Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “differential transformation”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 361 records · Page 20
All sourcesNASA NTRSDOE OSTI
NASA NTRS1980

Design Studies for a Far Infrared Absolute Spectrometer for the Cosmic Background Explorer

Unrelenting symmetry of design is required to assure the thermal balance of a cryogenically cooled, rapid scan interferometer spectrometer to be mounted in vacuum with the Cosmic Background Explorer liquid helium dewar. The instrument receives inputs from Winston cone optical flux collectors, one open to space and a second coupled to a black body reference source. A differential instrument, the spectrometer produces outputs corresponding to the Fourier transform of the spectral radiance difference between the two inputs. The two outputs are sensed by four detectors, two optimized for shorter wavelength response, and two optimized for longer wavelengths. The optical design, detector and signal channel, system sensitivity, mechanics, thermal control and cryogenics, electronics and power systems, command and control, calibration, system test requirements, and the instrument interface are discussed. Recommendations for continued work are indicated for the superconducting reflective horns, the motor bearing and drive, and design detail.

Johnson, N. J. E.↗
NASA NTRS1982

Boundary-fitted coordinate systems for numerical solution of partial differential equations - A review

A comprehensive review of methods of numerically generating curvilinear coordinate systems with coordinate lines coincident with all boundary segments is given. Some general mathematical framework and error analysis common to such coordinate systems is also included. The general categories of generating systems are those based on conformal mapping, orthogonal systems, nearly orthogonal systems, systems produced as the solution of elliptic and hyperbolic partial differential equations, and systems generated algebraically by interpolation among the boundaries. Also covered are the control of coordinate line spacing by functions embedded in the partial differential operators of the generating system and by subsequent stretching transformation. Dynamically adaptive coordinate systems, coupled with the physical solution, and time-dependent systems that follow moving boundaries are treated. References reporting experience using such coordinate systems are reviewed as well as those covering the system development.

Thompson, J. F.↗
NASA NTRSDate not supplied

Evaluating Crystallinity in Thermoplastic Composites for Aerospace Applications

Polymer matrix composites (PMCs) offer many benefits for the aerospace industry due to their potential for weight reduction when compared to metal or ceramic based materials. Most PMCs currently in flight use thermoset matrices, however, thermoplastic resins are being explored as alternatives due to their ability to be remelted, which is of particular interest due to the potential for in-situ repair and faster production. Most thermoplastic resins are semicrystalline polymers. The properties of semicrystalline thermoplastics are directly influenced by their crystallinity, which can vary due to many factors including thermal treatments, environmental conditions, and mechanical deformation. Within thermoplastic PMC parts, crystallinity gradients can arise due to variations in part geometry, across part thicknesses, and along bonded joints. Monitoring the crystallinity of thermoplastic composites is key to ensuring these materials meet the high demands required for aerospace. Several different analytical techniques exist that can be used to characterize the bulk crystallinity of thermoplastic materials. However, many existing methods lack the specificity required to identify the subtle variations in crystallinity that may play a significant role in the performance and durability of PMC parts. Because of this, a significant amount of work is still required to fully characterize and understand the crystallinity profiles of thermoplastic PMCs and the resulting impact to material properties. This talk discusses the use of multiple techniques such as Differential Scanning Calorimetry, Polarized Light Optical Microscopy, and Fourier-Transform Infrared Spectroscopy to characterize the crystallinity in carbon fiber/thermoplastic composites. Samples of different crystallinity profiles were manufactured using various cooling procedures. This work aims to provide the fundamental data necessary to understand the effects of crystallinity on thermoplastic PMCs, which is key to advancing their use in aerospace applications.

Thermoplastics↗
NASA NTRS1943

On a new method for calculating the potential flow past a body of revolution

A new method is presented for obtaining the velocity potential of the flow about a body of revolution moving uniformly in the direction of its axis of symmetry in a fluid otherwise at rest. This method is based essentially on the fact that the form of the differential equation for the velocity potential is invariant with regard to conformal transformation of the meridian plane. By means of the conformal transformation of the meridian profile into a circle a system of orthogonal curvilinear coordinates is obtained, the main feature of which is that one of the coordinate lines is the meridian profile itself. The use of this type of coordinate system yields a simple expression of the boundary condition at the surface of the solid and leads to a rational process of iteration for the solution of the differential equation for the velocity potential. It is shown that the velocity potential for an arbitrary body of revolution may be expressed in terms of universal functions which, although not normal, are obtainable by means of simple quadratures. The general results are applied to a body of revolution obtained by revolving a symmetrical Joukowski profile about its axis of symmetry. A numerical example further serves to illustrate the theory.

Kaplan, Carl↗
DOE OSTI2023

Grain size dependence of thermally induced oxidation in zirconium carbide

Here complementary analytical approaches were employed to probe the effect of grain size on thermally induced oxidation of zirconium carbide (ZrC) utilizing thermogravimetric analysis, differential scanning calorimetry, and Raman spectroscopy, as well as synchrotron-based and laboratory-based X-ray diffraction (XRD) experiments. The oxidation mechanism and phase behavior of nanocrystalline ZrC (grain size ~ 20 nm) were compared with that of the more documented microcrystalline ZrC (grain size ~ 1 µm). Synchrotron XRD at the Advanced Photon Source with a hydrothermal diamond anvil cell (HDAC) used as a sample chamber revealed that the onset of oxidation is at ~ 380 °C for microcrystalline ZrC which is in agreement with previous work. In contrast, the critical oxidation temperature was ~ 330 °C for nanocrystalline ZrC. Additional high-temperature synchrotron XRD experiments at the National Synchrotron Light Source II using a lamp furnace in combination with Raman analysis showed that tetragonal ZrO 2 forms as an initial oxidation product and transforms at higher temperatures to the monoclinic phase. Thermogravimetric analysis (TGA) coupled with differential scanning calorimetry (DSC) confirmed the X-ray results of a lower critical oxidation temperature for the nanocrystalline sample. The phase transformations in the oxide phase with associated critical temperatures were also evident in the thermodynamic data as exothermic heat events.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2025

Consistent performance of large language models in rare disease diagnosis across ten languages and 4917 cases

Background Large language models (LLMs) are increasingly used medicine for diverse applications including differential diagnostic support. The training data used to create LLMs such as the Generative Pretrained Transformer (GPT) predominantly consist of English-language texts, but LLMs could be used across the globe to support diagnostics if language barriers could be overcome. Initial pilot studies on the utility of LLMs for differential diagnosis in languages other than English have shown promise, but a large-scale assessment on the relative performance of these models in a variety of European and non-European languages on a comprehensive corpus of challenging rare-disease cases is lacking. Methods We created 4917 clinical vignettes using structured data captured with Human Phenotype Ontology (HPO) terms with the Global Alliance for Genomics and Health (GA4GH) Phenopacket Schema. These clinical vignettes span a total of 360 distinct genetic diseases with 2525 associated phenotypic features. We used translations of the Human Phenotype Ontology together with language-specific templates to generate prompts in English, Chinese, Czech, Dutch, French, German, Italian, Japanese, Spanish, and Turkish. We applied GPT-4o, version gpt-4o-2024-08-06, and the medically fine-tuned Meditron3-70B to the task of delivering a ranked differential diagnosis using a zero-shot prompt. An ontology-based approach with the Mondo disease ontology was used to map synonyms and to map disease subtypes to clinical diagnoses in order to automate evaluation of LLM responses. Findings For English, GPT-4o placed the correct diagnosis at the first rank 19.9% and within the top-3 ranks 27.0% of the time. In comparison, for the nine non-English languages tested here the correct diagnosis was placed at rank 1 between 16.9% and 20.6%, within top-3 between 25.4% and 28.6% of cases. The Meditron3 model placed the correct diagnosis within the first 3 ranks for 20.9% of cases in English and between 19.9% and 24.0% for the other nine languages. Interpretation The differential diagnostic performance of LLMs across a comprehensive corpus of rare-disease cases was largely consistent across the ten languages tested. This suggests that the utility of LLMs in clinical settings may extend to non-English clinical settings.

Artificial intelligence↗
DOE OSTI2020

Analysis of Material Properties of Aged Rigid Polyurethane Foam

Foams are classified as either rigid or flexible and are commonly used in applications such as pipe insulation, carpet fibers, and chair foam cushions. Polyurethane foams are excellent thermal insulators, have excellent mechanical properties, and are chemically inert. With the addition of fire protective additives, polyurethane foams have the ability to intumescence when exposed to high temperatures. For the future storage of the 9977 nuclear shipping packages, the material properties of the General Plastics 3716 rigid polyurethane foam were evaluated for long term performance by comparing aged samples to unaged samples. General Plastics 3716 rigid polyurethane foam was aged from November 2014 to June 2019 at varying temperatures of: 160 deg. F, 160 deg. F + 50% relative humidity (RH), 185 deg. F, 215 deg. F, and 250 deg. F. The foam was then analyzed via Fourier Transform Infrared Spectroscopy (FTIR), 3D Optical Microscope, Thermal Gravimetric Analysis (TGA), Differential Scanning Calorimetry (DSC). FTIR and Optical Microscope: No significant change to chemical or physical structure was evident by FTIR or optical microscope analysis. DSC: Decrease of heat capacity over time for all oven temperatures except a slight increase for the 185 deg. F conditioned foams. Largest decrease evident with foams conditioned at 250 deg. F. A slight increase in heat capacity for humid samples. Similar trends in heat capacities at 71 deg. C and 102 deg. C. TGA: As conditioning temperature increased, a decline was evident in the average percent weight loss, onset temperature, and first derivative peak temperature. No significant effect due to humidity except for a decrease in first derivative peak temperature. Unaged samples exhibited strong intumescence. Foams conditioned at 160 deg. F exhibited slight intumescence. Results suggest fire protective additives decomposed at higher temperatures, which contributed to a lower average percent mass loss in higher temperature conditioning. Further thermal degradation testing would be beneficial to evaluate the ability of the conditioned foams to intumesce. Evaluate irradiation dose rate effects on conditioned foam. Possible conditioning in an environment with a known humidity or analyzing a new aliquot of sample for each trial to achieve a more representative heat capacity calculation.

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗
DOE OSTI2021

Tbx18-positive cells-derived myofibroblasts contribute to renal interstitial fibrosis via transforming growth factor-β signaling

It has been demonstrated that the T-box family transcription factor 18 (Tbx18) -positive cells give rise to renal mesenchymal cells and contribute to the development of the urinary system. However, it is unclear whether Tbx18-positive cells are the origin of the myofibroblasts during renal fibrosis. The present study aimed to determine the contribution of Tbx18-positive cells in kidney fibrosis and their underlying mechanism. We show that Tbx18-positive cells contribute to the development of the urinary system, especially renal fibroblasts. Following unilateral ureteral obstruction (UUO), genetic fate tracing results demonstrated that Tbx18-positive cells not only proliferate but also expand and differentiate into fibroblasts and myofibroblasts, indicating that they may act as profibrotic progenitors. Cell culture results suggest that transforming growth factor (TGF)-β promotes Tbx18-positive cells differentiation into myofibroblasts and assist their contribution to kidney fibrosis. Overall, the present study demonstrated that Tbx18-positive cells may act as profibrotic progenitor cells in a pathological condition of UUO-induced injury. Moreover, TGF-β may play a role in differentiation of Tbx18-positive cells into myofibroblasts in kidney fibrosis. These findings may provide a potential target on Tbx18-positive myofibroblast progenitors in the treatment of renal fibrosis.

60 APPLIED LIFE SCIENCES↗
NASA NTRS1992

Advanced satellite communication system

The objective of this research program was to develop an innovative advanced satellite receiver/demodulator utilizing surface acoustic wave (SAW) chirp transform processor and coherent BPSK demodulation. The algorithm of this SAW chirp Fourier transformer is of the Convolve - Multiply - Convolve (CMC) type, utilizing off-the-shelf reflective array compressor (RAC) chirp filters. This satellite receiver, if fully developed, was intended to be used as an on-board multichannel communications repeater. The Advanced Communications Receiver consists of four units: (1) CMC processor, (2) single sideband modulator, (3) demodulator, and (4) chirp waveform generator and individual channel processors. The input signal is composed of multiple user transmission frequencies operating independently from remotely located ground terminals. This signal is Fourier transformed by the CMC Processor into a unique time slot for each user frequency. The CMC processor is driven by a waveform generator through a single sideband (SSB) modulator. The output of the coherent demodulator is composed of positive and negative pulses, which are the envelopes of the chirp transform processor output. These pulses correspond to the data symbols. Following the demodulator, a logic circuit reconstructs the pulses into data, which are subsequently differentially decoded to form the transmitted data. The coherent demodulation and detection of BPSK signals derived from a CMC chirp transform processor were experimentally demonstrated and bit error rate (BER) testing was performed. To assess the feasibility of such advanced receiver, the results were compared with the theoretical analysis and plotted for an average BER as a function of signal-to-noise ratio. Another goal of this SBIR program was the development of a commercial product. The commercial product developed was an arbitrary waveform generator. The successful sales have begun with the delivery of the first arbitrary waveform generator.

Staples, Edward J.↗
DOE OSTI2023

Infrared Signatures for Phase Identification in Hafnium Oxide Thin Films

Phase identification in HfO 2 -based thin films is a prerequisite to understanding the mechanisms stabilizing the ferroelectric phase in these materials, which hold great promise in next-generation non-volatile memory and computing technology. While grazing-incidence X-ray diffraction is commonly employed for this purpose, it has difficulty unambiguously differentiating between the ferroelectric phase and other metastable phases that may exist due to similarities in the d-spacings, their low intensities, and the overlapping of reflections. Infrared signatures provide an alternative route. However, their use in phase identification remains limited because phase control has overwhelmingly been accomplished via substituents, thereby convoluting infrared signatures between the substituent and the phase changes they induce. Herein, we report the infrared optical responses of three undoped hafnium oxide films where annealing conditions have been used to create films consisting primarily of the ferroelectric polar orthorhombic Pca2 1 , antipolar orthorhombic Pbca, and monoclinic P2 1 /c phases, as was confirmed via transmission electron microscopy (TEM), UVvisible optical properties, and electrical property measurements. Vibrational signatures acquired from synchrotron nano-Fourier transform infrared spectroscopy (nano-FTIR) are shown to be capable of differentiating between the phases in a non-destructive, rapid, and nanoscale manner. The utility of nano-FTIR is illustrated for a film exhibiting an antiferroelectric polarization response. In this sample, it is proven that this behavior results from the Pbca phase rather than the often-cited tetragonal phase. Here, by demonstrating that IRspectroscopy can unambiguously distinguish phases in this material, this work establishes a tool needed to isolate the factors dictating ferroelectric phase stability in HfO 2 -based materials.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
NASA NTRS2000

Polar Plate Theory for Orthogonal Anisotropy

Laminated fiber-reinforced (or filamentary) composites are used today for their high strength-to-weight and stiffness-to-weight ratios. However, because of the anisotropic behavior of composites, determining the response on a macroscopic scale is challenging. This is particularly evident in the evaluation of the governing differential equations of a circular disk with the fibers of the lamina oriented with rectilinear orthogonality. This includes any situation involving a composite plate of circular geometry in which out-of-plane displacements due to load are desired, such as fastener pull through loading of a composite plate. Current analysis techniques use numerical methods with rectilinear coordinate systems to solve problems with circular geometry. These analyses over predict plate stiffness by 20% and underpredict failure by 70%. Consequently, there is a need to transform classical composite plate theory to a polar coordinate system. In order to better analyze structures with circular geometries the classical composite plate equations are transformed into the plate equations for a rectilinearly anisotropic composite in polar coordinates. A composite plate is typically a laminate of fibers in rectilinear directions. Subsequent to the lay-tip the necessary geometry is cut out of a rectangular plate. In a similar manner, the derivation of the plate equation starts with the fundamental definitions of strain, displacement and curvature and incorporates the material property angular dependence into the equilibrium equations for a differential polar element. In the transformed state, the stiffness coefficients are no longer constant, adding to the complexity of the governing differential equations. This paper discusses the new derivation and evaluation of the plate equations for a circular composite disk with orthogonal rectilinear anisotropy. The resultant new three partial differential equations, which describe the circular anisotropic plate, can be used to evaluate out-of-plane displacements for given load conditions in design of composite Structures. Without the formation of these unique plate equations, design of such structures is not a precise engineering accomplishment due to the lack of precise design tools. Such structures are over designed to compensate and costly tests need to be performed after production to validate safety. Though impossible to present the mathematical solution within the limits of this paper, the application of the newly derived plate equation in its orthotropic form for a balanced symmetric laminate compared favorably with test results and surpassed the results of standard numerical methods. Within the limitations of the problem foundation the predictive model provides a mechanism for establishing out-of-plane deflection levels for a circular composite plate. Future work in this area should include incorporating the circular composite plate equations in Finite Element Models. The new anisotropic polar plate equations call be utilized as a design tool so that the design more accurately meets the requirements, thus taking full advantage of the weight savings of composites. These accomplishments have not previously been presented to the engineering community nor ever proposed to the IAF. Should the proposed paper be accepted for presentation, attendance is assured.

Bailey, Michelle D.↗
DOE OSTI2021

Improvements of pre-emptive identification of particle accelerator failures using binary classifiers and dimensionality reduction

In this paper we look at the properties of the Spallation Neutron Source (SNS) Differential Beam Current Monitor (DCM) data and various methods of data transformation to improve pre-emptive detection of machine trips. In this study, the foundation of the approach is the analysis of new underlying data and understanding various properties with the goal of faster classification, higher precision and higher recall with the aim to reduce false positives as low as required. The result of the research presented in this paper are a binary classifier capable of predicting accelerator failures with millisecond classification time, 96% precision, 58% true positive and 0% false positive rate and optimization techniques enabling real-time implementations.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2022

Multimodal imaging of hemorrhagic transformation biomarkers in an ischemic stroke model.

Abstract Hemorrhagic transformation of ischemic stroke has devastating consequences, with high mortality and poor functional outcomes. Animal models of ischemic stroke also demonstrate the potential for hemorrhagic transformation, which complicates biochemical characterization, treatment studies, and hinders poststroke functional outcomes in affected subjects. The incidence of hemorrhagic transformation of ischemic stroke in animal model research is not commonly reported. The postmortem brain of such cases presents a complex milieu of biomarkers due to the presence of healthy cells, regions of varying degrees of ischemia, dead and dying cells, dysregulated metabolites, and blood components (especially reactive Fe species released from lysed erythrocytes). To improve the characterization of hemorrhage biomarkers on an ischemic stroke background, we have employed a combination of histology, X-ray fluorescence imaging (XFI), and Fourier transform infrared (FTIR) spectroscopic imaging to assess 122 photothrombotic (ischemic) stroke brains. Rapid freezing preserves brain biomarkers in situ and minimizes metabolic artifacts due to postmortem ischemia. Analysis revealed that 25% of the photothrombotic models had clear signs of hemorrhagic transformation. The XFI and FTIR metabolites provided a quantitative method to differentiate key metabolic regions in these models. Across all hemorrhage cases, it was possible to consistently differentiate otherwise healthy tissue from other metabolically distinct regions, including the ischemic infarct, the ischemic penumbra, blood vessels, sites of hemorrhage, and a region surrounding the hemorrhage core that contained elevated lipid oxidation. Chemical speciation of deposited Fe demonstrates the presence of heme-Fe and accumulation of ferritin.

Pushie, M. J. (ORCID:0000000174945427)↗
DOE OSTI2021

A Comparison of Emerging Nonfluoropolymer-Based Coextruded PV Backsheets to Industry-Benchmark Technologies

As the photovoltaic (PV) industry is rapidly expanding around the world, there has been an increasing interest in extending the lifespan of PV modules. Concern has also emerged regarding the recyclability of modules and their component materials, including fluoropolymer-based backsheets. Laminated polyethylene-terephthalate (PET) core backsheets have traditionally been used in the PV industry, but new, coextruded polyolefin (PO) backsheets show promise as an improved alternative. In this work, minimodule and coupon samples of seven different backsheets (made of layers including contemporary PET and fluoropolymers, novel PO, and polyamide materials) were run through hygrometric- or UV photolytic-accelerated aging to identify and better understand each material's degradation modes and the backsheets' field reliability. In addition to the artificial aging, the natural weathering methods used in this article are described. The comprehensive set of chemical, mechanical, and structural characterizations at intermittent read points in this article is presented, including: visual appearance and color; gloss; mechanical tensile testing; I-V performance; electroluminescence (EL) imaging; dielectric breakdown; Fourier-transform infrared-chemical structure; X-ray-polymer structure; and differential scanning calorimetry-crystalline content. After 4000 h of aging, a strong correlation occurs between initial physical characteristics (mechanical tensile test) and operating performance (EL and I-V characteristics).

14 SOLAR ENERGY↗
DOE OSTI2020

Microstructure and Mechanical Properties of Intercritically Treated Grade 91 Steel

Premature creep failures at the intercritical heat affected zone (ICHAZ) of creep-resistant steel weldments have been frequently reported. However, the creep degradation mechanism of different microstructure constituents in ICHAZ is complicated and needs further clarification. In this work, Grade 91 steel was intercritically heat-treated at a temperature (860 °C) between the critical temperatures AC1 and AC3, and a correlation between microstructure and mechanical properties of the heat-treated specimen was built. The effects of austenitization and tempering resulting from the intercritical treatment (IT) differentiated the local strain energies between the two microstructure constituents: newly transformed martensite (NTM) and over-tempered martensite (OTM). The formation of NTM grains led to a hardness increase from 247 HV0.5 in the base metal to 332 HV0.5 in the IT specimen. The ultimate tensile strength (UTS) increased from 739 MPa in the base metal to 1054 MPa in the IT specimen. Extensive growth of the OTM grains and rapid recovery of NTM grains took place simultaneously in the IT specimen during a typical tempering at 760 °C. These microstructure degradations led to a lowered hardness of 178 HV0.5, a reduced UTS of 596 MPa, and a poor creep resistance with a minimum creep strain rate of 0.49 %/h at 650 °C in an IT + tempering (ITT) specimen.

36 MATERIALS SCIENCE↗
NASA NTRS1970

Discrete control of linear distributed systems with application to the deformable primary mirror of a large orbiting telescope

One of the more significant technological problems associated with the orbital operation of large astronomical telescope's is the fabrication and maintenance of the primary mirror surface to the tolerance required for diffraction-limited performance. An interesting approach to the solution of this problem involves continuously measuring and automatically correcting the optical surface of a thin deformable mirror by means of discrete actuators located on its rear surface: The realization of diffraction-limited performance from a telescope in space by this method rests on the ability of the designer to achieve extremely accurate control of a highly complex, interacting, multivariable system. This paper presents the results of a detailed study of the discrete control of linear distributed systems with specific application to the design of a practical controller for a plant representative of a telescope primary mirror for an orbiting astronomical observatory. The problem of controlling the distributed plant is treated by employing modal techniques to represent variations in the optical figure. Distortion of the mirror surface, which arises primarily from thermal gradients, is countered by actuators working against a backing structure to apply a corrective force distribution to the controlled surface. Each displacement actuator is in series with a spring attached to the mirror by means of a pad intentionally introduced to restrict the excitation of high-order modes. Control is then exerted over a finite number (equal.to the number of actuators) of the most significant modes. Through the application of the modal expansion technique the mirror equation of motion is transformed tb a set of uncoupled, linear, time-invariant, ordinary differential equations. The desired dynamic response and static accuracy may then be achieved by the application of classical single-variable design techniques. The formulation of a quadratic performance index which incorporates a measure of image quality permits determination of the trade-off between the-number of actuators and optical purity. A criterion for defining actuator placement and pad size is presented which minimizes the tendency of the controller to excite the unmonitored modes.

Jeremiah F Creedon↗
NASA NTRS1972

Longitudinal dynamic stability of a shuttle vehicle.

Analytical study of the longitudinal dynamic stability of a nonrolling, lifting vehicle gliding at hypersonic speeds. The analysis applies to shuttle vehicles designed for operating up to the rim of a planetary atmosphere. A general nondimensional time transformation is introduced to derive a unified second-order linear differential equation for the angle of attack, valid for all types of reentry of a general type of vehicle. The stability of motion is discussed for two fundamental regimes of flight that are based on widely different assumptions. For near ballistic entry along a straight line trajectory, the equation reduces to a confluent hypergeometric equation, the solution of which can be expressed in terms of Whittaker's function. Using a theorem in the theory of stability of differential equations, criteria for damped oscillations are derived. It is shown that the aerodynamic criteria for stability are the same as for the case of ballistic entry. In addition, for each vehicle configuration, and specified planetary atmosphere, there exists an altitude range where the angle of attack frequency is nearly equal to the orbital frequency causing instability in pitch. This resonance instability is due to the ellipticity of the orbit. Criteria for eccentricity instability are derived.

Vinh, N. X.↗