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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 19 records

Propagation of sound waves through a linear shear layer: A closed form solution

Closed form solutions are presented for sound propagation from a line source in or near a shear layer. The analysis was exact for all frequencies and was developed assuming a linear velocity profile in the shear layer. This assumption allowed the solution to be expressed in terms of parabolic cyclinder functions. The solution is presented for a line monopole source first embedded in the uniform flow and then in the shear layer. Solutions are also discussed for certain types of dipole and quadrupole sources. Asymptotic expansions of the exact solutions for small and large values of Strouhal number gave expressions which correspond to solutions previously obtained for these limiting cases.

Scott, J. N.↗

Propagation of sound waves through a linear shear layer - A closed form solution

Closed form solutions are presented for sound propagation from a line source in or near a shear layer. The analysis is exact for all frequencies and is developed assuming a linear velocity profile in the shear layer. This assumption allows the solution to be expressed in terms of parabolic cylinder functions. The solution is presented for a line monopole source first embedded in the uniform flow and then in the shear layer. Solutions are also discussed for certain types of dipole and quadrupole sources. Asymptotic expansions of the exact solutions for small and large values of Strouhal number give expressions which correspond to solutions previously obtained for these limiting cases.

Scott, J. N.↗

Closed-form solutions for atomspheric flight with applications to shuttle guidance

Closed-form solutions for the motion of a rocket-powered vehicle during atmospheric ascent and closed-form solutions for unpowered atmospheric reentry are developed. These closed-form solutions are then used to develop a simplified guidance scheme and to develop a variation-of-parameters integration of more accurate equations of motion with the closed-form solutions as base solutions. The variation-of-parameters integration of the more accurate equations of motion also allows the transition partial derivative matrices associated with these equations to be easily developed. Then the partial derivative transition matrices are used to develop a guidance scheme based on the more accurate equations of motion instead of the less accurate closed-form solutions.

Ingram, H. L.↗

An exact closed form solution for constant area compressible flow with friction and heat transfer

The well-known differential equation for the one-dimensional flow of a compressible fluid with heat transfer and wall friction has no known solution in closed form for the general case. This report presents a closed form solution for the special case of constant heat flux per unit length and constant specific heat. The solution was obtained by choosing the square of a dimensionless flow parameter as one of the independent variables to describe the flow. From this exact solution, an approximate simplified form is derived that is applicable for predicting subsonic flow performance characteristics for many types of constant area passages in internal flow. The data included in this report are considered sufficiently accurate for use as a guide in analyzing and designing internal gas flow systems.

Sturas, J. I.↗

Closed-form solutions of performability

Methods which yield closed form performability solutions for continuous valued variables are developed. The models are similar to those employed in performance modeling (i.e., Markovian queueing models) but are extended so as to account for variations in structure due to faults. In particular, the modeling of a degradable buffer/multiprocessor system is considered whose performance Y is the (normalized) average throughput rate realized during a bounded interval of time. To avoid known difficulties associated with exact transient solutions, an approximate decomposition of the model is employed permitting certain submodels to be solved in equilibrium. These solutions are then incorporated in a model with fewer transient states and by solving the latter, a closed form solution of the system's performability is obtained. In conclusion, some applications of this solution are discussed and illustrated, including an example of design optimization.

Meyer, J. F.↗

A General Closed-Form Solution for the Lunar Reconnaissance Orbiter (LRO) Antenna Pointing System

The National Aeronautics and Space Administration s (NASA) Lunar Reconnaissance Orbiter (LRO) launched on June 18, 2009 from the Cape Canaveral Air Force Station aboard an Atlas V launch vehicle into a direct insertion trajectory to the Moon LRO, designed, built, and operated by the NASA Goddard Space Flight Center in Greenbelt, MD, is gathering crucial data on the lunar environment that will help astronauts prepare for long-duration lunar expeditions. During the mission s nominal life of one year its six instruments and one technology demonstrator will find safe landing site, locate potential resources, characterize the radiation environment and test new technology. To date, LRO has been operating well within the bounds of its requirements and has been collecting excellent science data images taken from the LRO Camera Narrow Angle Camera (LROC NAC) of the Apollo landing sites have appeared on cable news networks. A significant amount of information on LRO s science instruments is provided at the LRO mission webpage. LRO s Attitude Control System (ACS), in addition to controlling the orientation of the spacecraft is also responsible for pointing the High Gain Antenna (HGA). A dual-axis (or double-gimbaled) antenna, deployed on a meter-long boom, is required to point at a selected Earth ground station. Due to signal loss over the distance from the Moon to Earth, pointing precision for the antenna system is very tight. Since the HGA has to be deployed in spaceflight, its exact geometry relative to the spacecraft body is uncertain. In addition, thermal distortions and mechanical errors/tolerances must be characterized and removed to realize the greatest gain from the antenna system. These reasons necessitate the need for an in-flight calibration. Once in orbit around the moon, a series of attitude maneuvers was conducted to provide data needed to determine optimal parameters to load onboard, which would account for the environmental and mechanical errors at any antenna orientation. The nominal geometry for the HGA involves an outer gimbal axis that is exactly perpendicular to the inner gimbal axis, and a target direction that is exactly perpendicular to the outer gimbal axis. For this nominal geometry, closed-form solutions of the desired gimbal angles are simple to get for a desired target direction specified in the spacecraft body fame. If the gimbal axes and the antenna boresight are slightly misaligned, the nominal closed-form solution is not sufficiently accurate for computing the gimbal angles needed to point at a target. In this situation, either a general closed-form solution has to be developed for a mechanism with general geometries, or a correction scheme has to be applied to the nominal closed-form solutions. The latter has been adopted for Solar Dynamics Observatory (SDO) as can be seen in Reference 1, and the former has been used for LRO. The advantage of the general closed-form solution is the use of a small number of parameters for the correction of nominal solutions, especially in the regions near singularities. Singularities here refer to cases when the nominal closed-form solutions have two or more solutions. Algorithm complexity, however, is the disadvantage of the general closed-form solution.

Shah, Neerav↗

Closed-form solutions for a class of optimal quadratic tracking problems

Closed-form solutions are derived for a class of tracking problems including a linear optimal regulator and a prefilter for a time-invariant plant. The solutions for the prefilter equation and state trajectory coupled by the Riccati equation are exponentially related to the stability matrix of the plant. A computational procedure is presented in recursive form when the desired output state dynamics is assumed linear and time-invariant. Several examples are given for illustration.

Turner, J. D.↗

Closed-form solutions for a class of optimal quadratic regulator problems with terminal constraints

Closed-form solutions are derived for coupled Riccati-like matrix differential equations describing the solution of a class of optimal finite time quadratic regulator problems with terminal constraints. Analytical solutions are obtained for the feedback gains and the closed-loop response trajectory. A computational procedure is presented which introduces new variables for efficient computation of the terminal control law. Two examples are given to illustrate the validity and usefulness of the theory.

Juang, J.-N.↗

Closed-form solutions of performability

It is noted that if computing system performance is degradable then system evaluation must deal simultaneously with aspects of both performance and reliability. One approach is the evaluation of a system's performability which, relative to a specified performance variable Y, generally requires solution of the probability distribution function of Y. The feasibility of closed-form solutions of performability when Y is continuous are examined. In particular, the modeling of a degradable buffer/multiprocessor system is considered whose performance Y is the (normalized) average throughput rate realized during a bounded interval of time. Employing an approximate decomposition of the model, it is shown that a closed-form solution can indeed be obtained.

Meyer, J. F.↗

Optimal slewing maneuvers for flexible spacecraft using a closed form solution for the linear tracking problem

The problem of maneuvering a flexible spacecraft through a large angle while requiring the feedback control system to track a desired output state is considered. The desired output state is assumed to be provided by the corresponding open-loop maneuvering solution for the linear system model. Closed form solutions are provided for the Riccati and prefilter equations defining the optimal control. Example maneuvers are presented where control-rate penalties have been included in the performance index, in order to smooth both the open- and closed-loop control commands.

Turner, J. D.↗

Closed-form solutions for linear regulator-design of mechanical systems including optimal weighting matrix selection

This paper addresses the restriction of Linear Quadratic Regulator (LQR) solutions to the algebraic Riccati Equation to design spaces which can be implemented as passive structural members and/or dampers. A general closed-form solution to the optimal free-decay control problem is presented which is tailored for structural-mechanical systems. The solution includes, as subsets, special cases such as the Rayleigh Dissipation Function and total energy. Weighting matrix selection is a constrained choice among several parameters to obtain desired physical relationships. The closed-form solution is also applicable to active control design for systems where perfect, collocated actuator-sensor pairs exist. Some examples of simple spring mass systems are shown to illustrate key points.

Hanks, Brantley R.↗

Closed-form solutions for linear regulator design of mechanical systems including optimal weighting matrix selection

Vibration in modern structural and mechanical systems can be reduced in amplitude by increasing stiffness, redistributing stiffness and mass, and/or adding damping if design techniques are available to do so. Linear Quadratic Regulator (LQR) theory in modern multivariable control design, attacks the general dissipative elastic system design problem in a global formulation. The optimal design, however, allows electronic connections and phase relations which are not physically practical or possible in passive structural-mechanical devices. The restriction of LQR solutions (to the Algebraic Riccati Equation) to design spaces which can be implemented as passive structural members and/or dampers is addressed. A general closed-form solution to the optimal free-decay control problem is presented which is tailored for structural-mechanical system. The solution includes, as subsets, special cases such as the Rayleigh Dissipation Function and total energy. Weighting matrix selection is a constrained choice among several parameters to obtain desired physical relationships. The closed-form solution is also applicable to active control design for systems where perfect, collocated actuator-sensor pairs exist.

Hanks, Brantley R.↗

Closed-form solutions of supersonic wing-body interference

Closed-form formulas are presented for the interference factors of a body in the presence of a wing at supersonic speeds, when a finite afterbody is introduced. In addition, formulas are obtained for those cases in which the base of the body is forward of the trailing edge of the exposed root chord. These formulas are shown to be valid, subject to the restrictions inherent in the formulation given by Pitts, Nielsen and Kaattari (1959).

Vira, N. R.↗

Analysis and closed-form solutions of circular and rectangular apertures in the transverse plane of a circular waveguide

An analysis is carried out to determine the discontinuity susceptance of a circular or rectangular aperture in the transverse plane of a circular waveguide. Closed-form solutions based on the scattered amplitude calculation and aperture coupling theory are derived. Example solutions for a small circular aperture and a resonant rectangular aperture are given. The calculated results agree very well with the experiments. The results have many applications in the design of circular waveguide components and circular waveguide-backed aperture antennas.

Eastham, Gary B.↗

Directivity optimization of a reflector antenna with cluster feeds - A closed-form solution

The directivity of a reflector antenna deteriorates as the feed moves away from the focal point for beam scanning. This deterioration can be substantially reduced if a cluster feed instead of single feed is used to control a beam. A closed-form solution is presented for the cluster excitation to achieve the optimum directivity. For an offset 108 lambda parabolic reflector scanning 10 beamwidths, the optimum directivity achieved by a 19-element (seven-element) cluster is 12 dB (8 dB) higher than that of a single element. Comparison of the optimum-directivity design and the popular conjugate field matching design is made. When the cluster spacing d is greater than 1 lambda, it is found that the optimum directivity is higher than that of conjugate field matching (CFM) scheme by an insignificant amount, although the excitations of two designs can be quite different. For d less than 0.5 lambda, the optimum design may exhibit the supergain phenomenon, namely extremely high directivities achieved by an oscillatory cluster excitation.

Lam, P. T.↗