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NASA NTRS1973

On the stability of numerical integration routines for ordinary differential equations.

Numerical integration methods for the solution of initial value problems for ordinary vector differential equations may be modelled as discrete time feedback systems. The stability criteria discovered in modern control theory are applied to these systems and criteria involving the routine, the step size and the differential equation are derived. Linear multistep, Runge-Kutta, and predictor-corrector methods are all investigated.

Glover, K.↗
NASA NTRS1974

A method of automatically stabilizing helicopter sling loads

The effect of geometric and aerodynamic characteristics on the stability of the lateral degrees of freedom of a typical helicopter sling load is examined. The feasibility of stabilizing the suspended load by controllable fins was also studied. Linear control theory was applied to the design of a simple control law that stabilized the load over a wide range of helicopter airspeeds.

Gera, J.↗
NASA NTRS1977

Unsteady linearized transonic flow analysis for slender bodies

An unsteady linearized formulation based on Oswatitsch-Keune's parabolic method is developed to analyze transonic flow past oscillating slender bodies. In contrast to the widely used integral transform method, it is shown that all solutions can be derived by a simpler method directly in the physical plane. By various expansion procedures, low-frequency solutions then are derived according to two clearly defined frequency ranges. Adams-Sears' iteration is employed to account for the second-order effects. Stability derivatives are compared with available theories and data. It is found that the derivatives depend more sensitively on thickness than on the reduced frequency. Finally, a critical assessment of the present method is given.

Liu, D. D.↗
NASA NTRS1977

Effects of periodic discharges

Periodic capacity checks are assessed as well as the effects of periodic discharges on the cycle life and the performance of cells during the cycle life. Topics discussed include the effect of the amount of electrolyte on cell capacity at 35 C; battery design for spacecraft; electrolyte starvation theory; battery separator degradation; negative electrode stability; voltage regulation; operating temperatures; and integration of reconditioning systems using microprocessors.

Ford, F. E.↗
NASA NTRS1978

Phase-locked loops and their application

A collection of papers is presented on the characteristics and capabilities of phase-locked loops (PLLs), along with some applications of interest. The discussion covers basic theory (linear and nonlinear); acquisition; threshold; stability; frequency demodulation and detection; tracking; cycle slipping and loss of lock; phase-locked oscillators; operation and performance in the presence of noise; AGC, AFC, and APC circuits and systems; digital PLL; and applications and miscellaneous. With the rapid development of IC technology, PLLs are expected to be used widely in consumer electronics.

Lindsey, W. C.↗
NASA NTRS1979

Long-term risk analysis associated with nuclear waste disposal in space

An assessment and verification of previous analytic results on the long term risk of earth reentry for hazardous payloads is presented. The two areas were studied: (1) stability of nominal, near-circular storage orbits in the regions between Venus and earth and between earth and Mars, and (2) probability of earth reentry for off-nominal planet-crossing orbits resulting from deployment system failures. In the first area, numerical integrations of the equations of motion are compared with stability predications based on secular perturbation theory. The agreement is good in terms of the heliocentric distances covered and the general behavior of the orbital history, although certain near-resonance situations can lead to difficulty. In the second area, a Monte Carlo simulation of orbital evolution is used and the results compared with Opik's analytic theory of planetary encounters and collision statistics, with data verified to within a close order-of-magnitude.

Friedlander, A. L.↗
NASA NTRS1981

Bifurcations from stationary to periodic solutions in a low-order model of forced, dissipative barotropic flow

The considered investigation is concerned with periodic solutions in the context of a forced, dissipative, barotropic spectral model truncated to three complex coefficients with constant forcing on only the intermediate scale. It is found that determining a periodic solution of this three-coefficient model also reduces to finding the algebraic roots of a real polynomial. In the derivation of this polynomial, a class of hydrodynamic spectral systems is described for which a periodic solution might be similarly specified. The existence of periodic solutions of the three-coefficient model is controlled by the roots of the stability polynomial of the basic stationary solution, which represents the simplest response to the constant forcing. When the forcing exceeds a critical value, the basic solution becomes unstable. Owing to the nature of the roots of the stability polynomial at critical forcing, bifurcation theory guarantees the existence of a periodic solution.

Mitchell, K. E.↗
NASA NTRS1982

Unsteady Newton-Busemann flow theory. III - Frequency dependence and indicial response

Hui and Tobak applied the complete unsteady Newton-Busemann flow theory to the study of dynamic stability of oscillating aerofoils and bodies in revolution. The present article extends the results to general frequencies that may be applicable to flutter analysis. The results are likewise applied to the indicial response fluctuations in unsteady flow at very high Mach numbers. The study shows that for a group of body shapes in Newtonian flow (including the cone and wedge), the aerodynamic response to a step change in angle of attack or pitching velocity contains an initial-instant impulse followed by a rapid adjustment to the new steady-flow conditions. The impulse component is in effect an apparent mass term analogous to that which occurs initially in the aerodynamic indicial response at the zero Mach number limit.

Hui, W. H.↗
NASA NTRS1983

MHD stability of incompressible coronal loops with radiative energy loss

Previous studies of the magnetohydrodynamic (MHD) stability of solar coronal loops have not taken into account the effects of radiative energy loss in the energy equation. However, since coronal loops continuously lose energy by radiation and heat conduction, it is important to understand how these energy loss mechanisms affect MHD stability. We investigate the problem assuming that a magnetic loop has cylindrical geometry. As a first step, stability is studied for a localized mode, and the result is applied to a specific equilibrium. We find that the radiative energy loss effect not only changes the growth rate of ideally unstable modes, but also alters the stability boundary predicted by ideal MHD theory.

An, C.-H.↗
NASA NTRS1986

Convection without eddy viscosity: An attempt to model the interiors of giant planets

In the theory of hydrostatic quasi-geostrophic flow in the Earth's atmosphere the principal results do not depend on the eddy viscosity. This contrasts with published theories of convection in deep rotating fluid spheres, where the wavelength of the fastest growing disturbance varies as E sup 1/3, where E, the Ekman number, is proportional to the eddy viscosity. A new theory of quasi-columnar motions in stably stratified fluid spheres attempts to capture the luck of the meteorologists. The theory allows one to investigate the stability of barotropic and baroclinic zonal flows that extend into the planetary interior. It is hypothesized that the internal heat Jupiter and Saturn comes out not radially but on sloping surfaces defined by the internal entropy distribution. To test the hypothesis one searches for basic states in which the wavelength of the fastest-growing disturbance remains finite as E tends to zero, and is which the heat flux vector is radially outward and poleward.

Ingersoll, A. P.↗
NASA NTRS1986

Stability of semidiscrete approximations for hyperbolic initial-boundary-value problems: An eigenvalue analysis

A hyperbolic initial-boundary-value problem can be approximated by a system of ordinary differential equations (ODEs) by replacing the spatial derivatives by finite-difference approximations. The resulting system of ODEs is called a semidiscrete approximation. A complication is the fact that more boundary conditions are required for the spatially discrete approximation than are specified for the partial differential equation. Consequently, additional numerical boundary conditions are required and improper treatment of these additional conditions can lead to instability. For a linear initial-boundary-value problem (IBVP) with homogeneous analytical boundary conditions, the semidiscrete approximation results in a system of ODEs of the form du/dt = Au whose solution can be written as u(t) = exp(At)u(O). Lax-Richtmyer stability requires that the matrix norm of exp(At) be uniformly bounded for O less than or = t less than or = T independent of the spatial mesh size. Although the classical Lax-Richtmyer stability definition involves a conventional vector norm, there is no known algebraic test for the uniform boundedness of the matrix norm of exp(At) for hyperbolic IBVPs. An alternative but more complicated stability definition is used in the theory developed by Gustafsson, Kreiss, and Sundstrom (GKS). The two methods are compared.

Warming, Robert F.↗
NASA NTRS1991

The stability of numerical boundary treatments for compact high-order finite-difference schemes

The stability characteristics of various compact fourth and sixth order spatial operators are assessed using the theory of Gustafsson, Kreiss and Sundstrom (G-K-S) for the semi-discrete Initial Boundary Value Problem (IBVP). These results are then generalized to the fully discrete case using a recently developed theory of Kreiss. In all cases, favorable comparisons are obtained between the G-K-S theory, eigenvalue determination, and numerical simulation. The conventional definition of stability is then sharpened to include only those spatial discretizations that are asymptotically stable. It is shown that many of the higher order schemes which are G-K-S stable are not asymptotically stable. A series of compact fourth and sixth order schemes, which are both asymptotically and G-K-S stable for the scalar case, are then developed.

Carpenter, Mark H.↗
NASA NTRS1990

Bearingless rotor aeromechanical stability measurements and correlations using nonlinear aerodynamics

The aeromechanical stability of a 1/8th Froude scaled bearingless rotor model was investigated experimentally in a wind tunnel. Both shaft-fixed and shaft-free conditions were examined to study the aeroelastic stability of a bearingless rotor without the incorporation of auxiliary dampers. This wind tunnel investigation generated a set of stability data for four different advance ratios, and a wide range of collective pitch settings. Theoretical analysis was performed using the newly developed University of Maryland Advanced Rotorcraft Code (UMARC). For analysis, the blade is modeled as an elastic beam undergoing flap bending, lag bending, elastic twist, and axial deformation. Blade response is calculated using a finite element method in time. Nonlinear aerodynamic effects are included by using a semiempirical stall modeling. The linearized periodic rotor perturbation equations in the nonrotating frame are solved for stability roots using Floquet transition matrix theory, as well as constant coefficient approximation. The predicted results are compared with the experimental data.

Wang, James M.↗
NASA NTRS1992

Nonlinear development and secondary instability of large-amplitude Goertler vortices in hypersonic boundary layers

The weakly nonlinear development of large-amplitude Goertler vortices in a small neighborhood of the neutral stability position given by the linear theory is considered. It is shown that if the basic state and the wall curvature satisfy a certain condition, the Goertler vortices will grow until they become so large as to drive a mean flow correction as large as the basic state. Attention is then given to the further downstream evolution of these large-amplitude vortices, and it is shown that the vortices have a triple layer structure which consists of a region of vortex activity bounded by two transition layers over which the amplitude of the harmonic part of the vortex decays to zero exponentially. The positions of the two transition layers are found to be governed by a free boundary problem, which is solved for a number of curvature distribution cases. The secondary instability of the two transition layers is investigated with respect to traveling waves which are pi/2 out of phase in the spanwise direction with the steady Goertler vortices.

Fu, Y.↗
NASA NTRS1994

The effects of Kelvin-Helmholtz instability on resonance absorption layers in coronal loops

One of the long-standing uncertainties in the wave-resonance theory of coronal heating is the stability of the resonance layer. The wave motions in the resonance layer produce highly localized shear flows which vary sinusoidally in time with the resonance period. This configuration is potentially susceptible to the Kelvin-Helmholtz instability (KHI), which can enhance small-scale structure and turbulent broadening of shear layers on relatively rapid ideal timescales. We have investigated numerically the response of a characteristic velocity profile, derived from resonance absorption models, to finite fluid perturbations comparable to photospheric fluctuations. We find that the KHI primarily should affect long (approximately greater than 6 x 10(exp 4) km) loops where higher velocity flows (M approximately greater than 0.2) exist in resonance layers of order 100 km wide. There, the Kelvin-Helmholtz growth time is comparable to or less than the resonance quarter-period, and the potentially stabilizing magnetic effects are not felt until the instability is well past the linear growth stage. Not only is the resonance layer broadened by the KHI, but also the convective energy transport out of the resonance layer is increased, thus adding to the efficiency of the wave-resonance heating process. In shorter loops, e.g., those in bright points and compact flares, the stabilization due to the magnetic field and the high resonance frequency inhibit the growth of the Kelvin-Helmholtz instability beyond a minimal level.

Karpen, Judith T.↗
NASA NTRS2007

Charges on Strange Quark Nuggets in Space

Since Witten's seminal 1984 paper on the subject, searches for evidence of strange quark nuggets (SQNs) have proven unsuccessful. In the absence of experimental evidence ruling out SQNs, the validity of theories introducing mechanisms that increase their stability should continue to be tested. To stimulate electromagnetic SQN searches, particularly space searches, we estimate the net charge that would develop on an SQN in space exposed to various radiation baths (and showers) capable of liberating their less strongly bound electrons, taking into account recombination with ambient electrons. We consider, in particular, the cosmic background radiation, radiation from the sun, and diffuse galactic and extragalactic gamma-ray backgrounds. A possible dramatic signal of SQNs in explosive astrophysical events is noted.

Abers, E. S.↗
NASA NTRS2011

Nonlinear Dynamic Inversion Baseline Control Law: Architecture and Performance Predictions

A model reference dynamic inversion control law has been developed to provide a baseline control law for research into adaptive elements and other advanced flight control law components. This controller has been implemented and tested in a hardware-in-the-loop simulation; the simulation results show excellent handling qualities throughout the limited flight envelope. A simple angular momentum formulation was chosen because it can be included in the stability proofs for many basic adaptive theories, such as model reference adaptive control. Many design choices and implementation details reflect the requirements placed on the system by the nonlinear flight environment and the desire to keep the system as basic as possible to simplify the addition of the adaptive elements. Those design choices are explained, along with their predicted impact on the handling qualities.

Miller, Christopher J.↗