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 721 records · Page 40
All sourcesNASA NTRSDOE OSTI
NASA NTRS1976

The dynamics of spin stabilized spacecraft with movable appendages, part 2

The dynamics and stability of a spin stabilized spacecraft with a hinged appendage system are treated analytically and numerically. The hinged system consists of a central hub with masses attached to (assumed) massless booms of fixed length whose orientation relative to the main part can change. The general three dimensional deployment dynamics of such a hinged system is considered without any restriction on the location of the hinge points. The equations of motion for the hinged system, with viscous damping at both hinge points, are linearized about the nominal equilibrium position where the booms are orthogonal to the nominal spin axis for the case of two dimensional and three dimensional motion. Analytic stability criteria are obtained from the necessary condition on the sign of all the coefficients in the system characteristic equation.

Bainum, P. M.↗
NASA NTRS1995

The large discretization step method for time-dependent partial differential equations

A new method for the acceleration of linear and nonlinear time dependent calculations is presented. It is based on the Large Discretization Step (LDS) approximation, defined in this work, which employs an extended system of low accuracy schemes to approximate a high accuracy discrete approximation to a time dependent differential operator. Error bounds on such approximations are derived. These approximations are efficiently implemented in the LDS methods for linear and nonlinear hyperbolic equations, presented here. In these algorithms the high and low accuracy schemes are interpreted as the same discretization of a time dependent operator on fine and coarse grids, respectively. Thus, a system of correction terms and corresponding equations are derived and solved on the coarse grid to yield the fine grid accuracy. These terms are initialized by visiting the fine grid once in many coarse grid time steps. The resulting methods are very general, simple to implement and may be used to accelerate many existing time marching schemes.

Haras, Zigo↗
NASA NTRS1992

Ion heating by kinetic cross-field streaming instability due to reflected ions at a quasiperpendicular shock

The present paper shows that the reflected ion at a supercritical quasi-perpendicular shock wave can excite a purely growing mode propagating parallel to the ambient magnetic field. To discuss the ion heating by such an unstable mode, the self-consistent quasi-linear kinetic equation is solved with the assumption that the present purely growing mode is the dominant unstable mode in the system. In the quasi-linear analysis of the instability, two particular cases are considered: the case of low initial ion beta and that of a high initial ion beta.

Yoon, Peter H.↗
NASA NTRS1987

Approximate polynomial preconditioning applied to biharmonic equations on vector supercomputers

Applying a finite difference approximation to a biharmonic equation results in a very ill-conditioned system of equations. This paper examines the conjugate gradient method used in conjunction with the generalized and approximate polynomial preconditionings for solving such linear systems. An approximate polynomial preconditioning is introduced, and is shown to be more efficient than the generalized polynomial preconditionings. This new technique provides a simple but effective preconditioning polynomial, which is based on another coefficient matrix rather than the original matrix operator as commonly used.

Wong, Yau Shu↗
NASA NTRS1987

Stability and robustness of slowly time-varying linear systems

A well-known result for finite-dimensional time-varying linear systems is that if each `frozen time' is stable, then the time-varying system is stable for sufficiently slow time-variations. These results are reviewed and extended to a class of Volterra integrodifferential equations, specifically, differential equations with a convolution operator in the right-hand-side. The results are interpreted in the context of robustness of time-varying linear systems with special emphasis on analysis of gain-scheduled control systems.

Shamma, Jeff S.↗
NASA NTRS1982

Linear Hamiltonian systems - The Riccati group and its invariants

The action of the Riccati group on the Riccati differential equation is associated with the action of a subgroup of the symplectic group on a set of Hamiltonian matrices. Within this framework canonical forms are developed for the matrix coefficients of the Riccati differential equation.

Garzia, M. R.↗
NASA NTRS1972

Dual characterizations of optimal systems.

The complementary variational principle developed in a Hilbert space setting provides a duality principle in the calculus of variations with dynamic constraints. This concept is adopted in this paper to investigate dual characterizations of optimal control systems. Systems under consideration include those with dynamics governed by linear ordinary differential equations, linear partial differential equations and non-linear ordinary differential equations.

Chan, W. L.↗
NASA NTRS1974

Use of Green's functions in the numerical solution of two-point boundary value problems

This study investigates the use of Green's functions in the numerical solution of the two-point boundary value problem. The first part deals with the role of the Green's function in solving both linear and nonlinear second order ordinary differential equations with boundary conditions and systems of such equations. The second part describes procedures for numerical construction of Green's functions and considers briefly the conditions for their existence. Finally, there is a description of some numerical experiments using nonlinear problems for which the known existence, uniqueness or convergence theorems do not apply. Examples here include some problems in finding rendezvous orbits of the restricted three body system.

Gallaher, L. J.↗
NASA NTRS1982

Short-term climatic fluctuations forced by thermal anomalies

A two level, global, spectral model using pressure as a vertical coordinate was developed. The system of equations describing the model is nonlinear and quasi-geostrophic (linear balance). Static stability is variable in the model. A moisture budget is calculated in the lower layer only. Convective adjustment is used to avoid supercritical temperature lapse rates. The mechanical forcing of topography is introduced as a vertical velocity at the lower boundary. Solar forcing is specified assuming a daily mean zenith angle. The differential diabatic heating between land and sea is paramterized. On land and sea ice surfaces, a steady state thermal energy equation is solved to calculate the surface temperature. On the oceans, the sea surface temperature is specified as the climatological average for January. The model is used to simulate the January, February and March circulations.

Hanna, A. F.↗
NASA NTRS1984

Short-term climatic fluctuations forced by thermal anomalies

A two level, global, spectral model using pressure as a vertical coordinate was developed. The system of equations describing the model is nonlinear and quasi-geostrophic (linear balance). Static stability is variable in the model. A moisture budget is calculated in the lower layer only. Convective adjustment is used to avoid supercritical temperature lapse rates. The mechanical forcing of topography is introduced as a vertical velocity at the lower boundary. Solar forcing is specified assuming a daily mean zenith angle. The differential diabatic heating between land and sea is parameterized. On land and sea ice surfaces, a steady state thermal energy equation is solved to calculate the surface temperature. On the oceans, the sea surface temperature is specified as the climatological average for January. The model is used to simulate the January, February and March circulations. Previously announced in STAR as N83-14822

Hanna, A. F.↗
NASA NTRS1983

Short-term climatic fluctuations forced by thermal anomalies

A two level, global, spectral model using pressure as a vertical coordinate was developed. The system of equations describing the model is nonlinear and quasi-geostrophic (linear balance). Static stability is variable in the model. A moisture budget is calculated in the lower layer only. Convective adjustment is used to avoid supercritical temperature lapse rates. The mechanical forcing of topography is introduced as a vertical velocity at the lower boundary. Solar forcing is specified assuming a daily mean zenith angle. The differential diabatic heating between land and sea is parameterized. On land and sea ice surfaces, a steady state thermal energy equation is solved to calculate the surface temperature. On the oceans, the sea surface temperature is specified as the climatological average for January. The model is used to simulate the January, February and March circulations. Previously announced in STAR as N83-14822

Hanna, A. F.↗
NASA NTRS1985

Direct finite element equation solving algorithms

This paper presents and examines direct solution algorithms for the linear simultaneous equations that arise when finite element models represent an engineering system. It identifies the mathematical processing of four solution methods and assesses their data processing implications using concurrent processing.

Melosh, R. J.↗
NASA NTRS1988

Approximations For Controls Of Hereditary Systems

Convergence properties of controls, trajectories, and feedback kernels analyzed. Report discusses use of factorization techniques to approximate optimal feedback gains in finite-time, linear-regulator/quadratic-cost-function problem of system governed by retarded-functional-difference equations RFDE's with control delays. Presents approach to factorization based on discretization of state penalty leading to simple structure for feedback control law.

Milman, Mark H.↗
NASA NTRS1988

The dynamics and control of the orbiting spacecraft control laboratory experiment (SCOLE) during station keeping

A mathematical model is developed to predict the dynamics of the proposed orbiting Spacecraft Control Laboratory Experiment during the station keeping phase. The Shuttle as well as the reflector are assumed to be rigid, the mast is flexible and is assumed to undergo elastic displacements very small as compared with its length. The equations of motion are derived using a Newton-Euler formulation. The model includes the effects of gravity, flexibility, and orbital dynamics. The control is assumed to be provided to the system through the Shuttle's three torquers, and through six actuators located by pairs at two points on the mast and at the mass center of the reflector. At each of the locations, an actuator acts parallel to the roll axis while the other one acts parallel to the pitch axis. It is seen that, in the presence of gravity-gradient torques in the system dynamics, the system assumes a new equilibrium position about which the equations must be linearized, primarily due to the offset in the mast attachment point to the reflector. The linear regulator theory is used to derive control laws for the linear model of the SCOLE including the first four flexible modes. Numerical results confirm the robustness of this control strategy for station keeping with maximum control efforts significantly below saturation levels.

Bainum, Peter M.↗
NASA NTRS1989

Integrated control-system design via generalized LQG (GLQG) theory

Thirty years of control systems research has produced an enormous body of theoretical results in feedback synthesis. Yet such results see relatively little practical application, and there remains an unsettling gap between classical single-loop techniques (Nyquist, Bode, root locus, pole placement) and modern multivariable approaches (LQG and H infinity theory). Large scale, complex systems, such as high performance aircraft and flexible space structures, now demand efficient, reliable design of multivariable feedback controllers which optimally tradeoff performance against modeling accuracy, bandwidth, sensor noise, actuator power, and control law complexity. A methodology is described which encompasses numerous practical design constraints within a single unified formulation. The approach, which is based upon coupled systems or modified Riccati and Lyapunov equations, encompasses time-domain linear-quadratic-Gaussian theory and frequency-domain H theory, as well as classical objectives such as gain and phase margin via the Nyquist circle criterion. In addition, this approach encompasses the optimal projection approach to reduced-order controller design. The current status of the overall theory will be reviewed including both continuous-time and discrete-time (sampled-data) formulations.

Bernstein, Dennis S.↗
NASA NTRS1990

A computational algorithm for spacecraft control and momentum management

Developments in the area of nonlinear control theory have shown how coordinate changes in the state and input spaces of a dynamical system can be used to transform certain nonlinear differential equations into equivalent linear equations. These techniques are applied to the control of a spacecraft equipped with momentum exchange devices. An optimal control problem is formulated that incorporates a nonlinear spacecraft model. An algorithm is developed for solving the optimization problem using feedback linearization to transform to an equivalent problem involving a linear dynamical constraint and a functional approximation technique to solve for the linear dynamics in terms of the control. The original problem is transformed into an unconstrained nonlinear quadratic program that yields an approximate solution to the original problem. Two examples are presented to illustrate the results.

Dzielski, John↗
NASA NTRS1993

Optimal fixed-finite-dimensional compensator for Burgers' equation with unbounded input/output operators

The problem of using reduced order dynamic compensators to control a class of nonlinear parabolic distributed parameter systems was considered. Concentration was on a system with unbounded input and output operators governed by Burgers' equation. A linearized model was used to compute low-order-finite-dimensional control laws by minimizing certain energy functionals. Then these laws were applied to the nonlinear model. Standard approaches to this problem employ model/controller reduction techniques in conjunction with linear quadratic Gaussian (LQG) theory. The approach used is based on the finite dimensional Bernstein/Hyland optimal projection theory which yields a fixed-finite-order controller.

Burns, John A.↗
NASA NTRS1994

Magnetotail dynamics under isobaric constraints

Using linear theory and nonlinear MHD simulations, we investigate the resistive and ideal MHD stability of two-dimensional plasma configurations under the isobaric constraint dP/dt = 0, which in ideal MHD is equivalent to conserving the pressure function P = P(A), where A denotes the magnetic flux. This constraint is satisfied for incompressible modes, such as Alfven waves, and for systems undergoing energy losses. The linear stability analysis leads to a Schroedinger equation, which can be investigated by standard quantum mechanics procedures. We present an application to a typical stretched magnetotail configuration. For a one-dimensional sheet equilibrium characteristic properties of tearing instability are rediscovered. However, the maximum growth rate scales with the 1/7 power of the resistivity, which implies much faster growth than for the standard tearing mode (assuming that the resistivity is small). The same basic eigen-mode is found also for weakly two-dimensional equilibria, even in the ideal MHD limit. In this case the growth rate scales with the 1/4 power of the normal magnetic field. The results of the linear stability analysis are confirmed qualitatively by nonlinear dynamic MHD simulations. These results suggest the interesting possibility that substorm onset, or the thinning in the late growth phase, is caused by the release of a thermodynamic constraint without the (immediate) necessity of releasing the ideal MHD constraint. In the nonlinear regime the resistive and ideal developments differ in that the ideal mode does not lead to neutral line formation without the further release of the ideal MHD constraint; instead a thin current sheet forms. The isobaric constraint is critically discussed. Under perhaps more realistic adiabatic conditions the ideal mode appears to be stable but could be driven by external perturbations and thus generate the thin current sheet in the late growth phase, before a nonideal instability sets in.

Birn, Joachim↗