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NASA NTRS2006

Multiple concurrent recursive least squares identification with application to on-line spacecraft mass-property identification

The present invention is a method for identifying unknown parameters in a system having a set of governing equations describing its behavior that cannot be put into regression form with the unknown parameters linearly represented. In this method, the vector of unknown parameters is segmented into a plurality of groups where each individual group of unknown parameters may be isolated linearly by manipulation of said equations. Multiple concurrent and independent recursive least squares identification of each said group run, treating other unknown parameters appearing in their regression equation as if they were known perfectly, with said values provided by recursive least squares estimation from the other groups, thereby enabling the use of fast, compact, efficient linear algorithms to solve problems that would otherwise require nonlinear solution approaches. This invention is presented with application to identification of mass and thruster properties for a thruster-controlled spacecraft.

Wilson, Edward↗

NASA NTRS2006

Demonstration of a Balloon Borne Arc-Second Pointer Design

Many designs for utilizing stratospheric balloons as low-cost platforms on which to conduct space science experiments have been proposed throughout the years. A major hurdle in extending the range of experiments for which these vehicles are useful has been the imposition of the gondola dynamics on the accuracy with which an instrument can be kept pointed at a celestial target. A significant number of scientists have sought the ability to point their instruments with jitter in the arc-second range. This paper presents the design and analysis of a stratospheric balloon borne pointing system that is able to meet this requirement. The test results of a demonstration prototype of the design with similar ability are also presented. Discussion of a high fidelity controller simulation for design analysis is presented. The flexibility of the flight train is represented through generalized modal analysis. A multiple controller scheme is utilized for coarse and fine pointing. Coarse azimuth pointing is accomplished by an established pointing system, with extensive flight history, residing above the gondola structure. A pitch-yaw gimbal mount is used for fine pointing, providing orthogonal axes when nominally on target. Fine pointing actuation is from direct drive dc motors, eliminating backlash problems. An analysis of friction nonlinearities and a demonstration of the necessity in eliminating static friction are provided. A unique bearing hub design is introduced that eliminates static friction from the system dynamics. A control scheme involving linear accelerometers for enhanced disturbance rejection is also presented. Results from a linear analysis of the total system and the high fidelity simulation are given. Results from a generalized demonstration prototype are presented. Commercial off-the-shelf (COTS) hardware was used to demonstrate the efficacy and performance of the pointer design for a mock instrument. Sub-arcsecond pointing ability from a ground hang test setup is shown from the testing results. This paper establishes that the proposed control strategy can be made robustly stable with significant design margins. Also demonstrated is the efficacy of the proposed system in rejecting disturbances larger than those considered realistic. The system is implemented and demonstrates sub arc second pointing ability using COTS hardware. Finally, we see that sub arc-second pointing stability can be achieved for a large instrument pointing at an inertial target.

DeWeese, Keith D.↗

NASA NTRS2006

Pattern-Recognition Algorithm for Locking Laser Frequency

A computer program serves as part of a feedback control system that locks the frequency of a laser to one of the spectral peaks of cesium atoms in an optical absorption cell. The system analyzes a saturation absorption spectrum to find a target peak and commands a laser-frequency-control circuit to minimize an error signal representing the difference between the laser frequency and the target peak. The program implements an algorithm consisting of the following steps: Acquire a saturation absorption signal while scanning the laser through the frequency range of interest. Condition the signal by use of convolution filtering. Detect peaks. Match the peaks in the signal to a pattern of known spectral peaks by use of a pattern-recognition algorithm. Add missing peaks. Tune the laser to the desired peak and thereafter lock onto this peak. Finding and locking onto the desired peak is a challenging problem, given that the saturation absorption signal includes noise and other spurious signal components; the problem is further complicated by nonlinearity and shifting of the voltage-to-frequency correspondence. The pattern-recognition algorithm, which is based on Hausdorff distance, is what enables the program to meet these challenges.

Karayan, Vahag↗

NASA NTRS2013

Sharp Eccentric Rings in Planetless Hydrodynamical Models of Debris Disks

Exoplanets are often associated with disks of dust and debris, analogs of the Kuiper Belt in our solar system. These "debris disks" show a variety of non-trivial structures attributed to planetary perturbations and utilized to constrain the properties of the planets. However, analyses of these systems have largely ignored the fact that, increasingly, debris disks are found to contain small quantities of gas, a component all debris disks should contain at some level. Several debris disks have been measured with a dust-to-gas ratio around unity where the effect of hydrodynamics on the structure of the disk cannot be ignored. Here we report that dust-gas interactions can produce some of the key patterns seen in debris disks that were previously attributed to planets. Through linear and nonlinear modeling of the hydrodynamical problem, we find that a robust clumping instability exists in this configuration, organizing the dust into narrow, eccentric rings, similar to the Fomalhaut debris disk. The hypothesis that these disks might contain planets, though thrilling, is not necessarily required to explain these systems.

debris disks↗

NASA NTRS1984

Transient Response Dynamic Module Modifications to Include Static and Kinetic Friction Effects

A methodology that supports forced transient response dynamic solutions when both static and kinetic friction effects are included in a structural system model is described. Modifications that support this type of nonlinear transient response solution are summarized for the transient response dynamics (TRD) NASTRAN module. An overview of specific modifications for the NASTRAN processing subroutines, INITL, TRD1C, and TRD1D, are described with further details regarding inspection of nonlinear input definitions to define the type of nonlinear solution required, along with additional initialization requirements and specific calculation subroutines to successfully solve the transient response problem. The extension of the basic NASTRAN nonlinear methodology is presented through several stages of development to the point where constraint equations and residual flexibility effects are introduced into the finite difference Newmark-Beta recurrsion formulas. Particular emphasis is placed on cost effective solutions for large finite element models such as the Space Shuttle with friction degrees of freedom between the orbiter and payloads mounted in the cargo bay. An alteration to the dynamic finite difference equations of motion is discussed, which allows one to include friction effects at reasonable cost for large structural systems such as the Space Shuttle. Data are presented to indicate the possible impact of transient friction loads to the payload designer for the Space Shuttle. Transient response solution data are also included, which compare solutions without friction forces and those with friction forces for payloads mounted in the Space Shuttle cargo bay. These data indicate that payload components can be sensitive to friction induced loads.

Misel, J. E.↗

NASA NTRS1991

Robust stabilization of the Space Station

A robust H-infinity control design methodology and its application to a Space Station Freedom (SSF) attitude and momentum control problem are presented. This approach incorporates nonlinear multi-parameter variations in the state-space formulation of H-infinity control theory. An application of this robust H-infinity control synthesis technique to the SSF control problem yields remarkable results in stability robustness with respect to moments of inertia variation of about 73 percent in one of the structured uncertainty directions. The performance and stability of this robust H-infinity controller for the SSF are compared to those of other controllers designed using a standard linear-quadratic-regulator synthesis technique.

Wie, Bong↗

NASA NTRS1992

Multi-dimensional high order essentially non-oscillatory finite difference methods in generalized coordinates

The nonlinear stability of compact schemes for shock calculations is investigated. In recent years compact schemes were used in various numerical simulations including direct numerical simulation of turbulence. However to apply them to problems containing shocks, one has to resolve the problem of spurious numerical oscillation and nonlinear instability. A framework to apply nonlinear limiting to a local mean is introduced. The resulting scheme can be proven total variation (1D) or maximum norm (multi D) stable and produces nice numerical results in the test cases. The result is summarized in the preprint entitled 'Nonlinearly Stable Compact Schemes for Shock Calculations', which was submitted to SIAM Journal on Numerical Analysis. Research was continued on issues related to two and three dimensional essentially non-oscillatory (ENO) schemes. The main research topics include: parallel implementation of ENO schemes on Connection Machines; boundary conditions; shock interaction with hydrogen bubbles, a preparation for the full combustion simulation; and direct numerical simulation of compressible sheared turbulence.

Shu, Chi-Wang↗

NASA NTRS1993

Using emergent order to shape a space society

A fast-growing movement in the scientific community is reshaping the way that we view the world around us. The short-hand name for this movement is 'chaos'. Chaos is a science of the global, nonlinear nature of systems. The center of this set of ideas is that simple, deterministic systems can breed complexity. Systems as complex as the human body, ecology, the mind or a human society. While it is true that simple laws can breed complexity, the other side is that complex systems can breed order. It is the latter that I will focus on in this paper. In the past, nonlinear was nearly synonymous with unsolvable because no general analytic solutions exist. Mathematically, an essential difference exists between linear and nonlinear systems. For linear systems, you just break up the complicated system into many simple pieces and patch together the separated solutions for each piece to form a solution to the full problem. In contrast, solutions to a nonlinear system cannot be added to form a new solution. The system must be treated in its full complexity. While it is true that no general analytical approach exists for reducing a complex system such as a society, it can be modeled. The technical involves a mathematical construct called phase space. In this space stable structures can appear which I use as analogies for the stable structures that appear in a complex system such as an ecology, the mind or a society. The common denominator in all of these systems is that they rely on a process called feedback loops. Feedback loops link the microscopic (individual) parts to the macroscopic (global) parts. The key, then, in shaping a space society, is in effectively using feedback loops. This paper will illustrate how one can model a space society by using methods that chaoticists have developed over the last hundred years. And I will show that common threads exist in the modeling of biological, economical, philosophical, and sociological systems.

Graps, Amara L.↗

NASA NTRS1973

Nonlinear gas oscillations in pipes. I - Theory.

The problem of forced acoustic oscillations in a pipe is studied theoretically. The oscillations are produced by a moving piston in one end of the pipe, while a variety of boundary conditions ranging from a completely closed to a completely open mouth at the other end are considered. The linear theory predicts large amplitudes near resonance and that nonlinear effects become crucially important. By expanding the equations of motion in a series in the Mach number, both the amplitude and waveform of the oscillation are predicted there. In both the open- and closed-end cases the need for shock waves in some range of parameters is found. The amplitude of the oscillation is different for the two cases, however, being proportional to the square root of the piston amplitude in the closed-end case and to the cube root for the open end.

Jimenez, J.↗

NASA NTRS1974

Nonlinear gas oscillations in pipes. II - Experiment

The problem of forced acoustic oscillations in a pipe was experimentally investigated, taking into account the response of both open and closed tubes to near-resonant excitation by large amplitude oscillations of a piston at one end of the tube. Attention was given to the effect of the orifice area on shock waves. By comparing the experimental results with nonlinear theory, wave reflection coefficients of the orifice plates were determined at both closed-tube and open-tube resonant frequencies. This approach can even be used when the terminating elements are subjected to intense periodic pressure pulses.

Sturtevant, B.↗

NASA NTRS1975

Application of the comparison principle to analysis of nonlinear systems

A comparison principle based on a Kamke theorem and Lipschitz conditions is presented along with its possible applications and modifications. It is shown that the comparison lemma can be used in the study of such areas as classical stability theory, higher order trajectory derivatives, Liapunov functions, boundary value problems, approximate dynamic systems, linear and nonlinear systems, and bifurcation analysis.

Gunderson, R. W.↗

NASA NTRS1993

An efficient model for coupling structural vibrations with acoustic radiation

The scattering of an incident wave by a flexible panel is studied. The panel vibration is governed by the nonlinear plate equations while the loading on the panel, which is the pressure difference across the panel, depends on the reflected and transmitted waves. Two models are used to calculate this structural-acoustic interaction problem. One solves the three dimensional nonlinear Euler equations for the flow-field coupled with the plate equations (the fully coupled model). The second uses the linear wave equation for the acoustic field and expresses the load as a double integral involving the panel oscillation (the decoupled model). The panel oscillation governed by a system of integro-differential equations is solved numerically and the acoustic field is then defined by an explicit formula. Numerical results are obtained using the two models for linear and nonlinear panel vibrations. The predictions given by these two models are in good agreement but the computational time needed for the 'fully coupled model' is 60 times longer than that for 'the decoupled model'.

Frendi, Abdelkader↗

NASA NTRS1974

Lagrangian formulation of the one-dimensional Vlasov equation

A new formulation of the one-dimensional Vlasov equation is derived which is analogous to the Kalman-transformed cold-plasma equations. The equations are shown to yield nonsecular, nonlinear approximations to a source or boundary-value problem. It is suggested that the formulation may have other applications in nonlinear plasma theory.

Lewak, G. J.↗

NASA NTRS1993

Variable-permittivity linear inverse problem for the H(sub z)-polarized case

The H(sub z)-polarized inverse problem has rarely been studied before due to the complicated way in which the unknown permittivity appears in the wave equation. This problem is equivalent to the acoustic inverse problem with variable density. We have recently reported the solution to the nonlinear variable-permittivity H(sub z)-polarized inverse problem using the Born iterative method. Here, the linear inverse problem is solved for permittivity (epsilon) and permeability (mu) using a different approach which is an extension of the basic ideas of diffraction tomography (DT). The key to solving this problem is to utilize frequency diversity to obtain the required independent measurements. The receivers are assumed to be in the far field of the object, and plane wave incidence is also assumed. It is assumed that the scatterer is weak, so that the Born approximation can be used to arrive at a relationship between the measured pressure field and two terms related to the spatial Fourier transform of the two unknowns, epsilon and mu. The term involving permeability corresponds to monopole scattering and that for permittivity to dipole scattering. Measurements at several frequencies are used and a least squares problem is solved to reconstruct epsilon and mu. It is observed that the low spatial frequencies in the spectra of epsilon and mu produce inaccuracies in the results. Hence, a regularization method is devised to remove this problem. Several results are shown. Low contrast objects for which the above analysis holds are used to show that good reconstructions are obtained for both permittivity and permeability after regularization is applied.

Moghaddam, M.↗

NASA NTRS1991

Finite difference time domain calculation of transients in antennas with nonlinear loads

In this paper transient fields for antennas with more general geometries are calculated directly using Finite Difference Time Domain methods. In each FDTD cell which contains a nonlinear load, a nonlinear equation is solved at each time step. As a test case the transient current in a long dipole antenna with a nonlinear load excited by a pulsed plane wave is computed using this approach. The results agree well with both calculated and measured results previously published. The approach given here extends the applicability of the FDTD method to problems involving scattering from targets including nonlinear loads and materials, and to coupling between antennas containing nonlinear loads. It may also be extended to propagation through nonlinear materials.

Luebbers, Raymond J.↗

NASA NTRS1981

Numerical treatment of nonlinear latent heat boundary conditions at moving interfaces in genuine two dimensional solidification problems

The proposed method for the treatment of two dimensional solidification problems is based on quasilinearization of the transformed heat conduction equation and latent heat condition at the interface and an iterative sequence in which these are solved simultaneously. Modern algorithms for solving such sparse systems mean that most of the storage advantage of other methods are reduced and the speed of solution can be improved.

Beckett, P. M.↗

NASA NTRS2008

Numerical Determination of Critical Conditions for Thermal Ignition

The determination of ignition or thermal explosion in an oxidizing porous body of material, as described by a dimensionless reaction-diffusion equation of the form .tu = .2u + .e-1/u over the bounded region O, is critically reexamined from a modern perspective using numerical methodologies. First, the classic stationary model is revisited to establish the proper reference frame for the steady-state solution space, and it is demonstrated how the resulting nonlinear two-point boundary value problem can be reexpressed as an initial value problem for a system of first-order differential equations, which may be readily solved using standard algorithms. Then, the numerical procedure is implemented and thoroughly validated against previous computational results based on sophisticated path-following techniques. Next, the transient nonstationary model is attacked, and the full nonlinear form of the reaction-diffusion equation, including a generalized convective boundary condition, is discretized and expressed as a system of linear algebraic equations. The numerical methodology is implemented as a computer algorithm, and validation computations are carried out as a prelude to a broad-ranging evaluation of the assembly problem and identification of the watershed critical initial temperature conditions for thermal ignition. This numerical methodology is then used as the basis for studying the relationship between the shape of the critical initial temperature distribution and the corresponding spatial moments of its energy content integral and an attempt to forge a fundamental conjecture governing this relation. Finally, the effects of dynamic boundary conditions on the classic storage problem are investigated and the groundwork is laid for the development of an approximate solution methodology based on adaptation of the standard stationary model.

Luo, W.↗

NASA NTRS1982

Aerodynamics of an airfoil with a jet issuing from its surface

A simple, two dimensional, incompressible and inviscid model for the problem posed by a two dimensional wing with a jet issuing from its lower surface is considered and a parametric analysis is carried out to observe how the aerodynamic characteristics depend on the different parameters. The mathematical problem constitutes a boundary value problem where the position of part of the boundary is not known a priori. A nonlinear optimization approach was used to solve the problem, and the analysis reveals interesting characteristics that may help to better understand the physics involved in more complex situations in connection with high lift systems.

Tavella, D. A.↗