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All sourcesNASA NTRSDOE OSTI
NASA NTRS1988

Nonlinear programming extensions to rational function approximation methods for unsteady aerodynamic forces

The approximation of unsteady generalized aerodynamic forces in the equations of motion of a flexible aircraft are discussed. Two methods of formulating these approximations are extended to include the same flexibility in constraining the approximations and the same methodology in optimizing nonlinear parameters as another currently used extended least-squares method. Optimal selection of nonlinear parameters is made in each of the three methods by use of the same nonlinear, nongradient optimizer. The objective of the nonlinear optimization is to obtain rational approximations to the unsteady aerodynamics whose state-space realization is lower order than that required when no optimization of the nonlinear terms is performed. The free linear parameters are determined using the least-squares matrix techniques of a Lagrange multiplier formulation of an objective function which incorporates selected linear equality constraints. State-space mathematical models resulting from different approaches are described and results are presented that show comparative evaluations from application of each of the extended methods to a numerical example.

Tiffany, Sherwood H.↗
NASA NTRS1987

Nonlinear programming extensions to rational function approximations of unsteady aerodynamics

This paper deals with approximating unsteady generalized aerodynamic forces in the equations of motion of a flexible aircraft. Two methods of formulating these approximations are extended to include both the same flexibility in constraining them and the same methodology in optimizing nonlinear parameters as another currently used 'extended least-squares' method. Optimal selection of 'nonlinear' parameters is made in each of the three methods by use of the same nonlinear (nongradient) optimizer. The objective of the nonlinear optimization is to obtain rational approximations to the unsteady aerodynamics whose state-space realization is of lower order than that required when no optimization of the nonlinear terms is performed. The free 'linear' parameters are determined using least-squares matrix techniques on a Lagrange multiplier formulation of an objective function which incorporates selected linear equality constraints. State-space mathematical models resulting from the different approaches are described, and results are presented which show comparative evaluations from application of each of the extended methods to a numerical example. The results obtained for the example problem show a significant (up to 63 percent) reduction in the number of differential equations used to represent the unsteady aerodynamic forces in linear time-invariant equations of motion as compared to a conventional method in which nonlinear terms are not optimized.

Tiffany, Sherwood H.↗
NASA NTRS1989

Aeroservoelastic modeling and applications using minimum-state approximations of the unsteady aerodynamics

Various control analysis, design, and simulation techniques for aeroelastic applications require the equations of motion to be cast in a linear time-invariant state-space form. Unsteady aerodynamics forces have to be approximated as rational functions of the Laplace variable in order to put them in this framework. For the minimum-state method, the number of denominator roots in the rational approximation. Results are shown of applying various approximation enhancements (including optimization, frequency dependent weighting of the tabular data, and constraint selection) with the minimum-state formulation to the active flexible wing wind-tunnel model. The results demonstrate that good models can be developed which have an order of magnitude fewer augmenting aerodynamic equations more than traditional approaches. This reduction facilitates the design of lower order control systems, analysis of control system performance, and near real-time simulation of aeroservoelastic phenomena.

Tiffany, Sherwood H.↗
NASA NTRS1993

Rational trigonometric approximations using Fourier series partial sums

A class of approximations (S(sub N,M)) to a periodic function f which uses the ideas of Pade, or rational function, approximations based on the Fourier series representation of f, rather than on the Taylor series representation of f, is introduced and studied. Each approximation S(sub N,M) is the quotient of a trigonometric polynomial of degree N and a trigonometric polynomial of degree M. The coefficients in these polynomials are determined by requiring that an appropriate number of the Fourier coefficients of S(sub N,M) agree with those of f. Explicit expressions are derived for these coefficients in terms of the Fourier coefficients of f. It is proven that these 'Fourier-Pade' approximations converge point-wise to (f(x(exp +))+f(x(exp -)))/2 more rapidly (in some cases by a factor of 1/k(exp 2M)) than the Fourier series partial sums on which they are based. The approximations are illustrated by several examples and an application to the solution of an initial, boundary value problem for the simple heat equation is presented.

Geer, James F.↗
NASA NTRS1993

Program Determines Minimum-State Approximations To Unsteady Aerodynamic Forces

MIST implements minimum-state method for determining rational approximations of matrix of aerodynamic force coefficients. Accepts complex tabular data representing generalized unsteady aerodynamic forces over set of reduced frequencies. Determines approximations to tabular data in Laplace domain by use of rational functions. Provides capability to select coefficients of denominator in rational approximations, to constrain approximation selectably without increasing size of problem, and determines and emphasizes criticial frequency ranges in determining approximations. Written in ANSI FORTRAN 77.

Karpel, Mordechay↗
NASA NTRS1990

Efficient solution of parabolic equations by Krylov approximation methods

Numerical techniques for solving parabolic equations by the method of lines is addressed. The main motivation for the proposed approach is the possibility of exploiting a high degree of parallelism in a simple manner. The basic idea of the method is to approximate the action of the evolution operator on a given state vector by means of a projection process onto a Krylov subspace. Thus, the resulting approximation consists of applying an evolution operator of a very small dimension to a known vector which is, in turn, computed accurately by exploiting well-known rational approximations to the exponential. Because the rational approximation is only applied to a small matrix, the only operations required with the original large matrix are matrix-by-vector multiplications, and as a result the algorithm can easily be parallelized and vectorized. Some relevant approximation and stability issues are discussed. We present some numerical experiments with the method and compare its performance with a few explicit and implicit algorithms.

Gallopoulos, E.↗
NASA NTRS1980

Design for active flutter suppression and gust alleviation using state-space aeroelastic modeling

An analytical design technique for an active flutter-suppression and gust-alleviation control system is presented. It is based on a rational approximation of the unsteady aerodynamic loads in the entire Laplace domain, which yields matrix equations of motion with constant coefficients. Some existing rational approximation schemes are reviewed, and a new technique which yields a minimal number of augmented states for a desired accuracy is presented. The state-space aeroelastic model is used to design a constant gain, partial-feedback control system, which simultaneously assures stability and optimizes any desired combination of gust response parameters throughout the entire flight envelope.

Karpel, M.↗
NASA NTRS2015

Rapid State Space Modeling Tool for Rectangular Wing Aeroservoelastic Studies

This paper introduces a modeling and simulation tool for aeroservoelastic analysis of rectangular wings with trailing-edge control surfaces. The inputs to the code are planform design parameters such as wing span, aspect ratio, and number of control surfaces. Using this information, the generalized forces are computed using the doublet-lattice method. Using Roger's approximation, a rational function approximation is computed. The output, computed in a few seconds, is a state space aeroservoelastic model which can be used for analysis and control design. The tool is fully parameterized with default information so there is little required interaction with the model developer. All parameters can be easily modified if desired. The focus of this paper is on tool presentation, verification, and validation. These processes are carried out in stages throughout the paper. The rational function approximation is verified against computed generalized forces for a plate model. A model composed of finite element plates is compared to a modal analysis from commercial software and an independently conducted experimental ground vibration test analysis. Aeroservoelastic analysis is the ultimate goal of this tool, therefore, the flutter speed and frequency for a clamped plate are computed using damping-versus-velocity and frequency-versus-velocity analysis. The computational results are compared to a previously published computational analysis and wind-tunnel results for the same structure. A case study of a generic wing model with a single control surface is presented. Verification of the state space model is presented in comparison to damping-versus-velocity and frequency-versus-velocity analysis, including the analysis of the model in response to a 1-cos gust.

Modeling and Simulation↗
NASA NTRS2015

Rapid State Space Modeling Tool for Rectangular Wing Aeroservoelastic Studies

This report introduces a modeling and simulation tool for aeroservoelastic analysis of rectangular wings with trailing-edge control surfaces. The inputs to the code are planform design parameters such as wing span, aspect ratio, and number of control surfaces. Using this information, the generalized forces are computed using the doublet-lattice method. Using Roger's approximation, a rational function approximation is computed. The output, computed in a few seconds, is a state space aeroservoelastic model which can be used for analysis and control design. The tool is fully parameterized with default information so there is little required interaction with the model developer. All parameters can be easily modified if desired. The focus of this report is on tool presentation, verification, and validation. These processes are carried out in stages throughout the report. The rational function approximation is verified against computed generalized forces for a plate model. A model composed of finite element plates is compared to a modal analysis from commercial software and an independently conducted experimental ground vibration test analysis. Aeroservoelastic analysis is the ultimate goal of this tool, therefore, the flutter speed and frequency for a clamped plate are computed using damping-versus-velocity and frequency-versus-velocity analysis. The computational results are compared to a previously published computational analysis and wind-tunnel results for the same structure. A case study of a generic wing model with a single control surface is presented. Verification of the state space model is presented in comparison to damping-versus-velocity and frequency-versus-velocity analysis, including the analysis of the model in response to a 1-cos gust.

Verification and Validation↗
NASA NTRS1979

Accurate numerical evaluation of modified Struve functions occurring in unsteady aerodynamics

A method is developed for computing the modified Struve functions that occur in unsteady aerodynamics. The method uses a rational approximation supplemented by an asymptotic series for large argument. Simple recursive formulas for generating the coefficients are derived. The method is capable of generating results of arbitrary accuracy. It can also be used for complex argument and order. For greater computing speed, a method is presented that uses the rational and asymptotic approximations to generate Chebyshev coefficients.

Desmarais, R. N.↗
NASA NTRS1991

Physically weighted approximations of unsteady aerodynamic forces using the minimum-state method

The Minimum-State Method for rational approximation of unsteady aerodynamic force coefficient matrices, modified to allow physical weighting of the tabulated aerodynamic data, is presented. The approximation formula and the associated time-domain, state-space, open-loop equations of motion are given, and the numerical procedure for calculating the approximation matrices, with weighted data and with various equality constraints are described. Two data weighting options are presented. The first weighting is for normalizing the aerodynamic data to maximum unit value of each aerodynamic coefficient. The second weighting is one in which each tabulated coefficient, at each reduced frequency value, is weighted according to the effect of an incremental error of this coefficient on aeroelastic characteristics of the system. This weighting yields a better fit of the more important terms, at the expense of less important ones. The resulting approximate yields a relatively low number of aerodynamic lag states in the subsequent state-space model. The formulation forms the basis of the MIST computer program which is written in FORTRAN for use on the MicroVAX computer and interfaces with NASA's Interaction of Structures, Aerodynamics and Controls (ISAC) computer program. The program structure, capabilities and interfaces are outlined in the appendices, and a numerical example which utilizes Rockwell's Active Flexible Wing (AFW) model is given and discussed.

Karpel, Mordechay↗
NASA NTRS1972

System evaluation from impulse response fragments

By use of rational approximation, a procedure was found which allows calculation of the z-transfer function from a restricted number of time sequence values. The order of the system need not be known. Transformation to the s-domain is easily provided.

Baumeister, J. K.↗
NASA NTRS1975

On the analytic structure of the Taylor-Maccoll conical-flow solution

The conical flow solution for axisymmetric supersonic flow past cones has been found to be virtually independent of the ratio of specific heats when normalized in a certain way. A simple rational approximation to this flow is derived. The important singularities and the limiting behavior of the solution are also discussed.

Schwartz, L. W.↗
NASA NTRS1981

Design for active and passive flutter suppression and gust alleviation

Analytical design techniques for active and passive control of aeroelastic systems are based on a rational approximation of the unsteady aerodynamic loads in the entire Laplace domain, which yields matrix equations of motion with constant coefficients. Some existing schemes are reviewed, the matrix Pade approximant is modified, and a technique which yields a minimal number of augmented states for a desired accuracy is presented. The state-space aeroelastic model is used to design an active control system for simultaneous flutter suppression and gust alleviation. The design target is for a continuous controller which transfers some measurements taken on the vehicle to a control command applied to a control surface. Structural modifications are formulated in a way which enables the treatment of passive flutter suppression system with the same procedures by which active control systems are designed.

Karpel, M.↗
NASA NTRS1982

Real time prediction of marine vessel motions using Kalman filtering techniques

The present investigation is concerned with the prediction of the future behavior of a vessel within some confidence bounds at a specific instant of time, taking into account an interval of a few seconds. The ability to predict accurately the motions of a vessel can reduce significantly the probability of failure of operations in rough seas. The investigation was started as part of an effort to ensure safe landing of aircraft on relatively small vessels. However, the basic principles involved in the study are the same for any offshore operation, such as carbo transfer in the open sea, structure installation, and floating crane operation. The Kalman filter is a powerful tool for achieving the goals of the prediction procedure. Attention is given to a linear optimal predictor, the equations of motion of the vessel, the wave spectrum, rational approximation, the use of Kalman filter and predictor in an application for a ship, and the motions of a semisubmersible.

Triantafyllou, M. S.↗
NASA NTRS1985

Adding computationally efficient realism to Monte Carlo turbulence simulation

Frequently in aerospace vehicle flight simulation, random turbulence is generated using the assumption that the craft is small compared to the length scales of turbulence. The turbulence is presumed to vary only along the flight path of the vehicle but not across the vehicle span. The addition of the realism of three-dimensionality is a worthy goal, but any such attempt will not gain acceptance in the simulator community unless it is computationally efficient. A concept for adding three-dimensional realism with a minimum of computational complexity is presented. The concept involves the use of close rational approximations to irrational spectra and cross-spectra so that systems of stable, explicit difference equations can be used to generate the turbulence.

Campbell, C. W.↗
NASA NTRS1986

A simple method for computing the relativistic Compton scattering kernel for radiative transfer

Correct computation of the Compton scattering kernel (CSK), defined to be the Klein-Nishina differential cross section averaged over a relativistic Maxwellian electron distribution, is reported. The CSK is analytically reduced to a single integral, which can then be rapidly evaluated using a power series expansion, asymptotic series, and rational approximation for sigma(s). The CSK calculation has application to production codes that aim at understanding certain astrophysical, laser fusion, and nuclear weapons effects phenomena.

Prasad, M. K.↗