaction rates which allow composition changes during flow through the nozzle.

Some factors in the Nessler method of nitrogen determination

Abstract: It has been shown that the amount of sulfuric acid used in the digestion of samples for the determination of nitrogen by the Nessler method may have a decided effect on the final color intensity. When 5.0 ml. of Nessler reagent are used for color development, a straight-line relationship between colorimeter readings and nitrogen concentrations is obtained, provided not more than 0.3 ml. of 1:1 sulfuric acid is used for the digestion. Furthermore, the method described herein can be applied to a wide range of nitrogen concentrations, from 5 to 400 micrograms, by appropriate choice of filters.

Ultrasonic delay lines II

Abstract: This paper deals with the experimental aspects of ultrasonic delay lines in liquid media and is a sequel to a previous paper treating the general theory of such devices. Delay devices employing ultrasonic propagation through liquids have proved satisfactory for delay times of the order of 3 milliseconds or less. Such devices can be made broadband and give excellent reproduction of pulse shape. The design considerations for such devices are discussed and illustrated by actual examples.

Inhibitors for ribonucleinase

Abstract: Reagents reacting with amino groups (formaldehyde, ninhydrin, phenylisocyanate) are inhibitory to ribonucleinase. Sodium chloride is also inhibitory.

Electron lenses of hyperbolic field structure.

Abstract: From the conditions of focus formation and Laplace's equation, a hyperbolic field structure is derived which permits a rigorous mathematical description and an explicit determination of electron trajectories of any amplitude. Various possible electron paths are formulated and charts are given for the most outstanding solutions, namely for symmetrical electron lenses and mirrors. The focal properties of such field formations are derived and the results condensed to closed formulations and plots easy to survey. Uniform rays of parallel incidence do not suffer any aberration within the volume of this field. The performance of the focal length, the principal planes and the back focus are discussed in detail, the latter being of great significance with respect to perfect image formation at high magnification. For thin lenses an approximation leads to a concise formulation which is in accord with published experiments.

Meteorological conditions accompanying mirages in the Salt Lake Desert

Abstract: Meteorological conditions commonly or invariably accompanying mirages in the Salt Lake Desert are here described, and the descriptions illustrated with photographs and instrumental records, where such are available and pertinent. Extended observation indicates that mirages are a standard and predictable phenomenon in the Salt Lake Desert. Less extended field studies show that similar meteorological conditions accompany mirages in all other American deserts.

Impact between chain roller and sprocket in a chain drive

Abstract: This paper presents some analytical relations which may help in developing a better understanding of the mechanics of roller-sprocket impact. Various relative impact velocities are derived. These velocities may be used in impact energy relations for determining limiting sprocket speeds on the basis of roller breakage, noise, heating, and sprocket wear. Some experimental data on roller breakage are presented and used to illustrate a method of organizing data. Impact energy using a certain velocity correlates to some extent with roller breakage.

Stabilization of carrier-frequency servomechanisms. I. Gain-phase margin diagrams of controller characteristics

Abstract: In this first part a method of analysis of linear carrier-frequency systems is indicated, and applied in particular to electrical servomechanisms using a two-phase alternating-current control motor. The performance of such a servo, as regards stability and fidelity of the output as a copy of the input, depends on the action of a controller-demodulator unit including the motor. A calculation of the steady-state effect of a linear transfer characteristic on the suppressed-carrier modulated error signal makes possible application of the Nyquist stability criterion. The method does not require symmetry of the characteristics about the carrier frequency; if such symmetry is present the system is similar to an ordinary or d-c. servo. Thus a band-rejection network tuned to the carrier frequency has an approximate proportional plus derivative action on the envelope of the error signal, resulting in an “anti-hunt” or stabilizing effect similar to that due to a translated band-rejection characteristic (center at zero-frequency) in a corresponding d-c. servo. A procedure is given for construction of transformed Nyquist diagrams, or gain-phase margin diagrams. The effect of the correctly tuned band-rejection network, as well as the effects of bandwidth, of the network being tuned off the carrier frequency, of increasing gain or velocity constant kν, and of the use of phase-shifting networks, are clearly shown by a series of such diagrams.

Science in national security

Abstract: In this issue’s cover story, “Rethinking the Unthinkable,” Houston T. Hawkins, a retired Air Force colonel and a Laboratory senior fellow, points out that since Vladimir Putin returned to power in Russia, relations with the United States and its NATO allies have cooled down like a thermometer in December. The biting off of Crimea from the Ukraine, the callous shooting down of a Malaysian passenger jet, and the frank words and deeds demonstrating Putin’s intention to consume more of the Ukraine and “other lands where people speak Russian” have raised concerns that, after two decades of hibernation, the Bear is awake— aggressive and very hungry. The relations have taken on a tone reminiscent of the Cold War.

Thermodynamics, part I: The second law from the standpoint of the equation of state

Abstract: In this paper the concepts “equilibrium” and “reversibility” are examined from the standpoint of the number of independent variables which define a thermodynamic system. The endeavor is made to show that the number of independent variables which are selected is crucial to the basic concepts as well as to the second law of thermodynamics. If this number is less than the minimum required to define the system completely, the equation of state for the internal energy will be incomplete. It is our practice to employ incomplete equations of state, and this results in processes which we consider to be irreversible. The Kelvin-Planck and Clausius principles are shown to be somewhat related to the first law of thermodynamics, and their validity arises from the fact that the equations of state which are used are incomplete. If these equations were complete, that is, if the number of independent variables were not less than the minimum required to describe the systems completely, we should have complete conversion of heat into work, as well as work into heat. The difficulties in realizing complete equations are considered. The entropy concept is considered in detail, and it is shown that there are serious limitations in its application.

A theory of the internal ballistics of the “hoch-und-niederdruck kanone”

Abstract: During the war the Germans developed guns with a constriction between chamber and bore, so that the pressure in the chamber was substantially higher than that in the bore. It is shown that a simple theory of such guns can be devised, which is adequate for analysis of firing trials and for testing the claims advanced for this idea. The claim of a high piezometric efficiency is not supported by this theoretical analysis.

Stabilization of carrier-frequency servomechanisms. II. Design formulae for proportional-derivative networks☆

Abstract: In an alternating current servomechanism, the error is proportional to the modulation envelope of a modulated-carrier error signal. It is shown in part I that for stability and fidelity of the servo, it is highly desirable that the effect of the controller includes a proportional-derivative action on the modulation envelope. This action may be obtained with various forms of RC networks, including the parallel “T,” bridge “T,” and Wien Bridge forms. This part contains detailed design procedures and tables of values for the various types of proportional-derivative networks. Several forms of parallel “T” networks arise from the fact that there are five independent time constants in the network, while in order to realize the desired transfer characteristic it is necessary to impose only four conditions. It is indicated how the remaining degree of freedom may be used to obtain the most suitable input and output impedances for the source and load impedances with which the parallel “T” is to be used. The derivations for the parallel “T” formulae are given in an Appendix. Tolerance requirements on the components of parallel “T” and bridge “T” networks are derived. If ±1 per cent components are used at 60 cycles, the resonant frequency will lie between 56.4 and 63.6 cycles, and the notch width (rejection band width) will be within ±0.99 cps. of the correct value. In order to guarantee that the phase shift at 60 cycles is within ±10°, the percentage deviation of each part must be less than ( 9.0 T d ω 0 ), where ω0 is the carrier angular frequency, Td the derivative time constant.