PARTIAL MATHEMATICAL SOLUTION OF THE THREE-DIMENSIONAL FOUR-BAR LINKAGE
Spacecraft design having four solar panels of which two adjacent hinges have skew axes in fixed coordinate system
SPACECRAFT DESIGN↗
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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
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Formal mathematical solutions of the force-free equations, spontaneous discontinuities, and dissipation in large-scale magnetic fields
Direct integration of the force-free field equation del x B = alpha B, in the simple case of the local deformation of a laminar field, produces field configurations containing tangential discontinuities (current sheets). Whereas continuous solutions allow only restricted field topologies, the discontinuities provide the necessary release from those restrictions in more general topologies. Magnetic fields in nature are strongly deformed by convection, so as to contain significant internal discontinuities. The bipolar magnetic fields containing the active X-ray corona of the sun are a case in point. It appears that the dissipation caused by the discontinuities may be the primary heat source producing the X-ray corona.
Parker, E. N.↗
Preprocessing Inconsistent Linear System for a Meaningful Least Squares Solution
Mathematical models of many physical/statistical problems are systems of linear equations. Due to measurement and possible human errors/mistakes in modeling/data, as well as due to certain assumptions to reduce complexity, inconsistency (contradiction) is injected into the model, viz. the linear system. While any inconsistent system irrespective of the degree of inconsistency has always a least-squares solution, one needs to check whether an equation is too much inconsistent or, equivalently too much contradictory. Such an equation will affect/distort the least-squares solution to such an extent that renders it unacceptable/unfit to be used in a real-world application. We propose an algorithm which (i) prunes numerically redundant linear equations from the system as these do not add any new information to the model, (ii) detects contradictory linear equations along with their degree of contradiction (inconsistency index), (iii) removes those equations presumed to be too contradictory, and then (iv) obtain the minimum norm least-squares solution of the acceptably inconsistent reduced linear system. The algorithm presented in Matlab reduces the computational and storage complexities and also improves the accuracy of the solution. It also provides the necessary warning about the existence of too much contradiction in the model. In addition, we suggest a thorough relook into the mathematical modeling to determine the reason why unacceptable contradiction has occurred thus prompting us to make necessary corrections/modifications to the models - both mathematical and, if necessary, physical.
Sen, Syamal K.↗
Preprocessing in Matlab Inconsistent Linear System for a Meaningful Least Squares Solution
Mathematical models of many physical/statistical problems are systems of linear equations~ Due to measurement and possible human errors/mistakes in modeling/data, as well as due to certain assumptions to reduce complexity, inconsistency (contradiction) is injected into the model, viz. the linear system. While any inconsistent system irrespective of the degree of inconsistency has always a least-squares solution, one needs to check whether an equation is too much inconsistent or, equivalently too much contradictory. Such an equation will affect/distort the least-squares solution to such an extent that renders it unacceptable/unfit to be used in a real-world application. We propose an algorithm which (i) prunes numerically redundant linear equations from the system as these do not add any new information to the model, (ii) detects contradictory linear equations along with their degree of contradiction (inconsistency index), (iii) removes those equations presumed to be too contradictory, and then (iv) obtain the . minimum norm least-squares solution of the acceptably inconsistent reduced linear system. The algorithm presented in Matlab reduces the computational and storage complexities and also improves the accuracy of the solution. It also provides the necessary warning about the existence of too much contradiction in the model. In addition, we suggest a thorough relook into the mathematical modeling to determine the reason why unacceptable contradiction has occurred thus prompting us to make necessary corrections/modifications to the models - both mathematical and, if necessary, physical.
Sen, Symal K.↗
The surface temperatures of solar system objects by thermal emission - A homogeneous model solution
Mathematical model for thermal behavior and surface temperatures of celestial bodies in solar system
Burnett, G. B., III↗
Assessment and Accommodation of Thermal Expansion of the Internal Active Thermal Control System Coolant During Launch to On-Orbit Activation of International Space Station Elements
The International Space Station (ISS) employs an Internal Active Thermal Control System (IATCS) comprised of several single-phase water coolant loops. These coolant loops are distributed throughout the ISS pressurized elements. The primary element coolant loops (i.e., US Laboratory module) contain a fluid accumulator to accommodate thermal expansion of the system. Other element coolant loops are parasitic (i.e., Airlock), have no accumulator, and require an alternative approach to insure that the system Maximum Design Pressure (MDP) is not exceeded during the Launch to Activation phase. During this time the element loop is a stand alone closed individual system. The solution approach for accommodating thermal expansion was affected by interactions of system components and their particular limitations. The mathematical solution approach was challenged by the presence of certain unknown or not readily obtainable physical and thermodynamic characteristics of some system components and processes. The purpose of this paper is to provide a brief description of a few of the solutions that evolved over time, a novel mathematical solution to eliminate some of the unknowns or derive the unknowns experimentally, and the testing and methods undertaken.
Edwards, J. Darryl↗
Assessment and Accommodation of Thermal Expansion of the Internal Active Thermal Control System Coolant During Launch to On-Orbit Activation of International Space Station Elements
The International Space Station (ISS) employs an Internal Active Thermal Control System (IATCS) comprised of several single-phase water coolant loops. These coolant loops are distributed throughout the ISS pressurized elements. The primary element coolant loops (i.e. U.S. Laboratory module) contain a fluid accumulator to accomodate thermal expansion of the system. Other element coolant loops are parasitic (i.e. Airlock), have no accumulator, and require an alternative approach to insure that the system maximum design pressure (MDP) is not exceeded during the Launch to Activation (LTA) phase. During this time the element loops is a stand alone closed system. The solution approach for accomodating thermal expansion was affected by interactions of system components and their particular limitations. The mathematical solution approach was challenged by the presence of certain unknown or not readily obtainable physical and thermodynamic characteristics of some system components and processes. The purpose of this paper is to provide a brief description of a few of the solutions that evolved over time, a novel mathematical solution to eliminate some of the unknowns or derive the unknowns experimentally, and the testing and methods undertaken.
Edwards, Darryl↗
Numerical Experiments in Error Control for Sound Propagation Using a Damping Layer Boundary Treatment
This paper presents results from numerical experiments for controlling the error caused by a damping layer boundary treatment when simulating the propagation of an acoustic signal from a continuous pressure source. The computations are with the 2D Linearized Euler Equations (LEE) for both a uniform mean flow and a steady parallel jet. The numerical experiments are with algorithms that are third, fifth, seventh and ninth order accurate in space and time. The numerical domain is enclosed in a damping layer boundary treatment. The damping is implemented in a time accurate manner, with simple polynomial damping profiles of second, fourth, sixth and eighth power. At the outer boundaries of the damping layer the propagating solution is uniformly set to zero. The complete boundary treatment is remarkably simple and intrinsically independant from the dimension of the spatial domain. The reported results show the relative effect on the error from the boundary treatment by varying the damping layer width, damping profile power, damping amplitude, propagtion time, grid resolution and algorithm order. The issue that is being addressed is not the accuracy of the numerical solution when compared to a mathematical solution, but the effect of the complete boundary treatment on the numerical solution, and to what degree the error in the numerical solution from the complete boundary treatment can be controlled. We report maximum relative absolute errors from just the boundary treatment that range from O[10-2] to O[10-7].
Algorithm↗
Numerical Experiments in Error Control for Sound Propagation Using a Damping Layer Boundary Treatment
This paper presents results from numerical experiments for controlling the error caused by a damping layer boundary treatment when simulating the propagation of an acoustic signal from a continuous pressure source. The computations are with the 2D Linearized Euler Equations (LEE) for both a uniform mean flow and a steady parallel jet. The numerical experiments are with algorithms that are third, fifth, seventh and ninth order accurate in space and time. The numerical domain is enclosed in a damping layer boundary treatment. The damping is implemented in a time accurate manner, with simple polynomial damping profiles of second, fourth, sixth and eighth power. At the outer boundaries of the damping layer the propagating solution is uniformly set to zero. The complete boundary treatment is remarkably simple and intrinsically independant from the dimension of the spatial domain. The reported results show the relative effect on the error from the boundary treatment by varying the damping layer width, damping profile power, damping amplitude, propagtion time, grid resolution and algorithm order. The issue that is being addressed is not the accuracy of the numerical solution when compared to a mathematical solution, but the effect of the complete boundary treatment on the numerical solution, and to what degree the error in the numerical solution from the complete boundary treatment can be controlled. We report maximum relative absolute errors from just the boundary treatment that range from O[10-2] to O[10-7].
Computational Aeroacoustics↗
Preliminary Thermal Analysis of a Mars Sample Return Earth Entry Vehicle
Thermal analysis of a vehicle designed to return samples from another planet, such as the Earth Entry vehicle for the Mars Sample Return mission, presents several unique challenges. The scientific purpose of a sample return mission is to return samples to Earth for detailed investigation. The Earth Entry Vehicle (EEV) must contain the samples after they have been collected and protect them from the high hearing rates of entry into the Earth's atmosphere. This requirement necessitates inclusion of detailed thermal analysis early in the design of the vehicle. This paper will describe the challenges and solutions for a preliminary thermal analysis of an Earth Entry Vehicle. The primary challenges included accurate updates of model .geometry, applying heat fluxes that change with position and time during exo-atmospheric cruise and entry, and incorporating orthotropic material properties. Many different scenarios were evaluated for the exo-atmospheric cruise to attain the desired thermal condition. The severity of the heat pulse during entry and the material response led to some unique modeling solutions. Overall, advanced modeling techniques and mathematical solutions were successfully used in predicting the thermal behavior of this complex system.
Amundsen, Ruth M.↗
Preliminary Thermal Analysis of a Mars Sample Return Earth Entry Vehicle
Thermal analysis of a vehicle designed to return samples from another planet, such as the Earth Entry vehicle for the Mars Sample Return mission, presents several unique challenges. The scientific purpose of a sample return mission is to return samples to Earth for detailed investigation. The Earth Entry Vehicle (EEV) must contain the samples after they have been collected and protect them from the high heating rates of entry into the Earth's atmosphere. This requirement necessitates inclusion of detailed thermal analysis early in the design of the vehicle. This paper will describe the challenges and solutions for a preliminary thermal analysis of an Earth Entry Vehicle. The primary challenges included accurate updates of model geometry, applying heat fluxes that change with position and time during exo-atmospheric cruise and entry, and incorporating orthotropic material properties. Many different scenarios were evaluated for the exoatmospheric cruise to attain the desired thermal condition. The severity of the heat pulse during entry and the material response led to some unique modeling solutions. Overall, advanced modeling techniques and mathematical solutions were successfully used in predicting the thermal behavior of this complex system.
Amundsen, Ruth M.↗
A photogrammetric solution to a particular problem
A closed-form mathematical solution to the classical photogrammetric problem is presented. Although quite general, the solution is more applicable to problems in which the image-space conjugates are very difficult to match but one of the elements of the pair is not. Additionally, observations are made that should make the solution to the general problem of automatic matching less computationally intensive. This approach was used to analyze flow visualization data for the F-18 High Alpha Research Vehicle. The conditions for this analysis were less than ideal for image-to-object-space transformation.
Hedgley, David R., Jr.↗
Solution of several problems for doubly connected regions with contiguous circular boundaries
Mathematical solution for doubly-connected regions with contiguous circular boundaries, and treatment of half-plane with one circular hole and plane with two equal holes
BOUNDARY VALUE↗
Methods of solution of the integral equation for the Foucault test Final report, 1 Jul. - 30 Sep. 1969
Mathematical solutions of integral equations for accurately adjusting astronomical mirrors in artificial satellites
Gatewood, B. E.↗
Proceeding of the Twenty-Fourth Seminar on Space Flight and Guidance Theory Progress report no. 8, 1 Apr. - 31 Dec. 1965
Seminar on space flight and guidance theory, with special attention to mathematical solutions of system stability
CONFERENCE↗
Solving Common Mathematical Problems
Mathematical Solutions Toolset is a collection of five software programs that rapidly solve some common mathematical problems. The programs consist of a set of Microsoft Excel worksheets. The programs provide for entry of input data and display of output data in a user-friendly, menu-driven format, and for automatic execution once the input data has been entered.
Luz, Paul L.↗
Characterization of Errors in Satellite-Based HCHO/NO 2 Tropospheric Column Ratios With Respect to Chemistry, Column-to-PBL Translation, Spatial Representation, and Retrieval Uncertainties
The availability of formaldehyde (HCHO) (a proxy for volatile organic compound reactivity) and nitrogen dioxide (NO 2 ) (a proxy for nitrogen oxides) tropospheric columns from ultraviolet–visible (UV–Vis) satellites has motivated many to use their ratios to gain some insights into the near-surface ozone sensitivity. Strong emphasis has been placed on the challenges that come with transforming what is being observed in the tropospheric column to what is actually in the planetary boundary layer (PBL) and near the surface; however, little attention has been paid to other sources of error such as chemistry, spatial representation, and retrieval uncertainties. Here we leverage a wide spectrum of tools and data to quantify those errors carefully. Concerning the chemistry error, a well-characterized box model constrained by more than 500 h of aircraft data from NASA's air quality campaigns is used to simulate the ratio of the chemical loss of HO 2 + RO 2 (LRO x ) to the chemical loss of NO x (LNO x ). Subsequently, we challenge the predictive power of HCHO/NO 2 ratios (FNRs), which are commonly applied in current research, in detecting the underlying ozone regimes by comparing them to LRO x /LNO x . FNRs show a strongly linear (R 2 =0.94) relationship with LRO x /LNO x , but only on the logarithmic scale. Following the baseline (i.e., ln(LRO x /LNO x ) = −1.0 ± 0.2) with the model and mechanism (CB06, r2) used for segregating NO x -sensitive from VOC-sensitive regimes, we observe a broad range of FNR thresholds ranging from 1 to 4. The transitioning ratios strictly follow a Gaussian distribution with a mean and standard deviation of 1.8 and 0.4, respectively. This implies that the FNR has an inherent 20 % standard error (1σ) resulting from not accurately describing the RO x –HO x cycle. We calculate high ozone production rates (PO 3 ) dominated by large HCHO × NO 2 concentration levels, a new proxy for the abundance of ozone precursors. The relationship between PO 3 and HCHO × NO 2 becomes more pronounced when moving towards NO x -sensitive regions due to nonlinear chemistry; our results indicate that there is fruitful information in the HCHO × NO 2 metric that has not been utilized in ozone studies. The vast amount of vertical information on HCHO and NO 2 concentrations from the air quality campaigns enables us to parameterize the vertical shapes of FNRs using a second-order rational function permitting an analytical solution for an altitude adjustment factor to partition the tropospheric columns into the PBL region. We propose a mathematical solution to the spatial representation error based on modeling isotropic semivariograms. Based on summertime-averaged data, the Ozone Monitoring Instrument (OMI) loses 12 % of its spatial information at its native resolution with respect to a high-resolution sensor like the TROPOspheric Monitoring Instrument (TROPOMI) (> 5.5 × 3.5 km 2 ). A pixel with a grid size of 216 km 2 fails at capturing ∼ 65 % of the spatial information in FNRs at a 50 km length scale comparable to the size of a large urban center (e.g., Los Angeles). We ultimately leverage a large suite of in situ and ground-based remote sensing measurements to draw the error distributions of daily TROPOMI and OMI tropospheric NO 2 and HCHO columns. At a 68 % confidence interval (1σ), errors pertaining to daily TROPOMI observations, either HCHO or tropospheric NO 2 columns, should be above 1.2–1.5 × 10 16 molec. cm −2 to attain a 20 %–30 % standard error in the ratio. This level of error is almost non-achievable with the OMI given its large error in HCHO. The satellite column retrieval error is the largest contributor to the total error (40 %–90 %) in the FNRs. Due to a stronger signal in cities, the total relative error (< 50 %) tends to be mild, whereas areas with low vegetation and anthropogenic sources (e.g., the Rocky Mountains) are markedly uncertain (> 100 %). Our study suggests that continuing development in the retrieval algorithm and sensor design and calibration is essential to be able to advance the application of FNRs beyond a qualitative metric.
Amir H. Souri↗
A qualitative study of the complete set of solutions of the differential equation of motion of a test particle in the equatorial plane of the Kerr gravitational field
A study is made of the mathematical solution of the differential equation of motion of a test particle in the equatorial plane of the Kerr gravitational field, using S (Schwarzschild-like) coordinates. A qualitative solution of this equation leads to the conclusion that there can only be 25 different types of orbits. For each value of a, the results are presented in a master diagram for which h and e are the parameters. A master diagram divides the h, e parameter space into regions such that at each point within one of these regions the types of admissible orbits are qualitatively the same. A pictorial representation of the physical orbits in the r, phi plane is also given.
Montgomery, H. E.↗