A Third-order and a Fourth-order Iteration Process for Nonlinear Equations
Convergence proof and procedure for third-order and fourth-order iteration process for nonlinear equations - Newton-Raphson second-order iteration
NEWTON-RAPHSON METHOD↗
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An equal-valued capacitor active RC network realization of a fourth-order low-pass Butterworth characteristic
This letter describes an active RC network which realizes a fourth-order low-pass Butterworth characteristic. It requires one active element and has the practical advantage that all capacitors have the same value. A design chart is included.
Huelsman, L. P.↗
Investigation of optimal control with a minimum-fuel consumption criterion for a fourth-order plant with two control inputs - synthesis of an efficient suboptimal control.
Pontryagin optimal principle applied to minimum fuel satellite attitude control with two feedback controls
OPTIMAL CONTROL↗
RC-RL decomposition of a fourth-order polynomial
Fourth order polynomial RC-RL decomposition conditions
Chan, S. P.↗
On the fourth-order anharmonic equation of state of solids
Elastic solids fourth order anharmonic equation of state from finite strain theory, resolving ambiguities by Mie-Gruneisen equation
Thomsen, L.↗
Application of the fourth-order anharmonic theory to the prediction of equations of state at high compressions and temperatures.
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Ahrens, T. J.↗
Higher order accurate partial implicitization: An unconditionally stable fourth-order-accurate explicit numerical technique
The previously obtained second-order-accurate partial implicitization numerical technique used in the solution of fluid dynamic problems was modified with little complication to achieve fourth-order accuracy. The Von Neumann stability analysis demonstrated the unconditional linear stability of the technique. The order of the truncation error was deduced from the Taylor series expansions of the linearized difference equations and was verified by numerical solutions to Burger's equation. For comparison, results were also obtained for Burger's equation using a second-order-accurate partial-implicitization scheme, as well as the fourth-order scheme of Kreiss.
Graves, R. A., Jr.↗
A pictorial study of an invariant torus in phase space of four dimensions.
An investigation was conducted with the aid of a computer graphics device at Goddard Space Flight Center to study the behavior of the invariant manifolds of a particular fourth-order equation, as a parameter in the equation is varied over the interval from 0 to 1. The equation consists of two coupled Van der Pol equations. For a small parameter value, the manifold is an asymptotically stable torus, where the flow on the torus is simply a rotation. As the value of the parameter is increased, the only thing that changes is the nature of the flow on the torus, which itself persists throughout the parameter variation. It is shown that ultimately the four periodic cycles which appear play a more significant part in the phase profile of the system than does the torus itself.
Baxter, R.↗
Asymptotic expansion of an impulse for an optimal finite burn
Closed form expressions are derived for the position and velocity of a spacecraft during a finite burn using the method involving the theory of asymptotic expansion of the optimal impulsive solution. The small parameter is given by the reciprocal of the mass flow rate. The expansion is given in terms of the impulsive solution and holds up through third-order for the velocity and fourth-order for the position.
Pines, S.↗
Undergraduate research report
Studies are reported of the probability distribution of yield strength for mild steel, and the application of numerical techniques to solve mechanical vibration problems. Tensile experiments were conducted for 50 cold rolled and 38 hot rolled mild steel rods. Chi-square, Kolmogro-Smirnov and goodness-of-fit methods are used to test the normality or lognormality of the experimental results. As an example of the application of numerical techniques, a nonlinear dash-pot-speing system subjected to seismic excitation is analyzed by using the fourth-order Runge-Kutta integrated method.
Yeh, H. Y.↗
Nearly spherical constant-power detonation waves as driven by focused radiation
An analysis is made of the flow within a three-dimensional explosion, or spark, created in a gas absorbing energy from a steady conical beam of radiation with nearly spherical symmetry. The radiation, typically from an array of lasers with a common focus, is assumed to be very intense and absorbed immediately behind an outwardly advancing strong shock. Departures of the laser power from spherical uniformity, which would result from practical problems of arrangement, are conveniently represented by an ascending series of Legendre polynomials in the polar angle. For nonuniformities of small amplitude, first-order perturbations of the flow field are analyzed in detail. Self-similarity is shown to be retained for zero counter-pressure and power constant with time. For the first five harmonics in power distortion, the resulting fourth-order system of equations is solved numerically for profiles of velocity components, density, pressure, and shock shape. Results are presented graphically.
George, Y. H.↗
A theory for predicting composite laminate warpage resulting from fabrication
Linear laminate theory is used with the moment-curvature relationship to derive equations for predicting end deflections due to warpage without solving the coupled fourth-order partial differential equations of the plate. Composite micro- and macrohyphenmechanics are used with laminate theory to assess the contribution of factors such as ply misorientation, fiber migration, and fiber and/or void volume ratio nonuniformity on the laminate warpage. Using these equations, it was found that a 1 deg error in the orientation angle of one ply was sufficient to produce warpage end deflection equal to two laminate thicknesses in a 10 inch by 10 inch laminate made from 8 ply Mod-I/epoxy. Using a sensitivity analysis on the governing parameters, it was found that a 3 deg fiber migration or a void volume ratio of three percent in some plies is sufficient to produce laminate warpage corner deflection equal to several laminate thicknesses. Tabular and graphical data are presented which can be used to identify possible errors contributing to laminate warpage and/or to obtain an a priori assessment when unavoidable errors during fabrication are anticipated.
Chamis, C. C.↗
First-order reactant in homogeneous turbulence before the final period of decay
A method is described for studying theoretically the concentration fluctuations of a dilute contaminate undergoing a first-order chemical reaction. The method is based on Deissler's (1958) theory for homogeneous turbulence for times before the final period, and it follows the approach used by Loeffler and Deissler (1961) to study temperature fluctuations in homogeneous turbulence. Four-point correlation equations are obtained; it is assumed that terms containing fifth-order correlation are very small in comparison with those containing fourth-order correlations, and can therefore be neglected. A spectrum equation is obtained in a form which can be solved numerically, yielding the decay law for the concentration fluctuations in homogeneous turbulence for the period much before the final period of decay.
Kumar, P.↗
Solid state thermal control for spacecraft
The steady-state heat-pumping capabilities of thermoelectric devices are examined for spacecraft in earth orbit, using a simple model. Thermoelectric parameters for a standard Bi2Te, alloy are fitted with fourth-order polynomials in temperature. Parameters are used which are the average for the temperature difference across the p-n couple. Internal spacecraft temperatures are varied from 270 to 330 K, and absorbed incident radiation fluxes are varied from zero to 700 W per sq m. It is found that heat rejection can be optimized with respect to thermoelectric device and radiator fin geometries, as well as with respect to input electrical power and that these devices are most useful on the surfaces with highest incident radiant power. Maximum heat fluxes which can be pumped out of the spacecraft are on the order of 50-160 W per sq m for the various thermal environments examined. Input electrical powers corresponding to these maximum heat fluxes range from about 50-300 W per sq m. However, efficiency is greatly improved by operation at lower input-power levels.
Harpster, J. W. C.↗
Directional spectra of ocean waves from microwave backscatter
The paper presents an analysis of two proposed microwave radar techniques for measuring ocean wave directional spectra. Tomiyasu's (1971) short pulse idea and Barrick's (1972) two-frequency correlation idea are regarded - independent of transmitted waveform - as essentially two alternative detection systems for modulated noise. Together, the two systems constitute a general detection system for modulated noise described some years ago by Parzen and Shiren (1956). A frequency domain analysis for backscatter on arbitrary incident waveform is given, and an interesting physical optics solution for the generalized fourth-order moments of the scattering matrix is obtained. It is shown that the present narrowband version of Barrick's two-frequency idea is impractical, and that the proper application of Barrick's idea is to wide band signals.
Jackson, F. C.↗
A theory for predicting composite laminate warpage resulting from fabrication
Linear laminate theory is used in conjunction with the moment-curvature relationship to derive equations for predicting end deflections due to warpage without solving the coupled fourth-order partial differential equations of the plate. Using these equations, it is found that a 1 deg error in the orientation angle of one ply is sufficient to produce warpage end deflection equal to two laminate thicknesses in a 10 inch by 10 inch laminate made from 8-ply Mod-I/epoxy. From a sensitivity analysis on the governing parameters, it is found that a 3 deg fiber migration or a void volume ratio of three percent in some plies is sufficient to produce laminate warpage corner deflection equal to several laminate thicknesses. Tabular and graphical data are presented which can be used to identify possible errors contributing to laminate warpage and/or to obtain an a priori assessment when unavoidable errors during fabrication are anticipated.
Chamis, C. C.↗
The influence of crack closure and elasto-plastic flow on the bending of a cracked plate
The influence of crack closure and elastoplastic flow on the bending behavior of thin cracked plates is investigated using an incremental elastoplastic plate bending finite element computer program. The finite element program was developed using assumptions consistent with Kirchhoff fourth-order plate theory while the material property treatment permits general isotropic work hardening with local elastic unloading. This technique is applied to the problem of a large centrally through cracked plate subject to remote circular bending. Comparison is drawn between two cases of the bending problem. The first neglects the possibility of crack face interference with bending, and the second includes a kinematic prohibition against the crack face from passing through the symmetry plane. Results are reported which isolate the effects of elastoplastic flow and crack closure.
Jones, D. P.↗
A critical study of higher-order numerical methods for solving the boundary-layer equations
A fourth-order box method is presented for calculating numerical solutions to parabolic, partial differential equations in two variables or ordinary differential equations. The method is the natural extension of the second-order Keller Box Scheme to fourth order and is demonstrated with application to the incompressible, laminar and turbulent boundary-layer equations. The efficiency of the present method is compared with other two-point and three-point higher-order methods; namely, the Keller Box Scheme with Richardson extrapolation, the method of deferred corrections, and the three-point spline methods. For equivalent accuracy, numerical results show the present method to be more efficient than the other higher-order methods for both laminar and turbulent flows.
Wornom, S. F.↗