Results for “generation equation” — Page 3

NASA NTRS1980

Numerical generation of two-dimensional orthogonal curvilinear coordinates in an Euclidean space

A noniterative method for the numerical generation of orthogonal curvilinear coordinates for plane annular regions between two arbitrary smooth closed curves was developed. The basic generating equation is the Gaussian equation for an Euclidean space which is solved analytically. The method is applied in many cases and these test results demonstrate that the proposed method can be readily applied to a wide variety of problems. The method can also be used for simply connected regions only by obtaining the solution of the linear equation under the changed boundary conditions.

Warsi, Z. U. A.↗

NASA NTRS1982

Grid generation by elliptic partial differential equations for a tri-element Augmentor-Wing airfoil

Two efforts to numerically simulate the flow about the Augmentor-Wing airfoil in the cruise configuration using the GRAPE elliptic partial differential equation grid generator algorithm are discussed. The Augmentor-Wing consists of a main airfoil with a slotted trailing edge for blowing and two smaller airfoils shrouding the blowing jet. The airfoil and the algorithm are described, and the application of GRAPE to an unsteady viscous flow simulation and a transonic full-potential approach is considered. The procedure involves dividing a complicated flow region into an arbitrary number of zones and ensuring continuity of grid lines, their slopes, and their point distributions across the zonal boundaries. The method for distributing the body-surface grid points is discussed.

Sorenson, R. L.↗

NASA NTRS1985

Space robot simulator vehicle

A Space Robot Simulator Vehicle (SRSV) was constructed to model a free-flying robot capable of doing construction, manipulation and repair work in space. The SRSV is intended as a test bed for development of dynamic and static control methods for space robots. The vehicle is built around a two-foot-diameter air-cushion vehicle that carries batteries, power supplies, gas tanks, computer, reaction jets and radio equipment. It is fitted with one or two two-link manipulators, which may be of many possible designs, including flexible-link versions. Both the vehicle body and its first arm are nearly complete. Inverse dynamic control of the robot's manipulator has been successfully simulated using equations generated by the dynamic simulation package SDEXACT. In this mode, the position of the manipulator tip is controlled not by fixing the vehicle base through thruster operation, but by controlling the manipulator joint torques to achieve the desired tip motion, while allowing for the free motion of the vehicle base. One of the primary goals is to minimize use of the thrusters in favor of intelligent control of the manipulator. Ways to reduce the computational burden of control are described.

Cannon, R. H., Jr.↗

NASA NTRS1986

Transformation of two and three-dimensional regions by elliptic systems

Grid smoothing and orthogonalization procedures were developed and implemented in the construction of two and three dimensional grids. The procedures are based on the variational methods of grid generation. The two-dimensional examples were computed using the MSU IRIS Graphics Workstation. It was demonstrated that the elliptic grid generation equations, with arbitrary forcing functions, can be solved, in their variational formulation, using a gradient method. Since gradient methods have a global convergence property, the divergence problems often encountered when using SOR iterative methods can be avoided. It is not to be concluded, however, that SOR methods should be abandoned, since gradient methods tend to converge very slowly. In fact, slow convergence was the major problem encountered in the three-dimensional grids. Further progress was made on the continuing effort to develop conservative interpolation formulas for overlapping grids.

Mastin, C. Wayne↗

NASA NTRS1988

A linearized model of helicopter dynamics including correlation with flight test

A linearized model for rotorcraft dynamics has been developed through the use of symbolic automatic equation generation techniques. The dynamic model has been formulated in a unique way such that it can be used to analyze a variety of rotor/body coupling problems including a rotor mounted on a flexible shaft with a number of modes as well as free flight stability and control characteristics. Direct comparison of the time response to longitudinaol, lateral and directional control inputs at various trim conditions shows that the linear model yields good to very good correlation with flight test. In particular it is also shown that a dynamic inflow model is essential to obtain good time response correlation especially for the hover trim condition and also that at translational flight trim conditions the main rotor wake interaction with the tail rotor and fixed tail surfaces is a significant contributor to the response. A relatively simple model for the downwash and sidewash at tail surfaces based on flat vortex wake theory is shown to produce good agreement. Finaly, the influence of rotor flap and lag dynamics on automatic control systems feedback gain limitations is investigated with the model.

Curtiss, H. C.↗

NASA NTRS1988

A study of helicopter stability and control including blade dynamics

A linearized model of rotorcraft dynamics has been developed through the use of symbolic automatic equation generating techniques. The dynamic model has been formulated in a unique way such that it can be used to analyze a variety of rotor/body coupling problems including a rotor mounted on a flexible shaft with a number of modes as well as free-flight stability and control characteristics. Direct comparison of the time response to longitudinal, lateral and directional control inputs at various trim conditions shows that the linear model yields good to very good correlation with flight test. In particular it is shown that a dynamic inflow model is essential to obtain good time response correlation, especially for the hover trim condition. It also is shown that the main rotor wake interaction with the tail rotor and fixed tail surfaces is a significant contributor to the response at translational flight trim conditions. A relatively simple model for the downwash and sidewash at the tail surfaces based on flat vortex wake theory is shown to produce good agreement. Then, the influence of rotor flap and lag dynamics on automatic control systems feedback gain limitations is investigated with the model. It is shown that the blade dynamics, especially lagging dynamics, can severly limit the useable values of the feedback gain for simple feedback control and that multivariable optimal control theory is a powerful tool to design high gain augmentation control system. The frequency-shaped optimal control design can offer much better flight dynamic characteristics and a stable margin for the feedback system without need to model the lagging dynamics.

Zhao, Xin↗

NASA NTRS1989

Parallel processing environment for multi-flexible body dynamics

The implementation of a dynamics solution algorithm with inherent parallelism which is applicable to the dynamics of large flexible space structures is described. The algorithm is unique in that parts of the solution can be computed simultaneously by working with different branches of its tree topology. The algorithm exhibits close to 0(n) type behavior. The data flow within the solution algorithm is discussed along with results from its implementation in a multiprocessing environment. A model of the United States Space Station is used as an example. The results show that, with fast multiple scalar processors, an efficient algorithm, and symbolically generated equations of motion, real-time performance can be achieved with present-day hardware technology, even with complex dynamical models.

Venugopal, Ravi↗

NASA NTRS2003

Using Plate Finite Elements for Modeling Fillets in Design, Optimization, and Dynamic Analysis

A methodology has been developed that allows the use of plate elements instead of numerically inefficient solid elements for modeling structures with 90 degree fillets. The technique uses plate bridges with pseudo Young's modulus (Eb) and thickness (tb) values placed between the tangent points of the fillets. These parameters are obtained by solving two nonlinear simultaneous equations in terms of the independent variables rlt and twallt. These equations are generated by equating the rotation at the tangent point of a bridge system with that of a fillet, where both rotations are derived using beam theory. Accurate surface fits of the solutions are also presented to provide the user with closed-form equations for the parameters. The methodology was verified on the subcomponent level and with a representative filleted structure, where the technique yielded a plate model exhibiting a level of accuracy better than or equal to a high-fidelity solid model and with a 90-percent reduction in the number of DOFs. The application of this method for parametric design studies, optimization, and dynamic analysis should prove extremely beneficial for the finite element practitioner. Although the method does not attempt to produce accurate stresses in the filleted region, it can also be used to obtain stresses elsewhere in the structure for preliminary analysis. A future avenue of study is to extend the theory developed here to other fillet geometries, including fillet angles other than 90 and multifaceted intersections.

Brown, A. M.↗

NASA NTRS2006

Calibration Technique for Polarization-Sensitive Lidars

Polarization-sensitive lidars have proven to be highly effective in discriminating between spherical and non-spherical particles in the atmosphere. These lidars use a linearly polarized laser and are equipped with a receiver that can separately measure the components of the return signal polarized parallel and perpendicular to the outgoing beam. In this work we describe a technique for calibrating polarization-sensitive lidars that was originally developed at NASA s Langley Research Center (LaRC) and has been used continually over the past fifteen years. The procedure uses a rotatable half-wave plate inserted into the optical path of the lidar receiver to introduce controlled amounts of polarization cross-talk into a sequence of atmospheric backscatter measurements. Solving the resulting system of nonlinear equations generates the system calibration constants (gain ratio, G, and offset angle, theta) required for deriving calibrated measurements of depolarization ratio from the lidar signals. In addition, this procedure also determines the mean depolarization ratio within the region of the atmosphere that is analyzed. Simulations and error propagation studies show the method to be both reliable and well behaved. Operational details of the technique are illustrated using measurements obtained as part of Langley Research Center s participation in the First ISCCP Regional Experiment (FIRE).

Alvarez, J. M.↗

NASA NTRSDate not supplied

Presentation material for Progress in Simulated Water Well Performance on Mars

This presentation reports on the results of some experimental work regarding the establishment and operation of water wells on Mars. These results were accomplished since this group reported on this concept at the AIAA Space conference in 2018. These experiments were conducted to improve the understanding of a Rodwell’s performance under Mars surface conditions. The motivation to improve this understanding results from the leverage that access to large quantities of water, already on Mars, gives to future human missions to that planet. Previous work by this group has described the likely location of these water deposits and the extensive history of terrestrial use of the Rodwell technique. The results reported in this presentation indicate several key points. First, it has been demonstrated that a stable pool of water can be maintained under Martian surface environmental conditions using an approach comparable to terrestrial Rodwells. Next, it was unexpectedly learned that the water pool exhibits some of the same structural characteristics – specifically an ice shelf just above the water pool – that are seen in terrestrial Rodwells. This finding directly affected the way in which test data was reduced. And finally the experimental data was used to generate equations for convection-driven evaporation and convective heat transfer. The evaporation results were found to follow the same trends as published results for experiments in terrestrial environments. But the results reported here have also extended the range of these trends into much smaller Rayleigh Number ranges than anything previously reported. It was also found that convective heat transfer does not following published results even in the same Rayleigh Number range. A suggested explanation for this result has been proposed based on phenomenon described in the literature, but this group continues to look at the results to see if a better explanation can be found.

Mars↗

NASA NTRSDate not supplied

A Survey of ISS and Visiting Vehicle Returned Surfaces for Environmental Characterization and Computer Model Development

The Orbital Debris Engineering Model (ORDEM) developed by the NASA Orbital Debris Program Office (ODPO) is a data-driven model — extensive radar, optical, laboratory, and in situ measurement data sets have been used to build the model since its earliest versions. A salient aspect of professional software development is the verification and validation (V&V) process. Verification answers the question “Is the model built correctly?” while validation addresses the question “Did we build the correct model?” Less extensive, reserved, or independent data sets serve the validation requirement. Due to the dynamic nature of the orbital debris environment, it is critical to use contemporaneous data sources that represent the current environment to support ORDEM development and validation. ORDEM has utilized in situ data collected from Space Shuttle and Hubble Space Telescope surface inspections, now over a decade old. This historical dataset is fundamental for providing baseline in situ measurement data for sizes between 10 to 300 microns, but new data sources are being evaluated using returned surfaces from or near the International Space Station (ISS). This paper reviews a general microscopic survey of ISS soft goods, the Pressurized Mating Adapter 2 (PMA-2) blanket, and a limited-scope feasibility study conducted on the Space Exploration Technologies Corporation (SpaceX) Dragon capsule’s Thermal Protection System (TPS) material. The PMA-2 blanket, exposed to the space environment between 09 July 2013 and 25 February 2015, is an approximately 3.7 m2-area blanket composed of a betacloth outer layer and multiple ballistic fabric inner layers. The SpaceX Cargo Dragon capsule regularly visited the ISS from 2012 through 2020 and potentially provides a timely and well-characterized source of data for modeling purposes. The capsule’s lateral surfaces use SpaceX Proprietary Ablative Material (SPAM) TPS material, a syntactic foam, for thermal management during all mission phases. Seven SPAM extracted samples have been analyzed to date. This paper will provide an overview of the characterization completed for impact features by size, depth, impactor diameter, and the impactor residues chemical analyses, allowing a differentiation between micrometeoroids and orbital debris and a categorization by mass density and density class. Impactor diameter is estimated using damage equations generated from ground-based hypervelocity impact testing. The orbital debris impactors are compared to the current ORDEM 3.2 model of the environment at ISS altitudes. We briefly discuss the meteoroid impactors, including constituents and mass densities, in the general context of current models.

Phillip Anz-meador↗

NASA NTRSDate not supplied

Statistical Process Control and Analysis on the Water Content Measurements in NASA Glenn’s Icing Research Tunnel

The Icing Research Tunnel at NASA Glenn follows the recommended practice for calibration outlined in SAE’s ARP5905. The calibration team has followed the schedule of a full calibration every five years with a check calibration done every six months following. The liquid water content of the IRT has maintained stability within the stated specifications of variation within +/- 10% of the curve fit equation generated from calibration data. Using past measurements and data trends, IRT characterization engineers wanted to develop methods for the ability to know when data were not within variation. Trends can be observed in the liquid water content measurement process by constructing statistical process control charts. This paper describes data processing procedures for the Multi-Element Sensor in the IRT, including collision efficiency corrections, canonical correlation analysis, process for rejection of data, and construction of control charts. Data are presented to display the control capability to meet defined liquid water content specifications of the IRT with the Multi-Element Sensor mounted in the center of the test section.

Statistical Process Control↗

NASA NTRSDate not supplied

Statistical Process Control and Analysis on the Water Content Measurements in NASA Glenn’s Icing Research Tunnel

The Icing Research Tunnel at NASA Glenn follows the recommended practice for calibration outlined in SAE’s ARP5905. The calibration team has followed the schedule of a full calibration every five years with a check calibration done every six months following. The liquid water content of the IRT has maintained stability within the stated specifications of variation within +/- 10% of the curve fit equation generated from calibration data. Using past measurements and data trends, IRT characterization engineers wanted to develop methods for the ability to know when data were not within variation. Trends can be observed in the liquid water content measurement process by constructing statistical process control charts. This paper describes data processing procedures for the Multi-Element Sensor in the IRT, including collision efficiency corrections, canonical correlation analysis, process for rejection of data, and construction of control charts. Data are presented to display the control capability to meet defined liquid water content specifications of the IRT with the Multi-Element Sensor mounted in the center of the test section.

Statistical Process Control↗

NASA NTRSDate not supplied

A Survey of ISS and Visiting Vehicle Returned Surfaces for Environmental Characterization and Computer Model Development

The Orbital Debris Engineering Model (ORDEM) developed by the NASA Orbital Debris Program Office (ODPO) is a data-driven model — extensive radar, optical, laboratory, and in situ measurement data sets have been used to build the model since its earliest versions. A salient aspect of professional software development is the verification and validation (V&V) process. Verification answers the question “Is the model built correctly?” while validation addresses the question “Did we build the correct model?” Less extensive, reserved, or independent data sets serve the validation requirement. Due to the dynamic nature of the orbital debris environment, it is critical to use contemporaneous data sources that represent the current environment to support ORDEM development and validation. ORDEM has utilized in situ data collected from Space Shuttle and Hubble Space Telescope surface inspections, now over a decade old. This historical dataset is fundamental for providing baseline in situ measurement data for sizes between 10 to 300 microns, but new data sources are being evaluated using returned surfaces from or near the International Space Station (ISS). This paper reviews a general microscopic survey of ISS soft goods, the Pressurized Mating Adapter 2 (PMA-2) blanket, and a limited-scope feasibility study conducted on the Space Exploration Technologies Corporation (SpaceX) Dragon capsule’s Thermal Protection System (TPS) material. The PMA-2 blanket, exposed to the space environment between 09 July 2013 and 25 February 2015, is an approximately 3.7 m2-area blanket composed of a betacloth outer layer and multiple ballistic fabric inner layers. The SpaceX Cargo Dragon capsule regularly visited the ISS from 2012 through 2020 and potentially provides a timely and well-characterized source of data for modeling purposes. The capsule’s lateral surfaces use SpaceX Proprietary Ablative Material (SPAM) TPS material, a syntactic foam, for thermal management during all mission phases. Seven SPAM extracted samples have been analyzed to date. This paper will provide an overview of the characterization completed for impact features by size, depth, impactor diameter, and the impactor residues chemical analyses, allowing a differentiation between micrometeoroids and orbital debris and a categorization by mass density and density class. Impactor diameter is estimated using damage equations generated from ground-based hypervelocity impact testing. The orbital debris impactors are compared to the current ORDEM 3.2 model of the environment at ISS altitudes. We briefly discuss the meteoroid impactors, including constituents and mass densities, in the general context of current models.

Phillip Anz-Meador↗

NASA NTRS1983

Derivation of the fundamental equation of sound generated by moving aerodynamic surfaces

Ffowcs Williams and Hawkings (1969) based their derivation of the fundamental equation of the sound generated by arbitrarily moving aerodynamic surfaces on the study of mass and momentum balance of a control volume imbedding a mathematical surface(s) exactly corresponding to real surface(s). These investigators also sketched an alternative method, employing generalized functions, for its derivation. This latter method, which was later developed by Farassat (1975), is purely mathematical and formal. Goldstein (1976) used the free-space Green function to produce an implicit derivation of the Ffowcs Williams and Hawkings equation. In the study presented here, Lowson's (1965) concept of moving point singularities is generalized to moving surface singularities, and a new derivation is given of the fundamental equation. The derivation is based on topological considerations of the underlying space, the fluid medium, and the integral properties of the Dirac delta function.

Aggarwal, H. R.↗

NASA NTRS1989

Systematic generation of multibody equations of motion suitable for recursive and parallel manipulation

The formulation of a method known as the joint coordinate method for automatic generation of the equations of motion for multibody systems is summarized. For systems containing open or closed kinematic loops, the equations of motion can be reduced systematically to a minimum number of second order differential equations. The application of recursive and nonrecursive algorithms to this formulation, computational considerations and the feasibility of implementing this formulation on multiprocessor computers are discussed.

Nikravesh, Parviz E.↗

NASA NTRS1976

The electron Boltzmann equation in a plasma generated by fission fragments

A Boltzmann equation formulation is presented for the determination of the electron distribution function in a plasma generated by fission fragments. The formulation takes into consideration ambipolar diffusion, elastic and inelastic collisions, recombination and ionization, and allows for the fact that the primary electrons are not monoenergetic. Calculations for He in a tube coated with fissionable material show that, over a wide pressure and neutron flux range, the distribution function is non-Maxwellian, but the electrons are essentially thermal. Moreover, about a third of the energy of the primary electrons is transferred into the inelastic levels of He. This fraction of energy transfer is almost independent of pressure and neutron flux but increases sharply in the presence of a sustainer electric field.

Hassan, H. A.↗

DOE OSTI2025

Gaussian-process generative model for the QCD equation of state

We develop a generative model for the nuclear matter equation of state at zero net baryon density using the Gaussian process regression method. We impose first-principles theoretical constraints from lattice quantum chromodynamics and hadron resonance gas at high- and low-temperature regions, respectively. By allowing the trained Gaussian process regression model to vary freely near the phase transition region, we generate random smooth crossover equations of state with different speeds of sound that do not rely on specific parametrizations. Here, we explore a collection of experimental observable dependencies on the generated equations of state, which paves the groundwork for future Bayesian inference studies to use experimental measurements from relativistic heavy-ion collisions to constrain the nuclear matter equation of state.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗