Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “linear equation systems”

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 289 records · Page 16
All sourcesNASA NTRSDOE OSTI
NASA NTRS1983

The effects of Ohmic heating and stable radiation on magnetic tearing

A study is made of the effect of a temperature-dependent Coulomb-like resistivity on the planar tearing mode. The local evolution of the temperature is described by an energy equation which includes Joule heating and optically thin radiation. The resulting system of coupled linear magnetohydrodynamic equations is solved numerically, and eigenfunctions and growth rates are obtained. In the absence of radiation, there are two distinct solutions above a critical value of the magnetic Reynolds number S, a tearing-like mode and a Joule-heating mode. Below this point, the growth rates coalesce into a conjugate-complex pair. When stable radiation (dR/dT greater than 0) is added, the heating mode disappears and a modified tearing excitation exists to much lower values of S before its growth is cut off by Ohmic heating. Examples are given for solar coronal parameters, and for those characteristic of fusion-research devices. The introduction of an effective value for the resistivity, in the presence of energy transport, allows a simple qualitative discussion of the different modes.

Tachi, T.↗
NASA NTRS1981

Time and frequency domain analysis of sampled data controllers via mixed operation equations

Specification of the mathematical equations required to define the dynamic response of a linear continuous plant, subject to sampled data control, is complicated by the fact that the digital components of the control system cannot be modeled via linear ordinary differential equations. This complication can be overcome by introducing two new mathematical operations; namely, the operation of zero order hold and digial delay. It is shown that by direct utilization of these operations, a set of linear mixed operation equations can be written and used to define the dynamic response characteristics of the controlled system. It also is shown how these linear mixed operation equations lead, in an automatable manner, directly to a set of finite difference equations which are in a format compatible with follow on time and frequency domain analysis methods.

Frisch, H. P.↗
NASA NTRS1991

A multi-point inverse airfoil design method based on conformal mapping

An exact method of multipoint inverse airfoil design for incompressible flow is presented. Multipoint design is handled by dividing the airfoil into a number of desired segments. For each segment the velocity distribution is prescribed together with an angle of attack at which the prescribed velocity distribution is to be achieved. In this manner, multipoint design objectives can be taken into account in the initial specification of the velocity distribution. In order for the multipoint inverse airfoil design problem to be well posed, three integral constraints and several conditions arise which must be satisfied. Further restrictions are imposed if the airfoil is to have a specified pitching moment, thickness ratio, or other constraints. The system of equations is solved partly as a linear system and partly through multidimensional Newton iteration. Since the velocity distribution is prescribed about the circle in the angular coordinate, specification of the velocity in terms of arc length is handled through the multidimensional Newton iteration as well. The current formulation sets the stage for a more general multipoint inverse airfoil design method in which it will be possible to specify the velocity distribution, some boundary-layer development, or the surface geometry along a segment.

Selig, Michael S.↗
NASA NTRS1995

An Adaptive Semi-Implicit Scheme for Simulations of Unsteady Viscous Compressible Flows

A numerical scheme for simulation of unsteady, viscous, compressible flows is considered. The scheme employs an explicit discretization of the inviscid terms of the Navier-Stokes equations and an implicit discretization of the viscous terms. The discretization is second order accurate in both space and time. Under appropriate assumptions, the implicit system of equations can be decoupled into two linear systems of reduced rank. These are solved efficiently using a Gauss-Seidel method with multigrid convergence acceleration. When coupled with a solution-adaptive mesh refinement technique, the hybrid explicit-implicit scheme provides an effective methodology for accurate simulations of unsteady viscous flows. The methodology is demonstrated for both body-fitted structured grids and for rectangular (Cartesian) grids.

Steinthorsson, Erlendur↗
NASA NTRS1977

Dynamic response and stability analysis of flexible, multibody systems

A general version of Lagrange's equations, including auxiliary nonholonomic, rheonomic conditions of constraint, is used in the dynamic simulation and stability analysis of interconnected flexible bodies. Modeling of the nonlinear flexible/rigid dynamic coupling effects, the interaction forces/torques, and the elastic deformation effects is discussed. A digital computer program is developed to obtain time-domain solution for the nonlinear response of systems represented as a collection of individual bodies, numerical linearization of system-governing equations, time-domain solution for the perturbation response about a nominal state, and a frequency-domain stability analysis corresponding to the linearization. The digital simulation code is employed to study the dynamic behavior of a typical satellite and a spacecraft with deployable experiment booms.

Bodley, C. S.↗
NASA NTRS1986

Identification of linearized equations of motion for the fixed wing configuration of the rotor systems research aircraft

The purpose of this report is to establish linear, decoupled models of rigid body motion for the fixed wing configuration of the Rotor Systems Research Aircraft (RSRA). Longitudinal and lateral control surface fixed linear models were created from aircraft time histories using current system identification techniques. Models were obtained from computer simulation at 160 KCAS and 200 KCAS, and from flight data at 160 KCAS. Comparisons were performed to examine modeling accuracy, variation of dynamics with airspeed and correlation of simulation and flight data results. The results showed that the longitudinal and lateral linear models accurately predicted RSRA dynamics. The flight data results showed that no significant handling qualities problems were present in the RSRA fixed wing aircraft at the flight speed tested.

Balough, D. L.↗
NASA NTRS1975

A study of attitude control concepts for precision-pointing non-rigid spacecraft

Attitude control concepts for use onboard structurally nonrigid spacecraft that must be pointed with great precision are examined. The task of determining the eigenproperties of a system of linear time-invariant equations (in terms of hybrid coordinates) representing the attitude motion of a flexible spacecraft is discussed. Literal characteristics are developed for the associated eigenvalues and eigenvectors of the system. A method is presented for determining the poles and zeros of the transfer function describing the attitude dynamics of a flexible spacecraft characterized by hybrid coordinate equations. Alterations are made to linear regulator and observer theory to accommodate modeling errors. The results show that a model error vector, which evolves from an error system, can be added to a reduced system model, estimated by an observer, and used by the control law to render the system less sensitive to uncertain magnitudes and phase relations of truncated modes and external disturbance effects. A hybrid coordinate formulation using the provided assumed mode shapes, rather than incorporating the usual finite element approach is provided.

Likins, P. W.↗
NASA NTRS1973

Hybrid-coordinate spacecraft dynamics using large-deformation modal coordinates

For a vehicle amenable to idealization as a set of n + 1 rigid bodies interconnected by n line hinges (implying tree topology), a set of minimum-dimension matrix equations of motion is recorded in terms of variables including the relative rotations of contiguous bodies. When a substructure or portion of the vehicle is subjected to large gross deformations but is restricted to 'small' strains or local deformations, the n-body model is (for sufficiently large n) assumed to experience 'small' relative rotations of contiguous bodies within the substructure, and the equations of motion are linearized in these variables. The application of coordinate transformations and truncations to the linearized subset of the system equations of motion then produces a hybrid-coordinate formulation of the problem involving both discrete and distributed coordinates.

Likins, P. W.↗
NASA NTRS2020

Model-Reference Adaptive Control of Distributed Lagrangian Infinite-Dimensional Systems Using Hamilton’s Principle

This paper presents a Hamilton's principle for distributed control of infinite-dimensional systems modeled by a distributed form of the Euler-Lagrange method. The distributed systems are governed by a system of linear partial differential equations in space and time. A generalized potential energy expression is developed that can capture most physical systems including those systems that have no spatial distribution. The Hamilton's principle is applied to derive distributed feedback control methods without resorting to the standard weak-form discretization approach to convert an infinite-dimensional systems to a finite-dimensional systems. It can be shown by the principle of least action that the distributed control synthesized by the Hamilton's principle is a minimum-norm control. A model-reference adaptive control framework is developed for distributed Lagrangian systems in the presence of uncertainty. The theory is demonstrated by an application of adaptive flutter suppression control of a flexible aircraft wing.

Nguyen, Nhan T.↗
NASA NTRS1996

First-Order System Least Squares for the Stokes Equations, with Application to Linear Elasticity

Following our earlier work on general second-order scalar equations, here we develop a least-squares functional for the two- and three-dimensional Stokes equations, generalized slightly by allowing a pressure term in the continuity equation. By introducing a velocity flux variable and associated curl and trace equations, we are able to establish ellipticity in an H(exp 1) product norm appropriately weighted by the Reynolds number. This immediately yields optimal discretization error estimates for finite element spaces in this norm and optimal algebraic convergence estimates for multiplicative and additive multigrid methods applied to the resulting discrete systems. Both estimates are uniform in the Reynolds number. Moreover, our pressure-perturbed form of the generalized Stokes equations allows us to develop an analogous result for the Dirichlet problem for linear elasticity with estimates that are uniform in the Lame constants.

Cai, Z.↗
NASA NTRS1985

The dynamics and control of large flexible space structures, 8

A development of the in plane open loop rotational equations of motion for the proposed Spacecraft Control Laboratory Experiment (SCOLE) in orbit configuration is presented based on an Eulerian formulation. The mast is considered to be a flexible beam connected to the (rigid) shuttle and the reflector. Frequencies and mode shapes are obtained for the mast vibrational appendage modes (assumed to be decoupled) for different boundary conditions based on continuum approaches and also preliminary results are obtained using a finite element representation of the mast reflector system. The linearized rotational in plane equation is characterized by periodic coefficients and open loop system stability can be examined with an application of the Floquet theorem. Numerical results are presented to illustrate the potential instability associated with actuator time delays even for delays which represent only a small fraction of the natural period of oscillation of the modes contained in the open loop model of the system. When plant and measurement noise effects are added to the previously designed deterministic model of the hoop column system, it is seen that both the system transient and steady state performance are degraded. Mission requirements can be satisfied by appropriate assignment of cost function weighting elements and changes in the ratio of plant noise to measurement noise.

Bainum, P. M.↗
NASA NTRS1989

The application of TSIM software to act design and analysis on flexible aircraft

The TSIM software is described. This is a package which uses an interactive FORTRAN-like simulation language for the simulation on nonlinear dynamic systems and offers facilities which include: mixed continuous and discrete time systems, time response calculations, numerical optimization, automatic trimming of nonlinear aircraft systems, and linearization of nonlinear equations for eigenvalues, frequency responses and power spectral response evaluation. Details are given of the application of TSIM to the analysis of aeroelastic systems under the RAE Farborough extension FLEX-SIM. The aerodynamic and structural data for the equations of motion of a flexible aircraft are prepared by a preprocessor program for incorporation in TSIM simulations. Within the simulation, the flexible aircraft model may then be selected interactively for different flight conditions and modal reduction techniques applied. The use of FLEX-SIM is demonstrated by an example of the flutter prediction for a simple aeroelastic model. By utilizing the numerical optimization facility of TSIM, it is possible to undertake identification of required parameters in the TSIM model within the simulation.

Kaynes, Ian W.↗
NASA NTRS1989

A new method for analysis of limit cycle behavior of the NASA/JPL 70-meter antenna axis servos

A piecewise linear method of analyzing the effects of discontinuous nonlinearities on control system performance is described. The limit cycle oscillatory behavior of the system resulting from the nonlinearities is described in terms of a sequence of linear system transient responses. The equations are derived which relate the initial and the terminal conditions of successive transients and the boundary conditions imposed by the non-linearities. The method leads to a convenient computation algorithm for prediction of limit cycle characteristics resulting from discontinuous nonlinearities such as friction, deadzones, and hysteresis.

Hill, R. E.↗
NASA NTRS1992

Analysis, modelling and simulation of the large-angle magnetic suspension test fixture

The large-angle magnetic suspension test fixture (LAMSTF) research project is described which is based on a sophisticated finite element computer program, VF/GFUN, for calculating magnetic fields. The LAMSTF design includes eddy current paths the effect of which on system dynamics is being studied to modify the system dynamic model and the digital controller. Linearized equations of motions have been developed for system modelling and analysis of controllers.

Britcher, Colin P.↗