Analytic power-series solution of the Schroedinger equation for the helium atom.
Analytic power series solution of nonrelativistic Schroedinger equation for two-electron atom, assuming fixed nucleus and singlet and triplet S states
POWER SERIES↗
SEARCH · Engineering Papers
Results for “State functions”
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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Ultrathin films as photomechanical transducers
In order to investigate the photomechanical effect in ultrathin films, several commercially available polymers of various structure and properties have been studied experimentally below the glass transition temperature. In the experiments using an Instron test machine equipped with a 200-g load cell and a microscope stage illuminator as the light source, the load at the fixed strain was recorded as a function of time. Plots of the steady-state change in stress as a function of the intensity of the incident radiation are presented for several of the materials studied. The response is found to be linear; the greater the intensity of the incident radiation, the greater is the change in stress.
Sarbolouki, M. N.↗
Reflection Modeling of the Black Hole Binary 4U 1630–47: The Disk Density and Returning Radiation
We present the analysis of X-ray observations of the black hole binary 4U 1630−47 using relativistic reflection spectroscopy. We use archival data from the Rossi X-ray Timing Explorer, Neil Gehrels Swift Observatory, and Nuclear Spectroscopic Telescope Array observatories, taken during different outbursts of the source between 1998 and 2015. Our modeling includes two relatively new advances in modern reflection codes: high-density disks, and returning thermal disk radiation. Accretion disks around stellar-mass black holes are expected to have densities well above the standard value assumed in traditional reflection models (i.e., n(e) ∼ 10^(15) per cu. cm). New high-density reflection models have important implications in the determination of disk truncation (i.e., the disk inner radius). This is because one must retain self-consistency in the irradiating flux and corresponding disk ionization state, which is a function of disk density and system geometry. We find that the disk density is n(e) ≥ 10^(20) per cu. cm across all spectral states. This density, combined with our constraints on the ionization state of the material, implies an irradiating flux impinging on the disk that is consistent with the expected theoretical estimates. Returning thermal disk radiation—the fraction of disk photons that bend back to the disk producing additional reflection components—is expected predominantly in the soft state. We show that returning radiation models indeed provide a better fit to the soft-state data, reinforcing previous results that show that in the soft state, the irradiating continuum may be blackbody emission from the disk itself.
Riley M. T. Connors↗
STEP and STEPSPL: Computer programs for aerodynamic model structure determination and parameter estimation
The successful parametric modeling of the aerodynamics for an airplane operating at high angles of attack or sideslip is performed in two phases. First the aerodynamic model structure must be determined and second the associated aerodynamic parameters (stability and control derivatives) must be estimated for that model. The purpose of this paper is to document two versions of a stepwise regression computer program which were developed for the determination of airplane aerodynamic model structure and to provide two examples of their use on computer generated data. References are provided for the application of the programs to real flight data. The two computer programs that are the subject of this report, STEP and STEPSPL, are written in FORTRAN IV (ANSI l966) compatible with a CDC FTN4 compiler. Both programs are adaptations of a standard forward stepwise regression algorithm. The purpose of the adaptation is to facilitate the selection of a adequate mathematical model of the aerodynamic force and moment coefficients of an airplane from flight test data. The major difference between STEP and STEPSPL is in the basis for the model. The basis for the model in STEP is the standard polynomial Taylor's series expansion of the aerodynamic function about some steady-state trim condition. Program STEPSPL utilizes a set of spline basis functions.
Batterson, J. G.↗
Linear canonical transformations of coherent and squeezed states in the Wigner phase space
It is shown that classical linear canonical transformations are possible in the Wigner phase space. Coherent and squeezed states are shown to be linear canonical transforms of the ground-state harmonic oscillator. It is therefore possible to evaluate the Wigner functions for coherent and squeezed states from that for the harmonic oscillator. Since the group of linear canonical transformations has a subgroup whose algebraic property is the same as that of the (2+1)-dimensional Lorentz group, it may be possible to test certain properties of the Lorentz group using optical devices. A possible experiment to measure the Wigner rotation angle is discussed.
Han, D.↗
Assessment of the Gaussian Covariance Approximation over an Earth-Asteroid Encounter Period
In assessing the risk an asteroid may pose to the Earth, the asteroids state is often predicted for many years, often decades. Only by accounting for the asteroids initial state uncertainty can a measure of the risk be calculated. With the asteroids state uncertainty growing as a function of the initial velocity uncertainty, orbit velocity at the last state update, and the time from the last update to the epoch of interest, the asteroids position uncertainties can grow to many times the size of the Earth when propagated to the encounter risk corridor. This paper examines the merits of propagating the asteroids state covariance as an analytical matrix. The results of this study help to bound the efficacy of applying different metrics for assessing the risk an asteroid poses to the Earth. Additionally, this work identifies a criterion for when different covariance propagation methods are needed to continue predictions after an Earth-encounter period.
planetary defense↗
Assessment of the Gaussian Covariance Approximation over an Earth-Asteroid Encounter Period
In assessing the risk an asteroid may pose to the Earth, the asteroids state is often predicted for many years, often decades. Only by accounting for the asteroids initial state uncertainty can a measure of the risk be calculated. With the asteroids state uncertainty growing as a function of the initial velocity uncertainty, orbit velocity at the last state update, and the time from the last update to the epoch of interest, the asteroids position uncertainties can grow to many times the size of the Earth when propagated to the encounter risk corridor. This paper examines the merits of propagating the asteroids state covariance as an analytical matrix. The results of this study help to bound the efficacy of applying different metrics for assessing the risk an asteroid poses to the Earth. Additionally, this work identifies a criterion for when different covariance propagation methods are needed to continue predictions after an Earth-encounter period.
uncertainty characterization↗
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.↗
Apollo flight crew vestibular assessment
Vestibular function in the weightless state of space flight is examined. Due to the lack of a systematic program to assess quantitatively the effects of space flight on crew vestibular function the analysis is based on qualitative information derived from motion sickness histories and subjective reporting by individual astronauts on the type and magnitude of vestibular disturbances experienced during and following their missions. It is concluded that the increased mobility afforded by the larger volume of the Apollo CM/LM resulted in a higher incidence of vestibular disturbances in the Apollo Program and that it is difficult to predict the likelihood of inflight vestibular problems. Quantitative examination of the effects of weightlessness on the vestibular function is recommended.
J.L. Homick↗
Optimal Feedback Control of Thermal Networks
An improved approach to the mathematical modeling of feedback control of thermal networks has been devised. Heretofore software for feedback control of thermal networks has been developed by time-consuming trial-and-error methods that depend on engineers expertise. In contrast, the present approach is a systematic means of developing algorithms for feedback control that is optimal in the sense that it combines performance with low cost of implementation. An additional advantage of the present approach is that a thermal engineer need not be expert in control theory. Thermal networks are lumped-parameter approximations used to represent complex thermal systems. Thermal networks are closely related to electrical networks commonly represented by lumped-parameter circuit diagrams. Like such electrical circuits, thermal networks are mathematically modeled by systems of differential-algebraic equations (DAEs) that is, ordinary differential equations subject to a set of algebraic constraints. In the present approach, emphasis is placed on applications in which thermal networks are subject to constant disturbances and, therefore, integral control action is necessary to obtain steady-state responses. The mathematical development of the present approach begins with the derivation of optimal integral-control laws via minimization of an appropriate cost functional that involves augmented state vectors. Subsequently, classical variational arguments provide optimality conditions in the form of the Hamiltonian equations for the standard linear-quadratic-regulator (LQR) problem. These equations are reduced to an algebraic Riccati equation (ARE) with respect to the augmented state vector. The solution of the ARE leads to the direct computation of the optimal proportional- and integral-feedback control gains. In cases of very complex networks, large numbers of state variables make it difficult to implement optimal controllers in the manner described in the preceding paragraph.
Papalexandris, Miltiadis↗
Some excited states of the helium molecule. i - the lowest super 1 sigma sub u plus and super 3 sigma sub g plus states and the first excited super 1 sigma sub g plus states.
Potential curves for excited states of helium molecule computed from variational wave functions
EXCITED STATE↗
Shuttle ascent trajectory optimization with function space quasi-Newton techniques
A Space Shuttle ascent trajectory optimization problem from lift-off to orbital insertion is solved with a function space version of a quasi-Newton parameter optimization method developed by Broyden. The problem includes five parameter and one bounded-function controls, two state-variable constraints, and four terminal conditions. The bounded controls are treated directly, while the remaining constraints are adjoined to the performance index (maximum payload) with penalty functions. The problem is formulated as a four-phase variational problem (liftoff, pitch-over, gravity-turn, linear tangent steering), and the appropriate gradients are developed by first variation theory. A projection operator is introduced to aid in the interpretation of the algorithm with mixed parameter and function controls.
Edge, E. R.↗
Stochastic estimation of organized turbulent structure - Homogeneous shear flow
A generalization of the conditional-eddy concept is proposed in which the conditional event specifies the local kinematic state in terms of the velocity and the deformation. Results are presented for stochastically estimated conditional eddies given the local kinematics. The equation governing the probability density function of a kinematic state has been derived for constant-property incompressible flow, providing a link between coherent flow structures corresponding to the conditional eddies and the modelling of turbulent transport. The primary contributions to the second-quadrant and fourth-quadrant Reynolds-stress events in homogeneous shear flow are shown to come from flow induced through the 'legs' and close to the 'heads' of upright and inverted 'hairpins', respectively.
Adrian, Ronald J.↗
A PDE Sensitivity Equation Method for Optimal Aerodynamic Design
The use of gradient based optimization algorithms in inverse design is well established as a practical approach to aerodynamic design. A typical procedure uses a simulation scheme to evaluate the objective function (from the approximate states) and its gradient, then passes this information to an optimization algorithm. Once the simulation scheme (CFD flow solver) has been selected and used to provide approximate function evaluations, there are several possible approaches to the problem of computing gradients. One popular method is to differentiate the simulation scheme and compute design sensitivities that are then used to obtain gradients. Although this black-box approach has many advantages in shape optimization problems, one must compute mesh sensitivities in order to compute the design sensitivity. In this paper, we present an alternative approach using the PDE sensitivity equation to develop algorithms for computing gradients. This approach has the advantage that mesh sensitivities need not be computed. Moreover, when it is possible to use the CFD scheme for both the forward problem and the sensitivity equation, then there are computational advantages. An apparent disadvantage of this approach is that it does not always produce consistent derivatives. However, for a proper combination of discretization schemes, one can show asymptotic consistency under mesh refinement, which is often sufficient to guarantee convergence of the optimal design algorithm. In particular, we show that when asymptotically consistent schemes are combined with a trust-region optimization algorithm, the resulting optimal design method converges. We denote this approach as the sensitivity equation method. The sensitivity equation method is presented, convergence results are given and the approach is illustrated on two optimal design problems involving shocks.
Borggaard, Jeff↗
An evaluation of random analysis methods for the determination of panel damping
An analysis is made of steady-state and non-steady-state methods for the measurement of panel damping. Particular emphasis is placed on the use of random process techniques in conjunction with digital data reduction methods. The steady-state methods considered use the response power spectral density, response autocorrelation, excitation-response crosspower spectral density, or single-sided Fourier transform (SSFT) of the response autocorrelation function. Non-steady-state methods are associated mainly with the use of rapid frequency sweep excitation. Problems associated with the practical application of each method are evaluated with specific reference to the case of a panel exposed to a turbulent airflow, and two methods, the power spectral density and the single-sided Fourier transform methods, are selected as being the most suitable. These two methods are demonstrated experimentally, and it is shown that the power spectral density method is satisfactory under most conditions, provided that appropriate corrections are applied to account for filter bandwidth and background noise errors. Thus, the response power spectral density method is recommended for the measurement of the damping of panels exposed to a moving airflow.
Bhat, W. V.↗
Barotropic instability of basic states with a realistic jet and a wave
The stability of basic states consisting of a jet similar to the stratospheric polar night jet and a traveling wave with a single zonal wavenumber is examined in a linearized nondivergent barotropic model on a sphere. Basic state waves are chosen to resemble observed traveling and stationary features in the winter stratosphere. Results are presented for disturbance growth rates, propagation characteristics, and energy conversion as a function of the basic state wave amplitude. The effects of small amplitude basic state waves on unstable disturbances arising from a zonally symmetric jet are discussed; results are shown where a small amplitude basic state wave dramatically affects the stability characteristics. Evidence is shown that the presence of a traveling wave may favor the appearance of disturbances that include other zonal wavenumbers which move with the basic state wave; this result is discussed in relation to the origin of observed quasi-nondispersive features in the polar winter stratosphere. Results for a stationary wavenumber 1 basic state wave suggest that a distorted polar vortex may be unstable to disturbances that would lead to further distortion. An unstable disturbance for a basic state with an eastward moving wavenumber 2 has components which resemble, in period and location, traveling waves that are observed in the winter stratosphere.
Manney, Gloria L.↗
Principle of corresponding states and equation of state of polymer liquids and glasses.
Isotherms and isobars for polymer liquids using corresponding states principle to establish function describing dependence of internal pressure on volume or temperature
INTERNAL PRESSURE↗
Ride comfort control in large flexible aircraft
The problem of ameliorating the discomfort of passengers on a large air transport subject to flight disturbances is examined. The longitudinal dynamics of the aircraft, including effects of body flexing, are developed in terms of linear, constant coefficient differential equations in state variables. A cost functional, penalizing the rigid body displacements and flexure accelerations over the surface of the aircraft is formulated as a quadratic form. The resulting control problem, to minimize the cost subject to the state equation constraints, is of a class whose solutions are well known. The feedback gains for the optimal controller are calculated digitally, and the resulting autopilot is simulated on an analog computer and its performance evaluated.
Warren, M. E.↗