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NASA NTRS1993

On the dynamics of approximating schemes for dissipative nonlinear equations

Since one can rarely write down the analytical solutions to nonlinear dissipative partial differential equations (PDE's), it is important to understand whether, and in what sense, the behavior of approximating schemes to these equations reflects the true dynamics of the original equations. Further, because standard error estimates between approximations of the true solutions coming from spectral methods - finite difference or finite element schemes, for example - and the exact solutions grow exponentially in time, this analysis provides little value in understanding the infinite time behavior of a given approximating scheme. The notion of the global attractor has been useful in quantifying the infinite time behavior of dissipative PDEs, such as the Navier-Stokes equations. Loosely speaking, the global attractor is all that remains of a sufficiently large bounded set in phase space mapped infinitely forward in time under the evolution of the PDE. Though the attractor has been shown to have some nice properties - it is compact, connected, and finite dimensional, for example - it is in general quite complicated. Nevertheless, the global attractor gives a way to understand how the infinite time behavior of approximating schemes such as the ones coming from a finite difference, finite element, or spectral method relates to that of the original PDE. Indeed, one can often show that such approximations also have a global attractor. We therefore only need to understand how the structure of the attractor for the PDE behaves under approximation. This is by no means a trivial task. Several interesting results have been obtained in this direction. However, we will not go into the details. We mention here that approximations generally lose information about the system no matter how accurate they are. There are examples that show certain parts of the attractor may be lost by arbitrary small perturbations of the original equations.

Jones, Donald A.↗

NASA NTRS1972

Study of waves in the earth's bow shock.

Results of an examination of the perturbation vectors of waves upstream and downstream from the region of maximum compression in the bow shock on Ogo 5 under particularly steady solar-wind conditions. The polarization of the upstream waves was right-hand circular, and that of the downstream waves left-hand elliptical in the spacecraft frame. By observing that the polarization of the waves remained unchanged as the shock motion swept the wave structure back and forth across the satellite three times in eight minutes, it was found that the waves were not stationary in the shock frame. A study of the methods of determining the shock normal indicates that the normal estimated from a shock model should be superior to the normal based on magnetic coplanarity. The propagation vectors of the waves examined did not coincide with the shock-model normal, the average magnetic field, or the plasma-flow velocity. However, the major axis of the polarization ellipse of the downstream wave was nearly parallel to the upstream propagation vector.

Holzer, R. E.↗

NASA NTRS1972

Stability of large-scale systems with stable and unstable subsystems.

The purpose of this paper is to develop new methods for constructing vector Liapunov functions and broaden the application of Liapunov's theory to stability analysis of large-scale dynamic systems. The application, so far limited by the assumption that the large-scale systems are composed of exponentially stable subsystems, is extended via the general concept of comparison functions to systems which can be decomposed into asymptotically stable subsystems. Asymptotic stability of the composite system is tested by a simple algebraic criterion. With minor technical adjustments, the same criterion can be used to determine connective asymptotic stability of large-scale systems subject to structural perturbations. By redefining the constraints imposed on the interconnections among the subsystems, the considered class of systems is broadened in an essential way to include composite systems with unstable subsystems. In this way, the theory is brought substantially closer to reality since stability of all subsystems is no longer a necessary assumption in establishing stability of the overall composite system.

Grujic, Lj. T.↗

NASA NTRS1986

Monitoring vegetation recovery patterns on Mount St. Helens using thermal infrared multispectral data

The Mount St. Helens 1980 eruption offers an opportunity to study vegetation recovery rates and patterns in a perturbed ecosystem. The eruptions of Mount St. Helens created new surfaces by stripping and implacing large volumes of eroded material and depositing tephra in the blast area and on the flanks of the mountain. Areas of major disturbance are those in the blast zone that were subject to debris avalanche, pyroclastic flows, mudflows, and blowdown and scorched timber; and those outside the blast zone that received extensive tephra deposits. It was observed that during maximum daytime solar heating, surface temperatures of vegetated areas are cooler than surrounding nonvegetated areas, and that surface temperature varies with percent vegetation cover. A method of measuring the relationship between effective radiant temperature (ERT) and percent vegetation cover in the thermal infrared (8 to 12 microns) region of the electromagnetic spectrum was investigated.

Langran, Kenneth J.↗

NASA NTRS2009

Method and system to perform energy-extraction based active noise control

A method to provide active noise control to reduce noise and vibration in reverberant acoustic enclosures such as aircraft, vehicles, appliances, instruments, industrial equipment and the like is presented. A continuous-time multi-input multi-output (MIMO) state space mathematical model of the plant is obtained via analytical modeling and system identification. Compensation is designed to render the mathematical model passive in the sense of mathematical system theory. The compensated system is checked to ensure robustness of the passive property of the plant. The check ensures that the passivity is preserved if the mathematical model parameters are perturbed from nominal values. A passivity-based controller is designed and verified using numerical simulations and then tested. The controller is designed so that the resulting closed-loop response shows the desired noise reduction.

Kelkar, Atul↗

NASA NTRS2014

Improving Sensorimotor Adaptation Following Long Duration Space Flight by Enhancing Vestibular Information Transfer

Crewmember adapted to the microgravity state may need to egress the vehicle within a few minutes for safety and operational reasons after gravitational transitions. The transition from one sensorimotor state to another consists of two main mechanisms: strategic and plastic-adaptive and have been demonstrated in astronauts returning after long duration space flight. Strategic modifications represent "early adaptation" - immediate and transitory changes in control that are employed to deal with short-term changes in the environment. If these modifications are prolonged then plastic-adaptive changes are evoked that modify central nervous system function, automating new behavioral responses. More importantly, this longer term adaptive recovery mechanism was significantly associated with their strategic ability to recover on the first day after return to Earth G. We are developing a method based on stochastic resonance to enhance information transfer by improving the brain's ability to detect vestibular signals (Vestibular Stochastic Resonance, VSR) especially when combined with balance training exercises such as sensorimotor adaptability (SA) training for rapid improvement in functional skill, for standing and mobility. This countermeasure to improve detection of vestibular signals is a stimulus delivery system that is wearable/portable providing low imperceptible levels of white noise based binaural bipolar electrical stimulation of the vestibular system (stochastic vestibular stimulation). To determine efficacy of vestibular stimulation on physiological and perceptual responses during otolith-canal conflicts and dynamic perturbations we have conducted a series of studies: We have shown that imperceptible binaural bipolar electrical stimulation of the vestibular system across the mastoids enhances balance performance in the mediolateral (ML) plane while standing on an unstable surface. We have followed up on the previous study showing VSR stimulation improved balance performance in both ML and anteroposterior planes while stimulating in the ML axis only. We have shown the efficacy of VSR stimulations on enhancing physiological and perceptual responses of whole-body orientation during low frequency perturbations (0.1 Hz) on the ocular motor system using a variable radius centrifuge on both physiological (using eye movements) and perceptual responses (using a joystick) to track imposed oscillations. The variable radius centrifuge provides a selective tilting sensation that is detectable only by the otolith organs providing conflicting information from the canal organs of the vestibular system (intra-vestibular conflict). These results indicate that VSR can improve performance in sensory conflict scenarios like that experienced during space flight. We have showed the efficacy of VSR stimulation to improve balance and locomotor control on subjects exposed to continuous, sinusoidal lateral motion of the support surface while walking on a treadmill while viewing perceptually matched linear optic flow. We have shown the safety of short term continuous use of up to 4 hours of VSR stimulation and its efficacy in improving balance and locomotor function in Parkinson's Disease patients. This technique for improving vestibular signal detection may thus provide additional information to improve strategic abilities. We hypothesize that VSR stimulation will act synergistically with SA training to improve adaptability by increased utilization of vestibular information and therefore serve to optimize and personalize the SA countermeasure prescription. This forms the basis of its usefulness both as a training modality and further help in significantly reducing the number of days required to recover functional performance to preflight levels after long duration space flight.

Mulavara, A. P.↗

NASA NTRS2023

Probabilistic Forecasting of Ground Magnetic Perturbation Spikes at Mid-Latitude Stations

The prediction of large fluctuations in the ground magnetic field (dB/dt) is essential for preventing damage from Geomagnetically Induced Currents. Directly forecasting these fluctuations has proven difficult, but accurately determining the risk of extreme events can allow for the worst of the damage to be prevented. Here we trained Convolutional Neural Network models for eight mid-latitude magnetometers to predict the probability that dB/dt will exceed the 99th percentile threshold 30–60 min in the future. Two model frameworks were compared, a model trained using solar wind data from the Advanced Composition Explorer (ACE) satellite, and another model trained on both ACE and SuperMAG ground magnetometer data. The models were compared to examine if the addition of current ground magnetometer data significantly improved the forecasts of dB/dt in the future prediction window. A bootstrapping method was employed using a random split of the training and validation data to provide a measure of uncertainty in model predictions. The models were evaluated on the ground truth data during eight geomagnetic storms and a suite of evaluation metrics are presented. The models were also compared to a persistence model to ensure that the model using both datasets did not over-rely on dB/dt values in making its predictions. Overall, we find that the models using both the solar wind and ground magnetometer data had better metric scores than the solar wind only and persistence models, and was able to capture more spatially localized variations in the dB/dt threshold crossings.

Michael Coughlan↗

NASA NTRS1975

Short-periodic variations and second-order numerical averaging

The principal disadvantage of the method of numerical averaging is that it provides only the average time history of the orbital elements and yields no information about the high-frequency or short-periodic variations that occur inside the averaging interval. This paper contains a description of a technique for recovering the short-periodic variations by minor modifications to the averaging process so as to permit the construction of a Fourier series for the osculating elements. The availability of this series permits the extension of the averaging technique to higher order and allows us to account for short-periodic coupling of the perturbations. Comparisons of the results with numerically integrated solutions are presented for three distinct orbit prediction problems.

Lutzky, D.↗

NASA NTRS2014

Stability Assessment and Tuning of an Adaptively Augmented Classical Controller for Launch Vehicle Flight Control

Recently, a robust and practical adaptive control scheme for launch vehicles [ [1] has been introduced. It augments a classical controller with a real-time loop-gain adaptation, and it is therefore called Adaptive Augmentation Control (AAC). The loop-gain will be increased from the nominal design when the tracking error between the (filtered) output and the (filtered) command trajectory is large; whereas it will be decreased when excitation of flex or sloshing modes are detected. There is a need to determine the range and rate of the loop-gain adaptation in order to retain (exponential) stability, which is critical in vehicle operation, and to develop some theoretically based heuristic tuning methods for the adaptive law gain parameters. The classical launch vehicle flight controller design technics are based on gain-scheduling, whereby the launch vehicle dynamics model is linearized at selected operating points along the nominal tracking command trajectory, and Linear Time-Invariant (LTI) controller design techniques are employed to ensure asymptotic stability of the tracking error dynamics, typically by meeting some prescribed Gain Margin (GM) and Phase Margin (PM) specifications. The controller gains at the design points are then scheduled, tuned and sometimes interpolated to achieve good performance and stability robustness under external disturbances (e.g. winds) and structural perturbations (e.g. vehicle modeling errors). While the GM does give a bound for loop-gain variation without losing stability, it is for constant dispersions of the loop-gain because the GM is based on frequency-domain analysis, which is applicable only for LTI systems. The real-time adaptive loop-gain variation of the AAC effectively renders the closed-loop system a time-varying system, for which it is well-known that the LTI system stability criterion is neither necessary nor sufficient when applying to a Linear Time-Varying (LTV) system in a frozen-time fashion. Therefore, a generalized stability metric for time-varying loop=gain perturbations is needed for the AAC.

VanZwieten, Tannen↗

NASA NTRS2003

A Study on the Effects of J2 Perturbations on a Drag-Free Control System for Spacecraft in Low Earth Orbit

Low Earth Orbit (LEO) missions provide a unique means of gathering information about many of Earth s aspects such as climate, atmosphere, and gravitational field. Among the greatest challenges of LEO missions are designing, predicting, and maintaining the spacecraft orbit. The predominant perturbative forces acting on a spacecraft in LEO are J2 and higher order gravitational components, the effects of which are fairly easy to predict, and atmospheric drag, which causes the greatest uncertainty in predicting spacecraft ephemeris. The continuously varying atmospheric drag requires increased spacecraft tracking in order to accurately predict spacecraft location. In addition, periodic propulsive maneuvers typically must be planned and performed to counteract the effects of drag on the spacecraft orbit. If the effects of drag could be continuously and autonomously counteracted, the uncertainty in ephemeris due to atmospheric drag would essentially be eliminated from the spacecraft dynamics. One method of autonomous drag compensation that has been implemented on some missions is drag-free control. Drag-free control of a spacecraft was initially proposed in the 1960's and is discussed extensively by Lange. His drag-free control architecture consists of a free-floating proof mass enclosed within a spacecraft, isolating it from external disturbance forces such as atmospheric drag and solar radiation pressure. Under ideal conditions, internal disturbance forces can be ignored or mitigated, and the orbit of the proof mass depends only on gravitational forces. A sensor associated with the proof mass senses the movement of the spacecraft relative to the proof mass. Using the sensor measurements, the spacecraft is forced to follow the orbit of the proof mass by using low thrust propulsion, thus counteracting any non-gravitational disturbance forces. If the non-gravitational disturbance forces are successfully removed, the spacecraft s orbit will be affected only by well-known gravitational forces and will thus be easier to predict.

Vess, Melissa Fleck↗

NASA NTRS1982

Theory of radio occultation by Saturn's rings

The radio occultation technique, as applied to Saturn's rings, is developed as a new method for the study of the physical properties of planetary ring systems. The rings are treated as a Doppler-spread radar target composed of an ensemble of discrete scatterers. The mathematical formulation of the received signal as a random-phasor-sum process is carried out following a conventional radar theory approach, providing a convenient starting point for deriving coherent signal parameters. A classical result is rederived for the equivalent refractive index of the medium. The analysis is generalized to include ringlets of arbitrary width and it is shown that when the width is such that two adjacent rays are differentially perturbed in phase, ray bending that causes focusing of the coherent signal may result. The diffuse component is also treated in detail.

Marouf, E. A.↗

NASA NTRS1974

Special perturbations employing osculating reference states

The concept of employing osculating reference position and velocity vectors in the numerical integration of the equations of motion of a satellite is examined. The choice of the reference point is shown to have a significant effect upon numerical efficiency and the class of trajectories described by the differential equations of motion. For example, when the position and velocity vectors on the osculating orbit at a fixed reference time are chosen, a universal formulation is yielded. For elliptical orbits, however, this formulation is unattractive for numerical integration purposes due to Poisson terms (mixed secular) appearing in the equations of motion. Other choices for the reference point eliminate this problem but usually at the expense of universality. A number of these formulations, including a universal one, are considered here. Comparisons of the numerical characteristics of these techniques with those of the Encke method are presented.

Born, G. H.↗

NASA NTRS2000

A Determination of the Precession and Rotation-Rate of Mars from Tracking and Altimetry Data

The doppler tracking of the MGS spacecraft and the laser altimetry acquired by the MOLA instrument on MGS have been used to measure the rotation of Mars. By observing the repeating pattern of gravity anomalies and the topography the direction of the rotation axis in both inertial space and body fixed coordinates, and the rate of rotation of the planet on its axis have been derived. The tracking of MGS in orbit about Mars by stations on Earth permit the orbit of MGS to be obtained in an inertial coordinate system enabling the periodic perturbation of the MGS orbit by Mars' gravity field to be used as a sensor to detect the vector rotation of Mars under the orbit of MGS. The altimetry data acquired by MOLA on MGS permit the topography to help define the orbit of MGS about Mars in a body fixed coordinate system thus enabling the pole of rotation of the planet to be determined in a coordinate system defined by the altimetry. Together they enable the precession of the rotation axis to be observed from which the moment of inertia can be derived, changes in length of day to be measured, and a method to check for possible Mars polar motion. Tracking data on the Mariner 9, Viking 1 and 2, from the 1970's have been analyzed with MGS data obtained in the mapping orbit in 1999 to provide a new estimate of the moment of inertia.

Smith, David E.↗

NASA NTRS1996

Excess chemical potential of small solutes across water--membrane and water--hexane interfaces

The excess chemical potentials of five small, structurally related solutes, CH4, CH3F, CH2F2, CHF3, and CF4, across the water-glycerol 1-monooleate bilayer and water-hexane interfaces were calculated at 300, 310, and 340 K using the particle insertion method. The excess chemical potentials of nonpolar molecules (CH4 and CF4) decrease monotonically or nearly monotonically from water to a nonpolar phase. In contrast, for molecules that possess permanent dipole moments (CH3F, CH2F, and CHF3), the excess chemical potentials exhibit an interfacial minimum that arises from superposition of two monotonically and oppositely changing contributions: electrostatic and nonelectrostatic. The nonelectrostatic term, dominated by the reversible work of creating a cavity that accommodates the solute, decreases, whereas the electrostatic term increases across the interface from water to the membrane interior. In water, the dependence of this term on the dipole moment is accurately described by second order perturbation theory. To achieve the same accuracy at the interface, third order terms must also be included. In the interfacial region, the molecular structure of the solvent influences both the excess chemical potential and solute orientations. The excess chemical potential across the interface increases with temperature, but this effect is rather small. Our analysis indicates that a broad range of small, moderately polar molecules should be surface active at the water-membrane and water-oil interfaces. The biological and medical significance of this result, especially in relation to the mechanism of anesthetic action, is discussed.

NASA Center ARC↗

NASA NTRS1973

Asymptotic stability and instability of large-scale systems

The purpose of this paper is to develop new methods for constructing vector Lyapunov functions and broaden the application of Lyapunov's theory to stability analysis of large-scale dynamic systems. The application, so far limited by the assumption that the large-scale systems are composed of exponentially stable subsystems, is extended via the general concept of comparison functions to systems which can be decomposed into asymptotically stable subsystems. Asymptotic stability of the composite system is tested by a simple algebraic criterion. By redefining interconnection functions among the subsystems according to interconnection matrices, the same mathematical machinery can be used to determine connective asymptotic stability of large-scale systems under arbitrary structural perturbations.

Grujic, L. T.↗

NASA NTRS1987

Solitary waves in the resonant phenomenon between a surface gravity wave packet and an internal gravity wave

A two-layer inviscid incompressible fluid system of intermediate depth is considered. A multiple-scales perturbation technique is applied to the basic equations and boundary conditions for a two-layer fluid system to derive a system of weakly nonlinear partial integrodifferential equations governing the resonant interaction between a surface gravity wave packet and an internal gravity wave at an intermediate depth, providing a bridge between the existing shallow and deep fluid theories. The convolution integral term in these equations accounts for the dispersion in the lower-layer fluid. An iterative fast Fourier transform scheme is developed to find solitary wave solutions to this system of equations. The overtaking collision of two pairs of solitary waves, simulated using a spectral method, is found to be inelastic. It is found that the amplitude of the solitary waves changes slightly after the collision. The phase shifts these solitary waves undergo was calculated numerically.

Sepulveda, Nicasio↗

NASA NTRS1971

On nonlinear effects in fracture mechanics.

Linear elastic treatment of fracture is considered applicable for net section stress up to about 0.8 the uniaxial tensile yield stress. Crack front plastic yield is still small enough to be viewed and treated as a small perturbation to the local crack front elastic stress field. Assuming these same circumstances and adopting the same point of view, an approach is presented for incorporating the nonlinear effects of small scale crack front plastic yield and slow crack extension in determination of the energy release rate and fracture toughness. Deviation from linearity of the load-displacement record in a fracture toughness test offers a quantifiable measure of these effects and is used to calculate the energy release rate. Fracture toughness values for one-eight inch thick 7075-T6 center cracked aluminum sheet are compared with uncorrected values and with values obtained by the Irwin method of plasticity correction.

Liebowitz, H.↗

NASA NTRS1987

Modeling of joints for the dynamic analysis of truss structures

An experimentally-based method for determining the stiffness and damping of truss joints is described. The analytical models use springs and both viscous and friction dampers to simulate joint load-deflection behavior. A least-squares algorithm is developed to identify the stiffness and damping coefficients of the analytical joint models from test data. The effects of nonlinear joint stiffness such as joint dead band are also studied. Equations for predicting the sensitivity of beam deformations to changes in joint stiffness are derived and used to show the level of joint stiffness required for nearly rigid joint behavior. Finally, the global frequency sensitivity of a truss structure to random perturbations in joint stiffness is discussed.

Belvin, W. Keith↗