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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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A method is developed for determining the stability margins of a simple Jeffcott model with deadband via analysis of the discrete Fourier transform of the system response. The model in question is of a uniform, unbalanced, flexible shaft that is supported by a bearing as it rotates about its x axis. This model is represented by a system of coupled nonlinear differential equations.
The problem of determining the acoustic field in an inviscid, isentropic fluid generated by a solid body whose surface executes prescribed vibrations is formulated and solved as a multiple scales perturbation problem, using the Mach number M based on the maximum surface velocity as the perturbation parameter. Following the idea of multiple scales, new 'slow' spacial scales are introduced, which are defined as the usual physical spacial scale multiplied by powers of M. The governing nonlinear differential equations lead to a sequence of linear problems for the perturbation coefficient functions. However, it is shown that the higher order perturbation functions obtained in this manner will dominate the lower order solutions unless their dependence on the slow spacial scales is chosen in a certain manner. In particular, it is shown that the perturbation functions must satisfy an equation similar to Burgers' equation, with a slow spacial scale playing the role of the time-like variable. The method is illustrated by a simple one-dimenstional example, as well as by three different cases of a vibrating sphere. The results are compared with solutions obtained by purely numerical methods and some insights provided by the perturbation approach are discussed.
The dynamics of radially expanding magnetic clouds is rigorously analyzed within the framework of ideal MHD. The cloud is modelled as a cylindrically symmetric magnetic flux rope. In the force balance we include the gas pressure gradient and the Lorentz force. Interaction with the ambient solar wind due to expansion of the magnetic cloud is represented by a drag force proportional to the bulk velocity. We consider the self-similar expansion of a polytrope, and reduce the problem to an ordinary nonlinear differential equation for the evolution function. Analyzing the asymptotic behavior of the evolution function, we formulate theoretical expectations for the long-term behavior of cloud parameters. We focus on the temporal evolution of (1) the magnetic field strength; (2) the twist of the field lines; (3) the asymmetry of the total field profile; and (4) the bulk flow speed. We present data from two magnetic clouds observed at 1 AU and 2 AU, respectively, and find good agreement with theoretical expectations. For a peak magnetic field strength at 1 AU of 25 nT and a polytropic index of 0.5, we find that a magnetic cloud can be distinguished from the background interplanetary field up to a distance of about 5 AU. Taking larger magnetic fields and bigger polytropic indices this distance can double.
A general method of analysis is developed for two-dimensional, steady, compressible flow in stators or rotors of radial and mixed flow turbomachines with conic flow surfaces (surfaces of right circular cones generated by center line of flow passage in the axial-radial plane). The variables taken into account are: (1) tip speed of the rotor, (2) flow rate, (3) blade shape, (4) variation in passage height with radius, (5) number of blades, and (6) cone angle of the flow surface. Relaxation methods are used to solve the nonlinear differential equation for the stream function. Two numerical examples are presented; one for compressible and the other for incompressible flow in a centrifugal compressor with thin, straight blades. The results of these examples are given by plots of the streamlines, constant velocity-ratio lines, and constant pressure-ratio line.
This report is an extension of a previous investigation (described in NACA rep. 1069) concerned with the solution of the nonlinear differential equation for transonic flow past a wavy wall. In the present work several new notions are introduced which permit the solution of the recursion formulas arising from the method of integration in series. In addition, a novel numerical tests of convergence, applied to the power series (in transonic similarity parameter) representing the local Mach number distribution at the boundary, indicates that smooth symmetrical potential flow past the wavy wall is no longer possible once the critical value of the stream Mach number has been exceeded.
Recently, system theorists have recognized that low order systems of nonlinear differential equations can give rise to solutions which are neither periodic, constant, nor predictable in steady state, but which are nonetheless bounded and deterministic. This behavior, which was first described in the study of weather systems, has been termed 'chaotic.' Much study of chaotic systems has concentrated on analysis of the systems' phase space attractors. It has been recognized that invariant measures of the attractor possess inherent information about the system. One such measure is the dimension of the attractors. The dimension of a chaotic attractor has been shown to be noninteger, leading to the term 'strange attractor;' the attractor is said to have a fractal structure. The correlation dimension has become one of the most popular measures of dimension. However, many problems have been identified in correlation dimension estimation from time sequences. The most common methods for obtaining the correlation dimension have been least squares curves fitting to find the slope of the correlation integral and the Takens Estimator. However, these estimates show unacceptable sensitivity to the upper limit on the distance chosen. Here, a new method is proposed which is shown to be rather insensitive to the upper limit and to perform in a very stable manner, at least in the absence of noise. The correlation dimension is also shown to be an effective discriminant in distinguishing between radar returns resulting from weather and those from the ground. The weather returns are shown to have a correlation dimension generally between 2.0 and 3.0, while ground returns have a correlation dimension exceeding 3.0.
This report presents the Space Station Furnace Facility (SSFF) thermal control system (TCS) preliminary control system design and analysis. The SSFF provides the necessary core systems to operate various materials processing furnaces. The TCS is defined as one of the core systems, and its function is to collect excess heat from furnaces and to provide precise cold temperature control of components and of certain furnace zones. Physical interconnection of parallel thermal control subsystems through a common pump implies the description of the TCS by coupled nonlinear differential equations in pressure and flow. This report formulates the system equations and develops the controllers that cause the interconnected subsystems to satisfy flow rate tracking requirements. Extensive digital simulation results are presented to show the flow rate tracking performance.
Paper discusses reliability of time-dependent approach to numerical solution of nonlinear differential equations (DEs) that describe steady-state behaviors of physical systems. Time-dependent approach followed in related study described in "Spurious Numerical Solutions of Differential Equations" (ARC-13209).
This paper presents an algorithm for estimating the angular rate vector of a satellite which is based on the time derivatives of vector measurements expressed in a reference and body coordinate. The computed derivatives are fed into a spacial Kalman filter which yields an estimate of the spacecraft angular velocity. The filter, named Extended Interlaced Kalman Filter (EIKF), is an extension of the Kalman filter which, although being linear, estimates the state of a nonlinear dynamic system. It consists of two or three parallel Kalman filters whose individual estimates are fed to one another and are considered as known inputs by the other parallel filter(s). The nonlinear dynamics stem from the nonlinear differential equation that describes the rotation of a three dimensional body. Initial results, using simulated data, and real Rossi X ray Timing Explorer (RXTE) data indicate that the algorithm is efficient and robust.
The dimensionless, transformed, nonlinear differential equation developed in NASA TR R-11 for describing the approximate motion and heating during entry into planetary atmospheres for constant aerodynamic coefficients and vehicle shape has been modified to include entries during which the aerodynamic coefficients and the vehicle shape are varied. The generality of the application of the original equation to vehicles of arbitrary weight, size, and shape and to arbitrary atmospheres is retained. A closed-form solution for the motion, heating, and the variation of drag loading parameter m/C(D)A has been obtained for the case of constant maximum resultant deceleration during nonlifting entries. This solution requires certain simplifying assumptions which do not compromise the accuracy of the results. The closed-form solution has been used to determine the variation of m/C(D)A required to reduce peak decelerations and to broaden the corridor for nonlifting entry into the earth's atmosphere at escape velocity. The attendant heating penalty is also studied.
The focus of our work has been on developing an insight into the physics that govern the optimum design of thermal insulation for use in thermal protection systems of launch vehicle. Of particular interest was to obtain optimality criteria for designing foam insulations that have density (or porosity) distributions through the thickness for optimum thermal performance. We investigate the optimum design of functionally graded thermal insulation for steady state heat transfer through the foam. We showed that the heat transfer in the foam has competing modes, of radiation and conduction. The problem assumed a fixed inside temperature of 400 K and varied the aerodynamic surface heating on the outside surface from 0.2 to 1.0 MW/sq m. The thermal insulation develops a high temperature gradient through the thickness. Investigation of the model developed for heat conduction in foams showed that at high temperatures (as on outside wall) intracellular radiation dominates the heat transfer in the foam. Minimizing radiation requires reducing the pore size, which increases the density of the foam. At low temperatures (as on the inside wall), intracellular conduction (of the metal and air) dominates the heat transfer. Minimizing conduction requires increasing the pore size. This indicated that for every temperature there was an optimum value of density that minimized the heat transfer coefficient. Two optimization studies were performed. One was to minimize the heat transmitted though a fixed thickness insulation by varying density profiles. The second was to obtain the minimum mass insulation for specified thickness. Analytical optimality criteria were derived for the cases considered. The optimality condition for minimum heat transfer required that at each temperature we find the density that minimizes the heat transfer coefficient. Once a relationship between the optimum heat transfer coefficient and the temperature was found, the design problem reduced to the solution of a simple nonlinear differential equation. Preliminary results of this work were presented at the American Society of Composites meeting, and the final version was submitted for publication in the AIAA Journal. In addition to minimizing the transmitted heat, we investigated the optimum design for minimum weight given an acceptable level of heat transmission through the insulation. The optimality criterion developed was different from that obtained for minimizing beat transfer coefficient. For minimum mass design, we had to find for a given temperature the optimum density, which minimized the logarithmic derivative of the insulation thermal conductivity with respect to its density. The logarithmic derivative is defined as the ratio of relative change in the dependent response (thermal conductivity) to the relative change in the independent variable (density). The results have been documented as a conference paper that will be presented at the upcoming AIAA.
We have undertaken the theoretical modelling of an existing liquid hydrogen (LH2) and gas hydrogen (GH2) mixer subsystem of the E1 Ground Test Facility at NASA John C. Stennis Space Center. The E1 test facility carries out comprehensive ground-based testing and certification of various liquid rocket engines and their components. The mixer described in this work is responsible for combining high pressure LH2 and GH2 to produce a hydrogen flow that meets certain thermodynamic properties before it is fed into a test article. The desired properties are maintained by precise control of the mixture of LH2 and GH2 flows. The mixer is modelled as a general multi-flow lumped volume for single constituent fluids using density and internal energy as states. The set of nonlinear differential equations is linearized about an equilibrium point and the resulting two-state, 3-input linear model is analyzed as a possible candidate for control design.
We have undertaken the theoretical modelling of an existing liquid hydrogen (LH2) and gas hydrogen (GH2) mixer subsystem of the E1 Ground Test Facility at NASA John C. Stennis Space Center. The E1 test facility carries out comprehensive ground-based testing and certification of various liquid rocket engines and their components. The mixer described in this work is responsible for combining high pressure before it is fed into a test article. The desired properties are maintained by precise control of the mixture of LH2 and GH2 flows. The mixer is modelled as a general multi-flow lumped volume for single constituent fluids using density and internal energy as states. The set of nonlinear differential equations is linearized about an equilibrium point and the resulting two-state, 3-input linear model is analyzed as a possible candidate for control design.
A novel approach is presented in this paper to analyze attitude precision and control for an instrument gimbaled to a spacecraft subject to an internal disturbance caused by a moving component inside the instrument. Nonlinear differential equations of motion for some sample cases are derived and solved analytically to gain insight into the influence of the disturbance on the attitude pointing error. A simple control law is developed to eliminate the instrument pointing error caused by the internal disturbance. Several cases are presented to demonstrate and verify the concept presented in this paper.
Mutually injection-locked arrays of electronic oscillators provide a novel means of controlling the aperture phase of a phased-array antenna, thus achieving the advantages of spatial power combining while retaining the ability to steer the radiated beam. In a number of design concepts, one or more of the oscillators are injection locked to a signal from an external master-oscillator. The behavior of such a system has been analyzed by numerical solution of a system of nonlinear differential equations which, due to its complexity, yields limited insight into the relationship between the injection signals and the aperture phase. In this paper, we develop a continuum model, which results in a single partial differential equation for the aperture phase as a function of time. Solution of the equation is effected by means of the Laplace transformation and yields detailed information concerning the dynamics of the array under the influence of the external injection signals.
The application of capillary screen liquid acquisition devices to space-based cryogenic propulsion systems is expected to necessitate thermodynamic conditioning in order to stabilize surface tension retention characteristics. The present results have been obtained in the framework of the research of low gravity condensation-flow processes for conditioning cryogenic liquid acquisition devices. The following system is studied: On the top of a subcooled horizontal disk, a liquid film condenses from the ambient saturated vapor. The liquid is forcedly removed at the disk edge, and there is an outward radial flow of the film. Stationary regimes of the flow are uncovered such that (i) the gravity is negligible, being eclipsed by the capillary forces; (ii) the film thickness is everywhere much smaller than the disk radius; and (iii) the slow-flow lubrication approximation is valid. A nonlinear differential equation for the film thickness as a function of the radial coordinate is obtained. The (two-dimensional) fields of velocities, temperature and pressure in the film are explicitly determined by the radial profile of its thickness. The equilibrium is controlled by two parameters: (i) the vapor-disk difference of temperatures and (ii) the liquid exhaust rate. For the flow regimes with a nearly uniform film thickness, the governing equation linearizes, and the film interface is analytically predicted to have a concave-up quartic parabola profile. Thus, perhaps counter-intuitively, the liquid film is thicker at the edge and thinner at the center of the disk.
A comprehensive analytical model of the interaction of the cantilever tip of the atomic force microscope (AFM) with the sample surface is developed that accounts for the nonlinearity of the tip-surface interaction force. The interaction is modeled as a nonlinear spring coupled at opposite ends to linear springs representing cantilever and sample surface oscillators. The model leads to a pair of coupled nonlinear differential equations that are solved analytically using a standard iteration procedure. Solutions are obtained for the phase and amplitude signals generated by various acoustic-atomic force microscope (A-AFM) techniques including force modulation microscopy, atomic force acoustic microscopy, ultrasonic force microscopy, heterodyne force microscopy, resonant difference-frequency atomic force ultrasonic microscopy (RDF-AFUM), and the commonly used intermittent contact mode (TappingMode) generally available on AFMs. The solutions are used to obtain a quantitative measure of image contrast resulting from variations in the Young modulus of the sample for the amplitude and phase images generated by the A-AFM techniques. Application of the model to RDF-AFUM and intermittent soft contact phase images of LaRC-cp2 polyimide polymer is discussed. The model predicts variations in the Young modulus of the material of 24 percent from the RDF-AFUM image and 18 percent from the intermittent soft contact image. Both predictions are in good agreement with the literature value of 21 percent obtained from independent, macroscopic measurements of sheet polymer material.
Fluid systems, or networks, consist of multiple components that work together to achieve some desired thermofluid state. For any generic application, this fluid state can be a combination of the fluid pressure, flow rate, enthalpy, or species concentration. Fluid system components, such as pumps and valves, are often governed by nonlinear differential equations, resulting in complex component-to-component interactions. System-level fluid network stability occurs when the flow through the system can maintain a steady-state solution in the presence of small perturbations, which depends on these component interactions. System instability, however, can go undetected until issues arise during integrated system testing. This presentation explores a method for system designers to think of the fluid network as an assembly of components, each with their own thermofluid surfaces of partial stability, called nullclines. The intersections of all nullclines yields system-level solutions, called equilibrium points. When designers define operating points, they are tuning system parameters so that these equilibrium points move to the desired location in the thermofluid state plane. However, linearization theory shows us that the dynamic behavior around these points can be unstable. The local stability of these equilibrium points can be assessed analytically with eigen-analysis, or numerically by propagating state-plane samples to construct a phase portrait. Investigating a phase portrait can help designers gain a qualitative understanding of a system’s dynamic performance. This understanding can then help inform requirement definitions, component selection, and operational procedures. This presentation includes an example of the phase portrait technique on a system featuring a centrifugal pump and a back-pressure regulator (BPR). Numerical modeling of this system suggests that equilibrium points on the left-hand side of the pump curve are dynamically unstable.