A note on the first passage time problem
Kolmogoroff backward equation for analyzing first passage time problem for linear single degree of freedom vibratory system with linear viscous damping
Shinozuka, M.↗
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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.
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Output feedback for linear multivariable systems with parameter uncertainty.
A minimax design method is applied to the problem of obtaining an acceptable output feedback matrix for linear multivariable systems with parameter uncertainty. The result is a set of nonlinear matrix equations (similar to those obtained by Levine and Athans (1970)), which must be solved for the feedback matrix. An example illustrates the technique and the fact that better results are achieved for large parameter variation than with a purely nominal design.
Basuthakur, S.↗
Acoustical theory of turbulence
Fluid functions are expanded in a series of functions constituting a complete and orthogonal system of wave solutions of the linearized Navier-Stokes system. The motion of characteristic waves is described by Hamiltonian equations of ray acoustics. Amplitude of the waves is described with the aid of nonlinear interaction terms. The distribution functions (squares of amplitudes) satisfy Boltzmann-type equations and completely describe the mean turbulence properties.
Kentzer, C. P.↗
Comparison of some optimal control methods for the design of turbine blades
This paper attempts a comparative study of some numerical methods for the optimal control design of turbine blades whose vibration characteristics are approximated by Timoshenko beam idealizations with shear and incorporating simple boundary conditions. The blade was synthesized using the following methods: (1) conjugate gradient minimization of the system Hamiltonian in function space incorporating penalty function transformations, (2) projection operator methods in a function space which includes the frequencies of vibration and the control function, (3) epsilon-technique penalty function transformation resulting in a highly nonlinear programming problem, (4) finite difference discretization of the state equations again resulting in a nonlinear program, (5) second variation methods with complex state differential equations to include damping effects resulting in systems of inhomogeneous matrix Riccatti equations some of which are stiff, (6) quasi-linear methods based on iterative linearization of the state and adjoint equation. The paper includes a discussion of some substantial computational difficulties encountered in the implementation of these techniques together with a resume of work presently in progress using a differential dynamic programming approach.
Desilva, B. M. E.↗
Character and stability of axisymmetric thermal convection in spheres and spherical shells
The influence of shell size and mode of heating on the behavior and stability of axisymmetric, infinite Prandtl number convection in a spherical geometry is studied. Heating from within and below features convection onset governed by a self-adjoint system of equations and boundary conditions. For heating only from within or from below, linearized equations and boundary conditions are non-self-adjoint. Identification of the parameter which initiates the departure from self-adjointness, together with the properties of the self-adjoint solution, provide a basis for calculating the heat transfer characteristics of the non-self-adjoint situations. The investigations are an effort to develop a model for heat transfer in planetary interiors. Further development of the technique by modifying the Galerkin method by the introduction of diagonal mode truncation is suggested to permit the consideration of higher values of the Rayleigh numbers, i.e., those more commensurate with terrestrial planet mantles.
Zebib, A.↗
Theory and modeling of atmospheric turbulence, part 1
The cascade transfer which is the only function to describe the mode coupling as the result of the nonlinear hydrodynamic state of turbulence is discussed. A kinetic theory combined with a scaling procedure was developed. The transfer function governs the non-linear mode coupling in strong turbulence. The master equation is consistent with the hydrodynamical system that describes the microdynamic state of turbulence and has the advantages to be homogeneous and have fewer nonlinear terms. The modes are scaled into groups to decipher the governing transport processes and statistical characteristics. An equation of vorticity transport describes the microdynamic state of two dimensional, isotropic and homogeneous, geostrophic turbulence. The equation of evolution of the macrovorticity is derived from group scaling in the form of the Fokker-Planck equation with memory. The microdynamic state of turbulence is transformed into the Liouville equation to derive the kinetic equation of the singlet distribution in turbulence. The collision integral contains a memory, which is analyzed with pair collision and the multiple collision. Two other kinetic equations are developed in parallel for the propagator and the transition probability for the interaction among the groups.
Source record↗
Relaxation methods for unfactored implicit upwind schemes
Relaxation methods are presented for unfactored implicit upwind schemes for hyperbolic equations. The theoretical bases are explained using linear and nonlinear scalar equations; construction of the method for the unsteady Euler equations (nonlinear system) is but a natural extension. One of the important advantages of the above methods vis a vis factored implicit schemes is the possibility of faster convergence to steady state, as illustrated by the results. Several classes of relaxation schemes such as pointwise, linewise, Gauss-Seidel, and non-Gauss-Seidel methods are discussed, along with various strategies for convergence.
Chakravarthy, S. R.↗
Nonlinear effects in the coupled response of tiles bonded to a plate
The coupled response to a large impulse on tiles bonded to a finite plate is studied. The analysis includes geometric nonlinearity born from boundary restraint in the plane, which stiffens the plate transversely. It also includes material nonlinearities born from plasticity of the plate's material and from properties of a polymer bond including memory and dissipation. The equations of motion are solved by the Galerkin method using linearized eigenfunctions of the system as trial functions. A strip of plate which is one tile wide is modeled.
El-Raheb, M.↗
Optimal orbital rendezvous using high and low thrust
Optimal control theory is used to examine a specific class of spacecraft trajectory problems where high- and low-thrust propulsion systems are utilized. These problems assume a spacecraft is in an established orbit about a planet. It is desired to execute an intercept of a pre-determined position in space in a specified amount of time using an optimal high-thrust program. The spacecraft then returns to the original orbit station using the low-thrust propulsion system in an optimal fashion. A minimum fuel solution is sought using the linearized equations of motion, known as the CW equations, which simplify the necessary computations. Solutions are obtained for problems with a fixed final time. However, for the time-open case, the optimal solution is for the final time to be infinite. With a weighted function of the final time in the performance index, a limited range of optimal single impulse solutions for the time -open case can also be found.
Prussing, John E.↗
Simulink Model of the Ares I Upper Stage Main Propulsion System
A numerical model of the Ares I upper stage main propulsion system is formulated based on first principles. Equation's are written as non-linear ordinary differential equations. The GASP fortran code is used to compute thermophysical properties of the working fluids. Complicated algebraic constraints are numerically solved. The model is implemented in Simulink and provides a rudimentary simulation of the time history of important pressures and temperatures during re-pressurization, boost and upper stage firing. The model is validated against an existing reliable code, and typical results are shown.
Burchett, Bradley T.↗
Radiation induced non-linear oscillations in ITER baseline scenario plasmas in DIII-D
Abstract This work shows how the radiation brought about by metals or metal-equivalent radiators such as Kr and Xe produces non-linear dynamics on otherwise stationary β N flattops of DIII-D ITER Baseline Scenario demonstration discharges. The Kr and Xe gases are used to reproduce the radiative loss rates of W in present machines that operate at core temperatures much lower than the expected ITER temperature. Experiments on DIII-D with injection of Kr and Xe, as well as with sources of intrinsic metals reach the range of radiated fraction values expected in the ITER core and experience slow oscillations in temperature and radiated power. In many cases of high radiated fraction, the core temperature decreases enough for the safety factor profile to rise above the 1/1 rational surface, naturally eliminating sawteeth and occasionally producing a persistent helical core. The oscillations can be reproduced by a modified Lotka–Volterra system for temperature and radiated fraction if diffusion and noise are included, which indicates that the interplay between temperature and radiation can be the main cause of the cyclic nature of the system. A new physics based model which includes equations for temperature, density and input power can also reproduce the oscillations observed in the experiments. The present results suggest that the non-linearity of the system can be increased by the inclusion of the inherently non-linear alpha heating term, which is proportional to ∼ n e 2 T i 2 , and obtains oscillations in the model when added to an otherwise more stationary system.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Operator inference for non-intrusive model reduction with quadratic manifolds
This paper proposes a novel approach for learning a data-driven quadratic manifold from high-dimensional data, then employing this quadratic manifold to derive efficient physics-based reduced-order models. The key ingredient of the approach is a polynomial mapping between high-dimensional states and a low-dimensional embedding. This mapping consists of two parts: a representation in a linear subspace (computed in this work using the proper orthogonal decomposition) and a quadratic component. The approach can be viewed as a form of data-driven closure modeling, since the quadratic component introduces directions into the approximation that lie in the orthogonal complement of the linear subspace, but without introducing any additional degrees of freedom to the low-dimensional representation. Combining the quadratic manifold approximation with the operator inference method for projection-based model reduction leads to a scalable non-intrusive approach for learning reduced-order models of dynamical systems. Further, applying the new approach to transport-dominated systems of partial differential equations illustrates the gains in efficiency that can be achieved over approximation in a linear subspace.
42 ENGINEERING↗
Bounds on the Fourier coefficients for the periodic solutions of non-linear oscillator equations
The differential equations describing nonlinear oscillations (as seen in mechanical vibrations, electronic oscillators, chemical and biochemical reactions, acoustic systems, stellar pulsations, etc.) are investigated analytically. The boundedness of the Fourier coefficients for periodic solutions is demonstrated for two special cases, and the extrapolation of the results to higher-dimensionsal systems is briefly considered.
Mickens, R. E.↗
Network-Level Optimization for Unbalanced Power Distribution System: Approximation and Relaxation
The nonlinear programming (NLP) problem to solve distribution-level optimal power flow (D-OPF) poses convergence issues and does not scale well for unbalanced distribution systems. The existing scalable D-OPF algorithms either use approximations that are not valid for an unbalanced power distribution system, or apply relaxation techniques to the nonlinear power flow equations that do not guarantee a feasible power flow solution. In this paper, we propose scalable D-OPF algorithms that simultaneously achieve optimal and feasible solutions by solving multiple iterations of approximate, or relaxed, D-OPF subproblems of low complexity. The first algorithm is based on a successive linear approximation of the nonlinear power flow equations around the current operating point, where the D-OPF solution is obtained by solving multiple iterations of a linear programming (LP) problem. The second algorithm is based on the relaxation of the nonlinear power flow equations as conic constraints together with directional constraints, which achieves optimal and feasible solutions over multiple iterations of a second-order cone programming (SOCP) problem. Finally, it is demonstrated that the proposed algorithms are able to reach an optimal and feasible solution while significantly reducing the computation time as compared to an equivalent NLPD-OPF model for the same distribution system.
24 POWER TRANSMISSION AND DISTRIBUTION↗
The three dimensional motion and stability of a rotating space station-cable-counterweight configuration
The three dimensional equations of motion for a cable connected space station-counterweight system are developed using a Lagrangian formulation. The system model employed allows for cable and end body damping and restoring effects. The equations are then linearized about the equilibrium motion and nondimensionalized. To first degree, the out-of-plane equations uncouple from the in-plane equations. From the general in-plane characteristic equation, necessary conditions for stability are obtained. The Routh-Hurwitz necessary and sufficient conditions for stability are derived for the general out-of-plane characteristic equation. Special cases of the in-plane and out-of-plane equations are then examined for stability. Time constants for the least damped mode are obtained for a range of system parameters by numerical examination of the roots of the in-plane and out-of-plane characteristic polynomials.
Bainum, P. M.↗
Equation modifying program, L219 (EQMOD). Volume 2: Supplemental system design and maintenance document
A digital computer program, L219 (EQMOD), available for execution on the CDC 6600 is described. The program modifies matrices according to card input instructions and prepares magnetic files of matrices suitable for use in the linear systems analysis program (QR) and the random harmonic analysis program L221 (TEV156). The particular field of application of the program is the modification of the theoretical equations of motion and load equations generated in DYLOFLEX by the equation of motion program (L217) and the load equation program (L218), respectively.
Hirayama, M. Y.↗
Linearization of the boundary-layer equations of the minimum time-to-climb problem
Ardema (1974) has formally linearized the two-point boundary value problem arising from a general optimal control problem, and has reviewed the known stability properties of such a linear system. In the present paper, Ardema's results are applied to the minimum time-to-climb problem. The linearized zeroth-order boundary layer equations of the problem are derived and solved.
Ardema, M. D.↗
Bounded Linear Stability Analysis - A Time Delay Margin Estimation Approach for Adaptive Control
This paper presents a method for estimating time delay margin for model-reference adaptive control of systems with almost linear structured uncertainty. The bounded linear stability analysis method seeks to represent the conventional model-reference adaptive law by a locally bounded linear approximation within a small time window using the comparison lemma. The locally bounded linear approximation of the combined adaptive system is cast in a form of an input-time-delay differential equation over a small time window. The time delay margin of this system represents a local stability measure and is computed analytically by a matrix measure method, which provides a simple analytical technique for estimating an upper bound of time delay margin. Based on simulation results for a scalar model-reference adaptive control system, both the bounded linear stability method and the matrix measure method are seen to provide a reasonably accurate and yet not too conservative time delay margin estimation.
Nguyen, Nhan T.↗