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FIT is computer program for interactively determining least-squares polynomial equations that fit user-supplied data. Finds leastsquares fits for functions of two independent variables. Interactive graphical and editing capabilities in FIT enables user to control polynomial equations to be fitted to data arising from most practical applications. FIT written in FORTRAN and COMPASS.
The geometry of general three-dimensional bodies is generated from coordinates of points in several cross sections. Since these points may not be smooth, they are divided into segments and general conic sections are curve fit in a least-squares sense to each segment of a cross section. The conic sections are then blended in the longitudinal direction by fitting parametric cubic-spline curves through coordinate points which define the conic sections in the cross-sectional planes. Both the cross-sectional and longitudinal curves may be modified by specifying particular segments as straight lines and slopes at selected points. Slopes may be continuous or discontinuous and finite or infinite. After a satisfactory surface fit has been obtained, cards may be punched with the data necessary to form a geometry subroutine package for use in other computer programs. At any position on the body, coordinates, slopes and second partial derivatives are calculated. The method is applied to a blunted 70 deg delta wing, and it was found to generate the geometry very well.
A system for measuring melting temperatures at high pressures is described. The sample is heated with radiation from a YAG laser. The beam is reflected downward through a microscope objective, through the upper diamond anvil, and focused onto the sample. Hense, intense heating is produced only at the sample and not within the diamond anvils. A vidicon system is used to observe the sample during heating. Incandescent light from the heated sample passes back through the objective lens into a grating spectrometer. The spectrum of the incandescent light is received by the photodiode array and stored in the multichannel analyzer. These data can then be transferred to floppy disk for analysis. A curve fitting program is used to compare the spectra with standard blackbody curves and to determine the temperature. Pressure is measured by the ruby fluorescence method. The system was used to study the melting behavior of natural troilite (FeS).
Chronological information about Martian processes comes from two sources: Crater-frequency studies and laboratory studies of Martian meteorites. Each has limitations that could be overcome by studies of returned Martian rocks and soils. Chronology of Martian volcanism: The currently accepted chronology of Martian volcanic surfaces relies on crater counts for different Martian stratigraphic units [1]. However, there is a large inherent uncertainty for intermediate ages near ~2 Ga ago. The effect of differing preferences for Martian cratering chronologies [1] is shown in Fig. 1. Stoeffler and Ryder [2] summarized lunar chronology, upon which Martian cratering chronology is based. Fig. 2 shows a curve fit to their data, and compares to it a corresponding lunar curve from [3]. The radiometric ages of some lunar and Martian meteorites as well as the crater-count delimiters for Martian epochs [4] also are shown for comparison to the craterfrequency curves. Scaling the Stoeffler-Ryder curve by a Mars/Moon factor of 1.55 [5] places Martian shergottite ages into the Early Amazonian to late Hesperian epochs, whereas using the lunar curve of [3] and a Mars/Moon factor ~1 consigns the shergottites to the Middle-to-Late Amazonian, a less probable result. The problem is worsened if a continually decreasing cratering rate since 3 Ga ago is accepted [6]. We prefer the adjusted St ffler-Ryder curve because it gives better agreement with the meteorite ages (Fig.
The PLTTER graphics system, which is part of CDDMS is discussed. CDDMS is a comprehensive system for data basing and subsequent plotting of data acquired during wind tunnel tests or from computational flow analyses. The PLTTER is a system which creates report-quality plots of data which is stored on a CDDMS data base. The Requests file system allows plot-controlling information to be arranged in the way which is most appropriate for any application. The PLTTER system features many capabilities which are especially useful when plotting wind tunnel data. The PLTTER offers a variety of page formats, different grid options and parametric curve fitting algorithms, and a powerful legend capability to identify relevant information about individual curves. One or more plots on a page can be suppressed if desired so that an established page format can be maintained. Final plot output may be standard Versatec plots, QMS Laser printer plots, or microfiche.
Ongoing integration of outrunner brushless electric motors into drones and Advanced Air Mobility aircraft presents a need for accurate acoustic predictions derived from prediction of the motor rotor vibrations. In outrunner motors, the external rotor vibrates and drives the acoustic field. At resonant frequencies of the rotor, the rotor displacements will be the largest. The vibrations can lead to acoustic tones that are relevant to the overall acoustic design of these types of vehicles. This study performs a finite element analysis to assess the modes in two electric motor rotors. This model is then refined into a simple parameterized geometry that can make the same predictions for motors of the same class. A parametric sweep and least-squares-curve fit to the finite element analysis data results in a series of simple curves that can predict the mode shapes and frequencies that are likely to appear in this class of motors without the need for any simulation. The simulations are validated by comparison to acoustic and experimental modal-analysis data.
Ongoing integration of outrunner brushless electric motors into drones and Advanced Air Mobility aircraft presents a need for accurate acoustic predictions derived from prediction of the motor rotor vibrations. In outrunner motors, the external rotor vibrates and drives the acoustic field. At resonant frequencies of the rotor, the rotor displacements will be the largest. The vibrations can lead to acoustic tones that are relevant to the overall acoustic design of these types of vehicles. This study performs a finite element analysis to assess the modes in two electric motor rotors. This model is then refined into a simple parameterized geometry that can make the same predictions for motors of the same class. A parametric sweep and least-squares-curve fit to the finite element analysis data results in a series of simple curves that can predict the mode shapes and frequencies that are likely to appear in this class of motors without the need for any simulation. The simulations are validated by comparison to acoustic and experimental modal-analysis data.
Visual absolute magnitudes of classical Cepheids, metal-poor RR Lyrae stars and short-period type II Cepheids have been determined with very high precision by combining a large number of old and new astrophysical data. Five independent methods (four observational and one theoretical) have been successfully used: (1) secular and statistical parallaxes; (2) moving-group parallaxes; (3) cluster main-sequence fitting; (4) the Baade-Wesselink method and its modifications; and (5) light-curve and velocity-curve fitting (the theoretical method). With these five adopted methods, the zero point of the galactic and extragalactic distance scale can be set on a relatively firm and self-consistent basis. Classical Cepheids and RR Lyrae stars now provide essentially identical distances to nearby galaxies: the distance modulus to the Large Magellanic Cloud is 18.5 and to the Small Magellanic Cloud is 18.8, both with an uncertainty of 0.1. The distance to the center of our Galaxy is 8.6 plus or minus 0.5 kiloparsecs. The major uncertainty in these values lies in the correction for interstellar extinction.
Numerical optimization techniques are playing an increasing role in curve and surface construction. Often difficult problems in curve and surface construction, especially when some aspect of shape control is involved, can be phrased as a constrained optimization problem. Four such classes of problems are explored: parametric curve fitting with non-linear shape constraints; explicit surface fitting with linear shape constraints; surface fitting to scattered data giving rise to ill-posed problems; finally, variable knot problems. In each of these problems there is a nonlinear aspect: either the shape of the curve or surface is important for manufacturing or engineering reasons or the shape affects the convergence of numerical algorithms which use the curve or surface or the placement of knots affects the accuracy of the fits. In all cases the class of functions used is that of parametric spline curves and tensor or direct product spline surfaces. The reason for choosing this class is that splines provide flexible models that are easily evaluated and stored. Furthermore, the B-spline representation of splines leads to convenient expressions for shape control over regions.
Numerical flowfield methods require a geometry subprogram which can calculate body coordinates, slopes, and radii of curvature for typical aircraft and spacecraft configurations. The objective of this paper is to develop a new surface-fitting technique which addresses two major problems with existing geometry packages: computer storage requirements and the time required of the user for the initial set-up of the geometry model. In the present method, coordinates of cross sections are fit in a least-squares sense using segments of general conic sections. After fitting each cross section, the next step is to blend the cross-sectional curve-fits in the longitudinal direction using general conics to fit specific meridional half-planes. For the initial setup of the geometry model, an interactive, completely menu-driven computer code has been developed to allow the user to make modifications to the initial fit for a given cross section or meridional cut. Graphic displays are provided to assist the user in the visualization of the effect of each modification. The completed model may be viewed from any angle using the code's three-dimensional graphics package. Geometry results for the modeling of the Space Shuttle and a proposed Aeroassist Flight Experiment (AFE) geometry are presented, in addition to calculated heat-transfer rates based on these models.
The geometry of general three-dimensional bodies was generated from coordinates of points in several cross sections. Since these points may not be on smooth curves, they are divided into groups forming segments and general conic sections are curve fit in a least-squares sense to each segment of a cross section. The conic sections are then blended in the longitudinal direction through longitudinal curves. Both the cross-sectional and longitudinal curves may be modified by specifying particular segments as straight lines or specifying slopes at selected points. This method was used to surface fit a 70 deg slab delta wing and the HL-10 Lifting Body. The results for the delta wing were very close to the exact geometry. Although there is no exact solution for the lifting body, the surface fit generated a smooth surface with cross-sectional planes very close to prescribed coordinate points.
The second phase of a joint National Aeronautics and Space Administration (NASA) and Federal Aviation Administration (FAA) program to study the high-altitude, low-frequency acoustic noise propagation characteristics of the Advanced Turboprop (propfan) Aircraft was conducted on April 3-13, 1989 at the White Sands Missile Range (WSMR), New Mexico. The first phase was conducted on October 26-31, 1987 in Huntsville, Alabama. NASA (Lewis) measured the source noise of the test aircraft during both phases while NASA (Langley) measured surface noise only during the second phase. FAA/NASA designed a program to obtain noise level data from the propfan test bed aircraft, both in the near field and at ground level, during simulated en route flights (35,000 and 20,000 feet ASL), and to test low frequency atmospheric absorption algorithms and prediction technology to provide insight into the necessity for regulatory measures. The curves of calculated source noise versus emission angle are based on a second order best-fit curve of the peak envelope of the adjusted ground data. Centerline and sideline derived source noise levels are shown to be in good agreement. A comparison of the Alabama chase plane source data and the calculated source noise at centerline for both the Alabama and New Mexico data shows good agreement for the 35,000 and the 20,000 feet (ASL) overflights. With the availability of the New Mexico in-flight data, further in depth comparisons will be made.
The derivation of design values for compression after impact strength for two types of honeycomb sandwich structures are presented. The sandwich structures in this study had an aluminum core and composite laminate facesheets of either 16-ply quasi or 18-ply directional lay-ups. The results show that a simple power law curve fit to the data can be used to create A- and B-basis residual strength curves.
A cross-power spectrum phase based adaptive technique is discussed which iteratively determines the time delay between two digitized signals that are coherent. The adaptive delay algorithm belongs to a class of algorithms that identifies a minimum of a pattern matching function. The algorithm uses a gradient technique to find the value of the adaptive delay that minimizes a cost function based in part on the slope of a linear function that fits the measured cross power spectrum phase and in part on the standard error of the curve fit. This procedure is applied to data from a Honeywell TECH977 static-engine test. Data was obtained using a combustor probe, two turbine exit probes, and far-field microphones. Signals from this instrumentation are used estimate the post-combustion residence time in the combustor. Comparison with previous studies of the post-combustion residence time validates this approach. In addition, the procedure removes the bias due to misalignment of signals in the calculation of coherence which is a first step in applying array processing methods to the magnitude squared coherence data. The procedure also provides an estimate of the cross-spectrum phase-offset.
The preparation of geometric data for input to three-dimensional potential flow programs was automated and simplified by a geometry package incorporated into the NASA Langley version of the 3-D lifting potential flow program. Input to the computer program for the geometry package consists of a very sparse set of coordinate data, often with an order of magnitude of fewer points than required for the actual potential flow calculations. Isolated components, such as wings, fuselages, etc. are paneled automatically, using one of several possible element distribution algorithms. Curves of intersection between components are calculated, using a hybrid curve-fit/surface-fit approach. Intersecting components are repaneled so that adjacent elements on either side of the intersection curves line up in a satisfactory manner for the potential-flow calculations. Many cases may be run completely (from input, through the geometry package, and through the flow calculations) without interruption. Use of the package significantly reduces the time and expense involved in making three-dimensional potential flow calculations.
The Laser Transit Anemometer (LTA) system is described. In the LTA system two parallel laser beams of known separation and cross sectional area are focussed at the same location or plane. When a particle in a flow field passes through both beams and the time is recorded for its transit (time of flight), its velocity can be calculated knowing the distance between the beams. By rotating the two beams (spots) around a common center and recording the number of valid events (a particle which passes through both spots in the proper sequence) at each angle the flow angle can be determined by curve fitting a predetermined number of angles or points and calculating the peak of what should be a Gaussian curve. The best angle or flow angle is defined as the angle at which the maximum number of valid events occurs. The LTA system functioned properly although conditions were less than desirable.
A surface-fitting technique has been developed which addresses two problems with existing geometry packages: computer storage requirements and the time required of the user for the initial setup of the geometry model. Coordinates of cross sections are fit using segments of general conic sections. The next step is to blend the cross-sectional curve-fits in the longitudinal direction using general conics to fit specific meridional half-planes. Provisions are made to allow the fitting of fuselages and wings so that entire wing-body combinations may be modeled. This report includes the development of the technique along with a User's Guide for the various menus within the program. Results for the modeling of the Space Shuttle and a proposed Aeroassist Flight Experiment geometry are presented.
Nine plates of the 7820-A CO2 band were taken in 1971. A curve-of-growth analysis of the CO2 lines indicates a rotational temperature of 241 + or - 2 K, with an average slope to the curve of growth of 0.60 + or - 0.03. The Venus phase angle ranged from 7.2 to 10.7 deg. The equivalent widths of the 1971 data fall on a smooth curve fit through the 1969 data for this band; there does not appear to be any discontinuity in the phase curve at small phase angles.