Iterative solution of nonlinear optimal control problems.
Mathematical programming methods for solving nonlinear state-constrained discrete optimal control problems
DYNAMIC PROGRAMMING↗
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On a transformation of the differential equations of the lunar theory
Hill-Brown differential equations for satellite coordinates transformed for integration with analytical programming language
Musen, P.↗
The influence of polarization on millimeter wave propagation through rain
A program for the measurement and analysis of the depolarization and differential attenuation that occur when millimeter wave signals propagate through rain is described. Initial data are taken along a 1.43 km path at 17.65 GHz and a supporting theoretical model is developed to relate the propagation effects to rainfall rate and wind velocity. A block diagram of the overall experiment is included. It consists of: (1) an RF system (millimeter wave transmitter and receiver), (2) transmitting and receiving antennas, (3) a weather system with rain gauges, wind sensors, and drop counters, and (4) a digital control, processing, and data storage system.
Bostian, C. W.↗
Analytical modeling of flash-back phenomena
To understand the flame flash-back phenomena more extensively, an analytical model was formed and a numerical program was written and tested to solve the set of differential equations describing the model. Results show that under a given set of conditions flame propagates in the boundary layer on a flat plate when the free stream is at or below 1.8 m/s.
Feng, C. C.↗
Thrust-vectored differential turns
Barrier surface construction in the joint space of the differential turning game for thrust-vectored vs. conventional aircraft is discussed. Differential-turn studies are based on modifications of existing computer programs including an energy-turn program, and one which generates hodograph data. Optimal turning flight in energy approximation is discussed for the conventional aircraft configurations. It is concluded that any advantages realized from thrust-vectoring are minor, unless hover is possible, where advantages would be major at low energies, and affect tactics at high energies as well.
Kelley, H. J.↗
Simulation of AFC for a DMSK Receiver
Low bit rates suitable for low-cost satellite communications. LOOP computer program written to simulate automatic frequency control (AFC) subsystem of differential minimum-shift keying (DMSK) receiver with bit rate of 2,400 baud. AFC simulated by LOOP is first-order loop configuration with first-order resistance-and-capacitance filter. Written in MS SuperSoft FORTRAN 77.
Davarian, Faramaz↗
Path-Following Solutions Of Nonlinear Equations
Report describes some path-following techniques for solution of nonlinear equations and compares with other methods. Use of multipurpose techniques applicable at more than one stage of path-following computation results in system relatively simple to understand, program, and use. Comparison of techniques with method of parametric differentiation (MPD) reveals definite advantages for path-following methods. Emphasis in investigation on multiuse techniques being applied at more than one stage of path-following computation. Incorporation of multipurpose techniques results in concise computer code relatively simple to use.
Barger, Raymond L.↗
Automatic differentiation as a tool in engineering design
Automatic Differentiation (AD) is a tool that systematically implements the chain rule of differentiation to obtain the derivatives of functions calculated by computer programs. AD is assessed as a tool for engineering design. The forward and reverse modes of AD, their computing requirements, as well as approaches to implementing AD are discussed. The application of two different tools to two medium-size structural analysis problems to generate sensitivity information typically necessary in an optimization or design situation is also discussed. The observation is made that AD is to be preferred to finite differencing in most cases, as long as sufficient computer storage is available; in some instances, AD may be the alternative to consider in lieu of analytical sensitivity analysis.
Barthelemy, Jean-Francois↗
Automatic Differentiation as a tool in engineering design
Automatic Differentiation (AD) is a tool that systematically implements the chain rule of differentiation to obtain the derivatives of functions calculated by computer programs. In this paper, it is assessed as a tool for engineering design. The paper discusses the forward and reverse modes of AD, their computing requirements, and approaches to implementing AD. It continues with application to two different tools to two medium-size structural analysis problems to generate sensitivity information typically necessary in an optimization or design situation. The paper concludes with the observation that AD is to be preferred to finite differencing in most cases, as long as sufficient computer storage is available.
Barthelemy, Jean-Francois M.↗
Single- and Multiple-Objective Optimization with Differential Evolution and Neural Networks
Genetic and evolutionary algorithms have been applied to solve numerous problems in engineering design where they have been used primarily as optimization procedures. These methods have an advantage over conventional gradient-based search procedures became they are capable of finding global optima of multi-modal functions and searching design spaces with disjoint feasible regions. They are also robust in the presence of noisy data. Another desirable feature of these methods is that they can efficiently use distributed and parallel computing resources since multiple function evaluations (flow simulations in aerodynamics design) can be performed simultaneously and independently on ultiple processors. For these reasons genetic and evolutionary algorithms are being used more frequently in design optimization. Examples include airfoil and wing design and compressor and turbine airfoil design. They are also finding increasing use in multiple-objective and multidisciplinary optimization. This lecture will focus on an evolutionary method that is a relatively new member to the general class of evolutionary methods called differential evolution (DE). This method is easy to use and program and it requires relatively few user-specified constants. These constants are easily determined for a wide class of problems. Fine-tuning the constants will off course yield the solution to the optimization problem at hand more rapidly. DE can be efficiently implemented on parallel computers and can be used for continuous, discrete and mixed discrete/continuous optimization problems. It does not require the objective function to be continuous and is noise tolerant. DE and applications to single and multiple-objective optimization will be included in the presentation and lecture notes. A method for aerodynamic design optimization that is based on neural networks will also be included as a part of this lecture. The method offers advantages over traditional optimization methods. It is more flexible than other methods in dealing with design in the context of both steady and unsteady flows, partial and complete data sets, combined experimental and numerical data, inclusion of various constraints and rules of thumb, and other issues that characterize the aerodynamic design process. Neural networks provide a natural framework within which a succession of numerical solutions of increasing fidelity, incorporating more realistic flow physics, can be represented and utilized for optimization. Neural networks also offer an excellent framework for multiple-objective and multi-disciplinary design optimization. Simulation tools from various disciplines can be integrated within this framework and rapid trade-off studies involving one or many disciplines can be performed. The prospect of combining neural network based optimization methods and evolutionary algorithms to obtain a hybrid method with the best properties of both methods will be included in this presentation. Achieving solution diversity and accurate convergence to the exact Pareto front in multiple objective optimization usually requires a significant computational effort with evolutionary algorithms. In this lecture we will also explore the possibility of using neural networks to obtain estimates of the Pareto optimal front using non-dominated solutions generated by DE as training data. Neural network estimators have the potential advantage of reducing the number of function evaluations required to obtain solution accuracy and diversity, thus reducing cost to design.
Rai, Man Mohan↗
Development of a Coherent Differential Absorption Lidar for Range Resolved Atmospheric CO2 Measurements
A pulsed, 2-m coherent Differential Absorption Lidar (DIAL) / Integrated Path Differential Absorption (IPDA) transceiver, developed under the Laser Risk Reduction Program (LRRP) at NASA, is integrated into a fully functional lidar instrument. This instrument will measure atmospheric CO2 profiles (by DIAL) initially from a ground platform, and then be prepared for aircraft installation to measure the atmospheric CO2 column densities in the atmospheric boundary layer (ABL) and lower troposphere. The airborne prototype CO2 lidar can measure atmospheric CO2 column density in a range bin of 1km with better than 1.5% precision at horizontal resolution of less than 50km. It can provide the image of the pooling of CO2 in lowlying areas and performs nighttime mass balance measurements at landscape scale. This sensor is unique in its capability to study the vertical ABL-free troposphere exchange of CO2 directly. It will allow the investigators to pursue subsequent in science-driven deployments, and provides a unique tool for Active Sensing of CO2 Emissions over Night, Days, and Seasons (ASCENDS) validation that was strongly advocated in the recent ASCENDS Workshop.
Yu, Jirong↗
Determination of flow areas for series-parallel compartment venting to satisfy pressure differential requirements.
A unique method for solving the design problem of determining the areas required to meet an imposed pressure differential limit is presented. The associated analytical technique for determining the transient pressure differentials in a multicompartment series-parallel connected system is also presented. In an effort to verify the results obtained by using the analytical methods presented, a test was conducted using a seven compartment system. It was found in the investigations that a computer program based on the analytical model described can determine the pressure, temperature, pressure differentials, and mass flow rates into and out of each compartment at subsonic or sonic velocities.
Kirby, C. E.↗
Spacecraft charging investigation - A joint research and technology program
A jointly planned U.S. Air Force-NASA program has been established to investigate the spacecraft charging phenomenon that has caused electronic anomalies in satellites in geosynchronous orbits. The objectives of this program are to provide design criteria, techniques, and test methods to insure control of absolute and differential charging of spacecraft surfaces. These objectives will be updated continuously over the next four years as data become available from the combined contractual and in-house programs. The geosynchronous altitude environment will be defined, ground and flight tests will be conducted, and materials and charge control techniques will be developed as required. The ultimate output of the program will be a spacecraft charging design criteria and test specification document. The program will be coordinated by a spacecraft charging program review group which has both Air Force and NASA representation.
Lovell, R. R.↗
Minimum fuel coplanar aeroassisted orbital transfer using collocation and nonlinear programming
The fuel optimal control problem arising in coplanar orbital transfer employing aeroassisted technology is addressed. The mission involves the transfer from high energy orbit (HEO) to low energy orbit (LEO) without plane change. The basic approach here is to employ a combination of propulsive maneuvers in space and aerodynamic maneuvers in the atmosphere. The basic sequence of events for the coplanar aeroassisted HEO to LEO orbit transfer consists of three phases. In the first phase, the transfer begins with a deorbit impulse at HEO which injects the vehicle into a elliptic transfer orbit with perigee inside the atmosphere. In the second phase, the vehicle is optimally controlled by lift and drag modulation to satisfy heating constraints and to exit the atmosphere with the desired flight path angle and velocity so that the apogee of the exit orbit is the altitude of the desired LEO. Finally, the second impulse is required to circularize the orbit at LEO. The performance index is maximum final mass. Simulation results show that the coplanar aerocapture is quite different from the case where orbital plane changes are made inside the atmosphere. In the latter case, the vehicle has to penetrate deeper into the atmosphere to perform the desired orbital plane change. For the coplanar case, the vehicle needs only to penetrate the atmosphere deep enough to reduce the exit velocity so the vehicle can be captured at the desired LEO. The peak heating rates are lower and the entry corridor is wider. From the thermal protection point of view, the coplanar transfer may be desirable. Parametric studies also show the maximum peak heating rates and the entry corridor width are functions of maximum lift coefficient. The problem is solved using a direct optimization technique which uses piecewise polynomial representation for the states and controls and collocation to represent the differential equations. This converts the optimal control problem into a nonlinear programming problem which is solved numerically by using a modified version of NPSOL. Solutions were obtained for the described problem for cases with and without heating constraints. The method appears to be more robust than other optimization methods. In addition, the method can handle complex dynamical constraints.
Shi, Yun Yuan↗
Computer program for the load and trajectory analysis of two DOF bodies connected by an elastic tether: Users manual
The derivation of the differential equations of motion of a 3 Degrees of Freedom body joined to a 3 Degrees of Freedom body by an elastic tether. The tether is represented by a spring and dashpot in parallel. A computer program which integrates the equations of motion is also described. Although the derivation of the equations of motions are for a general system, the computer program is written for defining loads in large boosters recovered by parachutes.
Doyle, G. R., Jr.↗
Random harmonic analysis program, L221 (TEV156). Volume 2: Supplemental system design and maintenenace document
Volume 2 of a two volume document is presented. A computer program, L222 (TEV 156), available for execution on the CDC 6600 computer is described. The program is capable of calculating steady-state solutions for linear second-order differential equations due to sinusoidal forcing functions. From this, steady-state solutions, generalized coordinates, and load frequency responses may be determined. Statistical characteristics of loads for the forcing function spectral shape may also be calculated using random harmonic analysis techniques. The particular field of application of the program is the analysis of airplane response and loads due to continuous random air turbulence.
Graham, M. L.↗
Dynamics of early planetary gear trains
A method to analyze the static and dynamic loads in a planetary gear train was developed. A variable-variable mesh stiffness (VVMS) model was used to simulate the external and internal spur gear mesh behavior, and an equivalent conventional gear train concept was adapted for the dynamic studies. The analysis can be applied either involute or noninvolute spur gearing. By utilizing the equivalent gear train concept, the developed method may be extended for use for all types of epicyclic gearing. The method is incorporated into a computer program so that the static and dynamic behavior of individual components can be examined. Items considered in the analysis are: (1) static and dynamic load sharing among the planets; (2) floating or fixed Sun gear; (3) actual tooth geometry, including errors and modifications; (4) positioning errors of the planet gears; (5) torque variations due to noninvolute gear action. A mathematical model comprised of power source, load, and planetary transmission is used to determine the instantaneous loads to which the components are subjected. It considers fluctuating output torque, elastic behavior in the system, and loss of contact between gear teeth. The dynamic model has nine degrees of freedom resulting in a set of simultaneous second order differential equations with time varying coefficients, which are solved numerically. The computer program was used to determine the effect of manufacturing errors, damping and component stiffness, and transmitted load on dynamic behavior. It is indicated that this methodology offers the designer/analyst a comprehensive tool with which planetary drives may be quickly and effectively evaluated.
August, R.↗
A users guide for A344: A program using a finite difference method to analyze transonic flow over oscillating airfoils
The design and usage of a pilot program for calculating the pressure distributions over harmonically oscillating airfoils in transonic flow are described. The procedure used is based on separating the velocity potential into steady and unsteady parts and linearizing the resulting unsteady differential equations for small disturbances. The steady velocity potential which must be obtained from some other program, was required for input. The unsteady equation, as solved, is linear with spatially varying coefficients. Since sinusoidal motion was assumed, time was not a variable. The numerical solution was obtained through a finite difference formulation and either a line relaxation or an out of core direct solution method.
Weatherill, W. H.↗