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Using stiffness matrix and lumped-mass representation specified number of natural frequencies are obtained using inverse iteration method. Mode shapes for each frequency are also obtained. These frequencies and mode shapes can be found in reasonable periods of computer time utilizing this code.
The manned simulation of a large transport aircraft is described. Aircraft and systems data necessary to implement the mathematical model described in Volume I and a discussion of how these data are used in model are presented. The results of the real-time computations in the NASA Ames Research Center Flight Simulator for Advanced Aircraft are shown and compared to flight test data and to the results obtained in a training simulator known to be satisfactory.
The nonlinear equations of motion are solved algebraically for conditions for which an airplane is in an equilibrium spin. Constrained minimization techniques are employed in obtaining the solution. Linear characteristics of the airplane about the equilibrium points are also presented and their significance in identifying the stability characteristics of the equilibrium points is discussed. Computer time requirements are small making the method appear potentially applicable in airplane design. Results are obtained for several configurations and are compared with other analytic-numerical methods employed in spin prediction. Correlation with experimental results is discussed for one configuration for which a rather extensive data base was available. A need is indicated for higher Reynolds number data taken under conditions which more accurately simulate a spin.
A detailed discussion is presented of the framelet joining, photometric data improvement, and statistical error analysis. The Lunar Orbiter film handling system, readout system, and the digitization are described, along with the technique of joining adjacent framelets by a using a digital computer. Time and cost estimates are given. The problems and techniques involved in improving the digitized data are discussed. It was found that spectacular improvements are possible. Program documentations are included.
The computer program described calculates the design-point characteristics of a gas generator or a turbojet lift engine for V/STOL applications. The program computes the dimensions and mass, as well as the thermodynamic performance of the model engine and its components. The program was written in FORTRAN 4 language. Provision has been made so that the program accepts input values in either SI Units or U.S. Customary Units. Each engine design-point calculation requires less than 0.5 second of 7094 computer time.
In order to provide greater flexibility, shorter computer time, and a parametric study, attention was devoted to an implicit numerical scheme. The experimental work involved outlining and monitoring approximately fourteen tests of an invertable capacitor. Surface and PCM temperatures were measured at several locations and photographs were made of the PCM during melting and solidification. Some typical experimental results are presented and discussed.
Work on the dynamics of cable-connected spinning spacecraft was completed by formulating the equations of motion by both the canonical equations and Lagrange's equations and programming them for numerical solution on a digital computer. These energy-based formulations will permit future addition of the effect of cable mass. Comparative runs indicate that the canonical formulation requires less computer time. Available literature on urban mass transportation was surveyed. Areas of the private rapid transit concept of urban transportation are also studied.
The equations of motion of a geodetic satellite in the earth's gravitational field expressed by gravity anomalies require the evaluation, amongst others, of the partial derivatives of the disturbing force with respect to individual gravity anomalies. Data are discussed on how anomaly blocks should be subdivided so that the partial derivatives may be numerically evaluated for each subdivision, and then finally meaned to give the value representative of the whole blocks, with accuracies better than 2 to 3 percent for all blocks. The number of subdivisions is large for the blocks nearest to the satellite subpoint and decreases away from it. The actual values of this spherical distance and the actual subdivision of the mean gravity anomaly blocks was determined numerically for 184 15 deg x 15 deg equal area blocks. Satellite heights above the earth of 400 km, 800 km and 1600 km were considered. The computer times for the suggested scheme were compared with alternative solutions.
A pilot computer program was developed for the design of minimum mass wing structures under flutter, strength, and minimum gage constraints. The wing structure is idealized by finite elements, and second-order piston theory aerodynamics is used in the flutter calculation. Mathematical programing methods are used for the optimization. Computation times during the design process are reduced by three techniques. First, iterative analysis methods used to reduce significantly reanalysis times. Second, the number of design variables is kept small by not using a one-to-one correspondence between finite elements and design variables. Third, a technique for using approximate second derivatives with Newton's method for the optimization is incorporated. The program output is compared witH previous published results. It is found that some flutter characteristics, such as the flutter speed, can display discontinous dependence on the design variables (which are the thicknesses of the structural elements). It is concluded that it is undesirable to use such quantities in the formulation of the flutter constraint.
A one-dimensional, two-constraint radiation sheild weight optimization procedure and a computer program, DOPEX-1D2C, is described. The DOPEX-1D2C uses the steepest descent method to alter a set of initial (input) thicknesses of a spherical shield configuration to achieve a minimum weight while simultaneously satisfying two dose-rate constraints. The code assumes an exponential dose-shield thickness relation with parameters specified by the user. Code input instruction, a FORTRAN-4 listing, and a sample problem are given. Typical computer time required to optimize a seven-layer shield is less than 1/2 minute on an IBM 7094.
A method for iterative reanalysis in automated structural design is presented for a finite-element analysis using the direct stiffness approach. A basic feature of the method is that the generalized stiffness and inertia matrices are expressed as functions of structural design parameters, and these generalized matrices are expanded in Taylor series about the initial design. Only the linear terms are retained in the expansions. The method is approximate because it uses static condensation, modal reduction, and the linear Taylor series expansions. The exact linear representation of the expansions of the generalized matrices is also described and a basis for the present method is established. Results of applications of the present method to the recalculation of the natural frequencies of two simple platelike structural models are presented and compared with results obtained by using a commonly applied analysis procedure used as a reference. In general, the results are in good agreement. A comparison of the computer times required for the use of the present method and the reference method indicated that the present method required substantially less time for reanalysis. Although the results presented are for relatively small-order problems, the present method will become more efficient relative to the reference method as the problem size increases. An extension of the present method to static reanalysis is described, ana a basis for unifying the static and dynamic reanalysis procedures is presented.
A systematic way of applying both perturbation methods and harmonic balance methods to nonlinear panel flutter problems is developed here. Results obtained by both these methods for two-dimensional simply supported and three-dimensional clamped-clamped plates with six modes agree well with those obtained by the straightforward direct integration method, yet require less computer time and provide better insight into the solutions. Effects of viscoelastic structural damping on the flutter stability boundary are generally found to be destabilizing and the postflutter behavior becomes more explosive. The methods developed here may be of interest in related vibration problems.
A new analytical technique for the solution of nonlinear longitudinal combustion instability problems in rocket combustors is developed. Using relatively little computation time, this technique is capable of predicting the transient and limit cycle behavior of the combustion instability oscillations as well as the disturbance amplitude required to trigger an instability in a linearly stable motor. The limit cycle waveforms are found to exhibit shock wave characteristics for most unstable engine operating conditions. It is shown that the characteristics of the resulting instability are independent of the nature of the initial disturbance and they depend solely upon the engine operating conditions and the characteristics of the unsteady combustion process.
An approach is made to filtering of two-dimensional steady-state problems based on the notion of nearest neighbor interaction, i.e., at a given point in the spatial grid, the value at the variable of interest can be assumed to depend only on the values at adjacent grid points. It is shown that for linear steady-state problems significant dimensionality reductions can be accomplished. It was possible to achieve the desired results using a small amount of computer time and without getting into stability difficulties.
The matrix operator approach to radiative transfer is shown to be a very powerful technique in establishing symmetry relations for multiple scattering in inhomogeneous atmospheres. Symmetries are derived for the reflection and transmission operators using only the symmetry of the phase function. These results will mean large savings in computer time and storage for performing calculations for realistic planetary atmospheres using this method. The results have also been extended to establish a condition on the reflection matrix of a boundary in order to preserve reciprocity. Finally energy conservation is rigorously proven for conservative scattering in inhomogeneous atmospheres.
This paper considers digitized Weaver modulators for single-sideband modulation that can be used in radio communications and telemetering. A system of 12 such modulators to process 12 analog signals digitally is replaced by six multiplex modulators at the cost of additional computation at the input but with an overall reduction in computation time and a higher degree of accuracy. A unique digital filter is developed that can interpolate and output samples at multiples of the input sampling rate and exhibits a passband that occurs above the Nyquist frequency with no apparent distortion effects. A discussion of the Fortran simulation of a combined modulator-demodulator structure is included to verify the effectiveness of this scheme.
A direct three-dimensional elliptic solver is presented for application in a wide class of numerical methods for solving partial differential equations in physics and engineering. The derived algorithm and FORTRAN code implement Buzbee, Golub and Nielson's proposed extension of Buneman's Cyclic-Reduction Poisson solver to three dimensions. Both a 'most direct' cyclic reduction and a revised method (to eliminate roundoff error difficulties) are derived. Tests on an IBM 360/67 computer, using various optional combinations of subroutines, showed significant differences in accuracy and computing time, with the optimum subroutine combination depending on mesh size.
An original technique for determining the optimal magnetic torque strategy for control of the attitude of spin stabilized spacecraft is presented. By employing Lagrange multipliers and the Calculus of Variations, optimal control equations are derived which define minimum time and minimum energy attitude maneuvers. Computer program algorithms to numerically solve these optimal control equations are also described. The performance of this technique is compared with a commonly employed planning method.