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NASA NTRS2006

Functional Requirements Document for HALE UAS Operations in the NAS: Step 1. Version 3

The purpose of this Functional Requirements Document (FRD) is to compile the functional requirements needed to achieve the Access 5 Vision of "operating High Altitude, Long Endurance (HALE) Unmanned Aircraft Systems (UAS) routinely, safely, and reliably in the national airspace system (NAS)" for Step 1. These functional requirements could support the development of a minimum set of policies, procedures and standards by the Federal Aviation Administration (FAA) and various standards organizations. It is envisioned that this comprehensive body of work will enable the FAA to establish and approve regulations to govern safe operation of UAS in the NAS on a routine or daily "file and fly" basis. The approach used to derive the functional requirements found within this FRD was to decompose the operational requirements and objectives identified within the Access 5 Concept of Operations (CONOPS) into the functions needed to routinely and safely operate a HALE UAS in the NAS. As a result, four major functional areas evolved to enable routine and safe UAS operations for an on-demand basis in the NAS. These four major functions are: Aviate, Navigate, Communicate, and Avoid Hazards. All of the functional requirements within this document can be directly traceable to one of these four major functions. Some functions, however, are traceable to several, or even all, of these four major functions. These cross-cutting functional requirements support the "Command / Control: function as well as the "Manage Contingencies" function. The requirements associated to these high-level functions and all of their supporting low-level functions are addressed in subsequent sections of this document.

Source record↗

NASA NTRS1995

Enhancements to the SSME transfer function modeling code

This report details the results of a one year effort by Ohio University to apply the transfer function modeling and analysis tools developed under NASA Grant NAG8-167 (Irwin, 1992), (Bartholomew, 1992) to attempt the generation of Space Shuttle Main Engine High Pressure Turbopump transfer functions from time domain data. In addition, new enhancements to the transfer function modeling codes which enhance the code functionality are presented, along with some ideas for improved modeling methods and future work. Section 2 contains a review of the analytical background used to generate transfer functions with the SSME transfer function modeling software. Section 2.1 presents the 'ratio method' developed for obtaining models of systems that are subject to single unmeasured excitation sources and have two or more measured output signals. Since most of the models developed during the investigation use the Eigensystem Realization Algorithm (ERA) for model generation, Section 2.2 presents an introduction of ERA, and Section 2.3 describes how it can be used to model spectral quantities. Section 2.4 details the Residue Identification Algorithm (RID) including the use of Constrained Least Squares (CLS) and Total Least Squares (TLS). Most of this information can be found in the report (and is repeated for convenience). Section 3 chronicles the effort of applying the SSME transfer function modeling codes to the a51p394.dat and a51p1294.dat time data files to generate transfer functions from the unmeasured input to the 129.4 degree sensor output. Included are transfer function modeling attempts using five methods. The first method is a direct application of the SSME codes to the data files and the second method uses the underlying trends in the spectral density estimates to form transfer function models with less clustering of poles and zeros than the models obtained by the direct method. In the third approach, the time data is low pass filtered prior to the modeling process in an effort to filter out high frequency characteristics. The fourth method removes the presumed system excitation and its harmonics in order to investigate the effects of the excitation on the modeling process. The fifth method is an attempt to apply constrained RID to obtain better transfer functions through more accurate modeling over certain frequency ranges. Section 4 presents some new C main files which were created to round out the functionality of the existing SSME transfer function modeling code. It is now possible to go from time data to transfer function models using only the C codes; it is not necessary to rely on external software. The new C main files and instructions for their use are included. Section 5 presents current and future enhancements to the XPLOT graphics program which was delivered with the initial software. Several new features which have been added to the program are detailed in the first part of this section. The remainder of Section 5 then lists some possible features which may be added in the future. Section 6 contains the conclusion section of this report. Section 6.1 is an overview of the work including a summary and observations relating to finding transfer functions with the SSME code. Section 6.2 contains information relating to future work on the project.

Irwin, R. Dennis↗

NASA NTRS1999

Generalized Functions for the Fractional Calculus

Previous papers have used two important functions for the solution of fractional order differential equations, the Mittag-Leffler functionE(sub q)[at(exp q)](1903a, 1903b, 1905), and the F-function F(sub q)[a,t] of Hartley & Lorenzo (1998). These functions provided direct solution and important understanding for the fundamental linear fractional order differential equation and for the related initial value problem (Hartley and Lorenzo, 1999). This paper examines related functions and their Laplace transforms. Presented for consideration are two generalized functions, the R-function and the G-function, useful in analysis and as a basis for computation in the fractional calculus. The R-function is unique in that it contains all of the derivatives and integrals of the F-function. The R-function also returns itself on qth order differ-integration. An example application of the R-function is provided. A further generalization of the R-function, called the G-function brings in the effects of repeated and partially repeated fractional poles.

Lorenzo, Carl F.↗

NASA NTRS2009

The Functional Breakdown Structure (FBS) and Its Relationship to Life Cycle Cost

The Functional Breakdown Structure (FBS) is a structured, modular breakdown of every function that must be addressed to perform a generic mission. It is also usable for any subset of the mission. Unlike a Work Breakdown Structure (WBS), the FBS is a function-oriented tree, not a product-oriented tree. The FBS details not products, but operations or activities that should be performed. The FBS is not tied to any particular architectural implementation because it is a listing of the needed functions, not the elements, of the architecture. The FBS for Space Transportation Systems provides a universal hierarchy of required functions, which include ground and space operations as well as infrastructure - it provides total visibility of the entire mission. By approaching the systems engineering problem from the functional view, instead of the element or hardware view, the SPST has created an exhaustive list of potential requirements which the architecture designers can use to evaluate the completeness of their designs. This is a new approach that will provide full accountability of all functions required to perform the planned mission. It serves as a giant check list to be sure that no functions are omitted, especially in the early architectural design phase. A significant characteristic of a FBS is that if architecture options are compared using this approach, then any missing or redundant elements of each option will be ' identified. Consequently, valid Life Cycle Costs (LCC) comparisons can be made. For example, one architecture option might not need a particular function while another option does. One option may have individual elements to perform each of three functions while another option needs only one element to perform the three functions. Once an architecture has been selected, the FBS will serve as a guide in development of the work breakdown structure, provide visibility of those technologies that need to be further developed to perform required functions, and help identify the personnel skills required to develop and operate the architecture. It also wifi allow the systems engineering activities to totally integrate each discipline to the maximum extent possible and optimize at the total system level, thus avoiding optimizing at the element level (stove-piping). In addition, it furnishes a framework that wifi help prevent over or under specifying requirements because all functions are identified and all elements are aligned to functions.

DeHoff, Bryan↗

NASA NTRS2013

Goal-Function Tree Modeling for Systems Engineering and Fault Management

The draft NASA Fault Management (FM) Handbook (2012) states that Fault Management (FM) is a "part of systems engineering", and that it "demands a system-level perspective" (NASAHDBK- 1002, 7). What, exactly, is the relationship between systems engineering and FM? To NASA, systems engineering (SE) is "the art and science of developing an operable system capable of meeting requirements within often opposed constraints" (NASA/SP-2007-6105, 3). Systems engineering starts with the elucidation and development of requirements, which set the goals that the system is to achieve. To achieve these goals, the systems engineer typically defines functions, and the functions in turn are the basis for design trades to determine the best means to perform the functions. System Health Management (SHM), by contrast, defines "the capabilities of a system that preserve the system's ability to function as intended" (Johnson et al., 2011, 3). Fault Management, in turn, is the operational subset of SHM, which detects current or future failures, and takes operational measures to prevent or respond to these failures. Failure, in turn, is the "unacceptable performance of intended function." (Johnson 2011, 605) Thus the relationship of SE to FM is that SE defines the functions and the design to perform those functions to meet system goals and requirements, while FM detects the inability to perform those functions and takes action. SHM and FM are in essence "the dark side" of SE. For every function to be performed (SE), there is the possibility that it is not successfully performed (SHM); FM defines the means to operationally detect and respond to this lack of success. We can also describe this in terms of goals: for every goal to be achieved, there is the possibility that it is not achieved; FM defines the means to operationally detect and respond to this inability to achieve the goal. This brief description of relationships between SE, SHM, and FM provide hints to a modeling approach to provide formal connectivity between the nominal (SE), and off-nominal (SHM and FM) aspects of functions and designs. This paper describes a formal modeling approach to the initial phases of the development process that integrates the nominal and off-nominal perspectives in a model that unites SE goals and functions of with the failure to achieve goals and functions (SHM/FM). This methodology and corresponding model, known as a Goal-Function Tree (GFT), provides a means to represent, decompose, and elaborate system goals and functions in a rigorous manner that connects directly to design through use of state variables that translate natural language requirements and goals into logical-physical state language. The state variable-based approach also provides the means to directly connect FM to the design, by specifying the range in which state variables must be controlled to achieve goals, and conversely, the failures that exist if system behavior go out-of-range. This in turn allows for the systems engineers and SHM/FM engineers to determine which state variables to monitor, and what action(s) to take should the system fail to achieve that goal. In sum, the GFT representation provides a unified approach to early-phase SE and FM development. This representation and methodology has been successfully developed and implemented using Systems Modeling Language (SysML) on the NASA Space Launch System (SLS) Program. It enabled early design trade studies of failure detection coverage to ensure complete detection coverage of all crew-threatening failures. The representation maps directly both to FM algorithm designs, and to failure scenario definitions needed for design analysis and testing. The GFT representation provided the basis for mapping of abort triggers into scenarios, both needed for initial, and successful quantitative analyses of abort effectiveness (detection and response to crew-threatening events).

Patterson, Jonathan D.↗

NASA NTRS2009

Physiological Factors Contributing to Postflight Changes in Functional Performance

Astronauts experience alterations in multiple physiological systems due to exposure to the microgravity conditions of space flight. These physiological changes include sensorimotor disturbances, cardiovascular deconditioning and loss of muscle mass and strength. These changes might affect the ability of crewmembers to perform critical mission tasks immediately after landing on lunar and Martian surfaces. To date, changes in functional performance have not been systematically studied or correlated with physiological changes. To understand how changes in physiological function impact functional performance an interdisciplinary pre/postflight testing regimen (Functional Task Test, FTT) has been developed that systematically evaluates both astronaut postflight functional performance and related physiological changes. The overall objectives of the FTT are to: Develop a set of functional tasks that represent critical mission tasks for Constellation. Determine the ability to perform these tasks after flight. Identify the key physiological factors that contribute to functional decrements. Use this information to develop targeted countermeasures. The functional test battery was designed to address high priority tasks identified by the Constellation program as critical for mission success. The set of functional tests making up the FTT include the: 1) Seat Egress and Walk Test, 2) Ladder Climb Test, 3) Recovery from Fall/Stand Test, 4) Rock Translation Test, 5) Jump Down Test, 6) Torque Generation Test, and 7) Construction Activity Board Test. Corresponding physiological measures include assessments of postural and gait control, dynamic visual acuity, fine motor control, plasma volume, orthostatic intolerance, upper and lower body muscle strength, power, fatigue, control and neuromuscular drive. Crewmembers will perform both functional and physiological tests before and after short (Shuttle) and long-duration (ISS) space flight. Data will be collected on R+0 (Shuttle only), R+1, R+6 and R+30. Using a multivariate regression model we will identify which physiological systems contribute the most to impaired performance on each functional test. This will allow us to identify the physiological systems that play the largest role in decrement in functional performance. Using this information we can then design and implement countermeasures that specifically target the physiological systems most responsible for the altered functional performance associated with space flight.

Bloomberg, J. J.↗

NASA NTRS2009

Defining the Physiological Factors that Contribute to Postflight Changes in Functional Performance

Astronauts experience alterations in multiple physiological systems due to exposure to the microgravity conditions of space flight. These physiological changes include sensorimotor disturbances, cardiovascular deconditioning and loss of muscle mass and strength. These changes might affect the ability of crewmembers to perform critical mission tasks immediately after landing on lunar and Martian surfaces. To date, changes in functional performance have not been systematically studied or correlated with physiological changes. To understand how changes in physiological function impact functional performance an interdisciplinary pre/postflight testing regimen (Functional Task Test, FTT) has been developed that systematically evaluates both astronaut postflight functional performance and related physiological changes. The overall objective of the FTT is to identify the key underlying physiological factors that contribute to performance of functional tests that are representative of critical mission tasks. This study will identify which physiological systems contribute the most to impaired performance on each functional test. This will allow us to identify the physiological systems that play the largest role in decrement in functional performance. Using this information we can then design and implement countermeasures that specifically target the physiological systems most responsible for the altered functional performance associated with space flight. The functional test battery was designed to address high priority tasks identified by the Constellation program as critical for mission success. The set of functional tests making up the FTT include the: 1) Seat Egress and Walk Test, 2) Ladder Climb Test, 3) Recovery from Fall/Stand Test, 4) Rock Translation Test, 5) Jump Down Test, 6) Torque Generation Test, and 7) Construction Activity Board Test. Corresponding physiological measures include assessments of postural and gait control, dynamic visual acuity, fine motor control, plasma volume, orthostatic intolerance, upper and lower body muscle strength, power, fatigue, control and neuromuscular drive. Crewmembers perform both functional and physiological tests before and after short (Shuttle) and long-duration (ISS) space flight. Data are collected on R+0 (Shuttle only), R+1, R+6 and R+30.

Bloomberg, J. J.↗

NASA NTRS2013

Goal-Function Tree Modeling for Systems Engineering and Fault Management

Johnson, Stephen B.↗

NASA NTRS2016

Exercise Effects on the Brain and Sensorimotor Function in Bed Rest

Long duration spaceflight microgravity results in cephalad fluid shifts and deficits in posture control and locomotion. Effects of microgravity on sensorimotor function have been investigated on Earth using head down tilt bed rest (HDBR). HDBR serves as a spaceflight analogue because it mimics microgravity in body unloading and bodily fluid shifts. Preliminary results from our prior 70 days HDBR studies showed that HDBR is associated with focal gray matter (GM) changes and gait and balance deficits, as well as changes in brain functional connectivity. In consideration of the health and performance of crewmembers we investigated whether exercise reduces the effects of HDBR on GM, functional connectivity, and motor performance. Numerous studies have shown beneficial effects of exercise on brain health. We therefore hypothesized that an exercise intervention during HDBR could potentially mitigate the effects of HDBR on the central nervous system. Eighteen subjects were assessed before (12 and 7 days), during (7, 30, and ~70 days) and after (8 and 12 days) 70 days of 6-degrees HDBR at the NASA HDBR facility in UTMB, Galveston, TX, US. Each subject was randomly assigned to a control group or one of two exercise groups. Exercise consisted of daily supine exercise which started 20 days before the start of HDBR. The exercise subjects participated either in regular aerobic and resistance exercise (e.g. squat, heel raise, leg press, cycling and treadmill running), or aerobic and resistance exercise using a flywheel apparatus (rowing). Aerobic and resistance exercise intensity in both groups was similar, which is why we collapsed the two exercise groups for the current experiment. During each time point T1-weighted MRI scans and resting state functional connectivity scans were obtained using a 3T Siemens scanner. Focal changes over time in GM density were assessed using voxel based morphometry (VBM8) under SPM. Changes in resting state functional connectivity was assessed using both a region of interest (ROI, or seed-to-voxel) approach as well as a whole brain intrinsic connectivity (i.e., voxel-to-voxel) analysis. For the ROI analysis we selected 11 ROIs of brain regions that are involved in sensorimotor function (i.e., L. Insular C., L. Putamen, R. Premotor C., L.+R. Primary Motor C., R. Vestibular C., L. Posterior Cingulate G., R. Cerebellum Lobule V + VIIIb + Crus I, and the R. Superior Parietal G.) and correlated their time course of brain activation during rest with all other voxels in the brain. The whole brain connectivity analysis tests changes in the strength of the global connectivity pattern between each voxel and the rest of the brain. Functional mobility was assessed using an obstacle course. Vestibular contribution to balance was measured using Neurocom Sensory Organization Test 5. Behavioral measures were assessed pre-HDBR, and 0, 8 and 12 days post-HDBR. Linear mixed models were used to test for effects of time, group, and group-by-time interactions. Family-wise error corrected VBM revealed significantly larger increases in GM volume in the right primary motor cortex in bed rest control subjects than in bed rest exercise subjects. No other significant group by time interactions in gray matter changes with bed rest were observed. Functional connectivity MRI revealed that the increase in connectivity during bed rest of the left putamen with the bilateral midsagittal precunes and the right cingulate gyrus was larger in bed rest control subjects than in bed rest exercise subjects. Furthermore, the increase in functional connectivity with bed rest of the right premotor cortex with the right inferior frontal gyrus and the right primary motor cortex with the bilateral premotor cortex was smaller in bed rest control subjects than in bed rest exercise subjects. Functional mobility performance was less affected by HDBR in exercise subjects than in control subjects and post HDBR exercise subjects recovered faster than control subjects. The group performance differences and GM changes were not related. Exercise in HDBR partially mitigates the adverse effect of HDBR on functional mobility, particularly during the post-bed rest recovery phase. In addition, exercise appears to result in differential brain structural and functional changes in motor regions such as the primary motor cortex, the premotor cortex and the putamen. Whether these central nervous system changes are related to motor behavioral changes including gait and balance warrants further research.

Koppelmans, V.↗

NASA NTRS2020

Evaluation of Fatigue Damage Accumulation Functions for Delamination Initiation and Propagation

The present report follows on the cohesive fatigue damage model methodology proposed in NASA-TP-2018-219838. In that report, an empirical function describing the incremental damage due to cyclic loading was used to calculate fatigue damage within a cohesive formulation. The form of the function was developed such that, when integrated at a constant stress amplitude from no damage to failure, it produces a life versus load response that is consistent with an S-N diagram. Therefore, the parameters of the damage model could be obtained by fitting the model predictions to an S-N diagram. The finite element analyses performed demonstrate that the cohesive fatigue accumulation function provides a link between the S-N diagram that describes crack initiation, and the Paris law that characterizes the rate of crack propagation. However, when the model was proposed, it was not known whether the form of the damage accumulation function associated with a desired S-N diagram is unique and, if not, if the link between S-N and the Paris law is unique and independent of the fatigue function selected. In the effort described herein, several alternative forms of the damage function that reproduce the desired features of S-N diagrams were found and evaluated. The effects of each of these functions on the predicted parameters of the Paris law and the propagation threshold are discussed. The results indicate that the predicted exponent m of the Paris law is indeed independent of the damage accumulation function. However, different functions predict different values for the pre-factor C of the Paris law. Therefore, the proper damage accumulation function must be selected by comparison with experiments. One of the new damage accumulation functions proposed herein was found to be particularly useful for analysis because of the ease with which the model parameters can be determined with a minimal amount of experimental information. The effectiveness of the proposed methodology and damage function was demonstrated by conducting analyses of a double cantilever beam test, a mixed-mode bending test, and a three-point bending test of a skin/doubler specimen. The results indicate that the same set of model parameters can provide accurate predictions of the rate of fatigue crack propagation for a variety of material interfaces, mode mixities, load levels, and stress ratios.

Cohesive elements↗

NASA NTRS1971

A table of integrals of the error function. II - Additions and corrections.

Integrals of products of error functions with other functions are presented, taking into account a combination of the error function with powers, a combination of the error function with exponentials and powers, a combination of the error function with exponentials of more complicated arguments, definite integrals from Laplace transforms, and a combination of the error function with trigonometric functions. Other integrals considered include a combination of the error function with logarithms and powers, a combination of two error functions, and a combination of the error function with other special functions.

Geller, M.↗

NASA NTRS2000

R-Function Relationships for Application in the Fractional Calculus

The F-function, and its generalization the R-function, are of fundamental importance in the fractional calculus. It has been shown that the solution of the fundamental linear fractional differential equation may be expressed in terms of these functions. These functions serve as generalizations of the exponential function in the solution of fractional differential equations. Because of this central role in the fractional calculus, this paper explores various intrarelationships of the R-function, which will be useful in further analysis. Relationships of the R-function to the common exponential function, e(t), and its fractional derivatives are shown. From the relationships developed, some important approximations are observed. Further, the inverse relationships of the exponential function, el, in terms of the R-function are developed. Also, some approximations for the R-function are developed.

Lorenzo, Carl F.↗

NASA NTRS2013

Arctic Tundra Vegetation Functional Types Based on Photosynthetic Physiology and Optical Properties

Non-vascular plants (lichens and mosses) are significant components of tundra landscapes and may respond to climate change differently from vascular plants affecting ecosystem carbon balance. Remote sensing provides critical tools for monitoring plant cover types, as optical signals provide a way to scale from plot measurements to regional estimates of biophysical properties, for which spatial-temporal patterns may be analyzed. Gas exchange measurements were collected for pure patches of key vegetation functional types (lichens, mosses, and vascular plants) in sedge tundra at Barrow, AK. These functional types were found to have three significantly different values of light use efficiency (LUE) with values of 0.013 plus or minus 0.0002, 0.0018 plus or minus 0.0002, and 0.0012 plus or minus 0.0001 mol C mol (exp -1) absorbed quanta for vascular plants, mosses and lichens, respectively. Discriminant analysis of the spectra reflectance of these patches identified five spectral bands that separated each of these vegetation functional types as well as nongreen material (bare soil, standing water, and dead leaves). These results were tested along a 100 m transect where midsummer spectral reflectance and vegetation coverage were measured at one meter intervals. Along the transect, area-averaged canopy LUE estimated from coverage fractions of the three functional types varied widely, even over short distances. The patch-level statistical discriminant functions applied to in situ hyperspectral reflectance data collected along the transect successfully unmixed cover fractions of the vegetation functional types. The unmixing functions, developed from the transect data, were applied to 30 m spatial resolution Earth Observing-1 Hyperion imaging spectrometer data to examine variability in distribution of the vegetation functional types for an area near Barrow, AK. Spatial variability of LUE was derived from the observed functional type distributions. Across this landscape, a fivefold variation in tundra LUE was observed. LUE calculated from the functional type cover fractions was also correlated to a spectral vegetation index developed to detect vegetation chlorophyll content. The concurrence of these alternate methods suggest that hyperspectral remote sensing can distinguish functionally distinct vegetation types and can be used to develop regional estimates of photosynthetic LUE in tundra landscapes.

VEGETATION↗

NASA NTRS2015

NASA's Functional Task Test: Providing Information for an Integrated Countermeasure System

Exposure to the microgravity conditions of spaceflight causes astronauts to experience alterations in multiple physiological systems. These physiological changes include sensorimotor disturbances, cardiovascular deconditioning, and loss of muscle mass and strength. Some or all of these changes might affect the ability of crewmembers to perform critical mission tasks immediately after landing on a planetary surface. The goals of the Functional Task Test (FTT) study were to determine the effects of spaceflight on functional tests that are representative of critical exploration mission tasks and to identify the key physiological factors that contribute to decrements in performance. The FTT was comprised of seven functional tests and a corresponding set of interdisciplinary physiological measures targeting the sensorimotor, cardiovascular and muscular changes associated with exposure to spaceflight. Both Shuttle and ISS crewmembers participated in this study. Additionally, we conducted a supporting study using the FTT protocol on subjects before and after 70 days of 6 head-down bed rest. The bed rest analog allowed us to investigate the impact of body unloading in isolation on both functional tasks and on the underlying physiological factors that lead to decrements in performance, and then to compare them with the results obtained in our spaceflight study. Spaceflight data were collected on three sessions before flight, on landing day (Shuttle only) and 1, 6 and 30 days after landing. Bed rest subjects were tested three times before bed rest and immediately after getting up from bed rest as well as 1, 6, and 12 days after reambulation. We have shown that for Shuttle, ISS and bed rest subjects, functional tasks requiring a greater demand for dynamic control of postural equilibrium (i.e. fall recovery, seat egress/obstacle avoidance during walking, object translation, jump down) showed the greatest decrement in performance. Functional tests with reduced requirements for postural stability (i.e. hatch opening, ladder climb, manual manipulation of objects and tool use) showed little reduction in performance. These changes in functional performance were paralleled by similar decrements in sensorimotor tests designed to specifically assess postural equilibrium and dynamic gait control. Bed rest subjects experienced similar deficits both in functional tests with balance challenges and in sensorimotor tests designed to evaluate postural and gait control as spaceflight subjects indicating that body support unloading experienced during spaceflight plays a central role in post-flight alteration of functional task performance. To determine how differences in body-support loading experienced during in-flight treadmill exercise affect postflight functional performance, the loading history for each subject during in-flight treadmill (T2) exercise was correlated with postflight measures of performance. ISS crewmembers who walked on the treadmill with higher pull-down loads had enhanced post-flight performance on tests requiring mobility. Taken together the spaceflight and bed rest data point to the importance of supplementing inflight exercise countermeasures with balance and sensorimotor adaptability training. These data also support the notion that inflight treadmill exercise performed with higher body loading provides sensorimotor benefits leading to improved performance on functional tasks that require dynamic postural stability and mobility.

Bloomberg, J. J.↗

NASA NTRS2018

Modified Geometric Truncation of the Scattering Phase Function

Phase function of light scattering on large atmospheric particles has very strong peak in forward direction constituting a challenge for accurate numerical calculations of radiance required in remote sensing problems. Scaling transformation replaces original phase function with a sum of the delta function and a new regular smooth phase function. Geometric truncation is one of the ways to construct such a smooth function. The replacement phase function coincides with the original one outside the forward cone and preserves the asymmetry parameter. It has discontinuity at the cone. Another simple functional form of the replacement phase function within the cone is suggested. It enables continuity and allows for a number of modifications. Three of them are considered in this study: preserving asymmetry parameter, providing continuity of the 1st derivative of the phase function, and preserving mean scattering angle. Yet another problem addressed in this study is objective selection of the width of the forward cone. That angle affects truncation fraction and values of the phase function within the cone. A heuristic approach providing unambiguous criterion of selection of the truncation angle is proposed. The approach has easy numerical implementation. Suggested modifications were tested on cloud phase function using discrete ordinates and Monte Carlo methods. It was shown that the modifications provide better accuracy of the radiance computation compare to the original geometric truncation with discrete ordinates while continuous derivative approach provides significant gain in computer time with Monte Carlo simulations.

Alexander Radkevich↗

NASA NTRSDate not supplied

Transfer Function Models and Sensitivity Analysis

In many situations, real or induced flaws such as tight cracks with known morphology cannot be manufactured in part configuration specimens or in real parts. Typically, fatigue cracks are manufactured in simple geometry specimens such as flat plates, dog-bone shaped flat or round specimens. If a nondestructive evaluation (NDE) technique is required to provide a reliably detectable target flaw size denoted as α_(90/95) for induced flaws in real part, then the direct method for qualifying the NDE procedure is to use the appropriate induced flaw specimens and perform probability of detection analysis using these flaws. This can be described as direct POD demonstration testing, which may follow guidelines of MIL-HDBK-1823. This paper considers a case, where induced flaws are not available in part configuration specimens. Therefore, a direct POD demonstration study cannot be undertaken. In such situation, general practice for NDE procedure qualification is to use artificial flaws in simple geometry and part configuration specimens, and induced flaws in the chosen simple geometry specimens. Signal response data is taken on all sets of artificial and induced flaws. NDE procedure testing on induced flaw in simple geometry specimen is called NDE demonstration testing here. A transfer function NDE procedure qualification method for forward case calculates predicted induced flaw size for demonstration using a chosen target flaw size. Another transfer function method for inverse case, calculates the target flaw size using a given demonstration flaw size. The transfer function analysis assumes relationship of artificial flaw signal responses in real parts and simple geometry specimens; and induced flaw responses in simple geometry specimens to induced flaws in real parts. The signal response transfer relationships model should be defined before transfer function models can be devised. Assuming that the signal response transfer relationships model is valid, forward and inverse case transfer function methods have been devised. Because of lack of signal response data from induced flaws in real part, 90/95% POD/confidence (P/C) cannot be demonstrated directly. However, the transfer function method may be assessed using simulation to evaluate whether the resulting target flaw size or demonstration flaw size provides adequate confidence to the assumed signal response transfer relationships model. Therefore, the transfer function approach is a risk assessment approach. Both the signal response transfer relationships model and the transfer function model are important in managing risk in results provided by the transfer function NDE technique qualification or assessment. The signal response transfer relationships model needs to be validated with empirical data and then transfer function model needs to be validated for desired P/C on case-by-case basis.

Nondestructive evaluation↗

NASA NTRS1983

Supplemental basis functions for the second transition row elements

It is noted that for molecular calculations, the basis sets presented by Huzinaga (1977) need to be augmented by (1) 5p functions to describe the 5s - 5p near degeneracy; (2) a diffuse 4d function to provide for a balanced description of the 5s2 4dn, 5s1 4d(n+1), and 4d(n+2) states of the atom; and (3) a set of 4f functions to correlate the 4d functions. Here, the diffuse 4d function is similar in function to the diffuse 3d function for the first transition row elements recommended by Hay (1977). A table is included giving the optimized values for the diffuse 4d, the 5p, and 4f (STO exponent) functions. The diffuse 4d function and the 5p functions are optimized at the SCF level on the basis of the 5s1 4d(n+1) state (except for Pd, which is optimized for the 4d10 state) and the 5s1 5p1 4dn state, respectively. The table also gives the energies and the atomic symmetries for each of the SCF calculations.

Walch, S. P.↗

NASA NTRS1994

Introduction to Generalized Functions with Applications in Aerodynamics and Aeroacoustics

Generalized functions have many applications in science and engineering. One useful aspect is that discontinuous functions can be handled as easily as continuous or differentiable functions and provide a powerful tool in formulating and solving many problems of aerodynamics and acoustics. Furthermore, generalized function theory elucidates and unifies many ad hoc mathematical approaches used by engineers and scientists. We define generalized functions as continuous linear functionals on the space of infinitely differentiable functions with compact support, then introduce the concept of generalized differentiation. Generalized differentiation is the most important concept in generalized function theory and the applications we present utilize mainly this concept. First, some results of classical analysis, are derived with the generalized function theory. Other applications of the generalized function theory in aerodynamics discussed here are the derivations of general transport theorems for deriving governing equations of fluid mechanics, the interpretation of the finite part of divergent integrals, the derivation of the Oswatitsch integral equation of transonic flow, and the analysis of velocity field discontinuities as sources of vorticity. Applications in aeroacoustics include the derivation of the Kirchhoff formula for moving surfaces, the noise from moving surfaces, and shock noise source strength based on the Ffowcs Williams-Hawkings equation.

Farassat, F.↗