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Results for “Non-linear ordinary differential equations”

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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All sourcesNASA NTRSDOE OSTI
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 NTRS2008

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.↗
NASA NTRS1993

Dynamics and control of coherent structures in the turbulent wall layer: An overview

We expand the velocity field in the vicinity of the wall in empirical eigenfunctions obtained from experiment. Truncating our system and using Galerkin projection, we obtain a closed set of non-linear ordinary differential equations with ten degrees of freedom. We find a rich dynamical behavior, including in particular a heteroclinic attracting orbit giving rise to intermittency. The intermittent jump from one attracting point to the other resembles in many respects the bursts observed in experiments. Specifically, the time between jumps and the duration of the jumps, is approximately that observed in a burst; the jump begins with the formation of a narrowed and intensified updraft, like the ejection phase of a burst, and is followed by a gentle, diffuse downdraft like the sweep phase of a burst. The magnitude of the Reynolds stress spike produced during a burst is limited by our truncation. The behavior is quite robust, much of it being due to the symmetries present (Aubry's group has examined dimensions up to 128 with persistence of the global behavior). We have examined eigenvalues and coefficients obtained from experiment, and from exact simulation, which differ in magnitude. Similar behavior is obtained in both cases; in the latter case, the heteroclinic orbits connect limit cycles instead of fixed points, corresponding to cross-stream waving of the streamwise rolls. The bifurcation diagram remains structurally similar, but somewhat distorted. The role of the pressure term is made clear - it triggers the intermittent jumps, which otherwise would occur at longer and longer intervals, as the system trajectory is attracted closer and closer to the heteroclinic cycle. The pressure term results in the jumps occurring at essentially random times, and the magnitude of the signal determines the average timing. Stretching of the wall region shows that the model is consistent with observations of polymer drag reduction. Change of the third order coefficients, corresponding to acceleration or deceleration of the mean flow, changes the heteroclinic cycles from attracting to repelling, increasing or decreasing the stability, in agreement with observations. The existence of fixed points is an artifact introduced by the projection; however, a decoupled model still displays the rich dynamics. Numerous assumptions made in Aubry et al. (1988) can now be proved exactly. Feeding back eigenfuncitons with the proper phase can delay the bursting, (the heteroclinic jump to the other fixed point), decreasing the drag. It is also possible to speed up the bursting, increasing mixing to control separation. Our approach is optimal for short time tracking in control.

Berkooz, Gal↗
NASA NTRS2004

Continuation Methods for Qualitative Analysis of Aircraft Dynamics

A class of numerical methods for constructing bifurcation curves for systems of coupled, non-linear ordinary differential equations is presented. Foundations are discussed, and several variations are outlined along with their respective capabilities. Appropriate background material from dynamical systems theory is presented.

Cummings, Peter A.↗
NASA NTRS2007

Biochemical Kinetics Model of DSB Repair and GammaH2AX FOCI by Non-homologous End Joining

We developed a biochemical kinetics approach to describe the repair of double strand breaks (DSB) produced by low LET radiation by modeling molecular events associated with the mechanisms of non-homologous end-joining (NHEJ). A system of coupled non-linear ordinary differential equations describes the induction of DSB and activation pathways for major NHEJ components including Ku(sub 70/80), DNA-PK(sub cs), and the Ligase IV-XRCC4 hetero-dimer. The autophosphorylation of DNA-PK(sub cs and subsequent induction of gamma-H2AX foci observed after ionizing radiation exposure were modeled. A two-step model of DNA-PK(sub cs) regulation of repair was developed with the initial step allowing access of other NHEJ components to breaks, and a second step limiting access to Ligase IV-XRCC4. Our model assumes that the transition from the first to second-step depends on DSB complexity, with a much slower-rate for complex DSB. The model faithfully reproduced several experimental data sets, including DSB rejoining as measured by pulsed-field electrophoresis (PFGE), quantification of the induction of gamma-H2AX foci, and live cell imaging of the induction of Ku(sub 70/80). Predictions are made for the behaviors of NHEJ components at low doses and dose-rates, where a steady-state is found at dose-rates of 0.1 Gy/hr or lower.

Cucinotta, Francis, A.↗
NASA NTRS2012

Droplet Deformation Prediction With the Droplet Deformation and Breakup Model (DDB)

The Droplet Deformation and Breakup Model was used to predict deformation of droplets approaching the leading edge stagnation line of an airfoil. The quasi-steady model was solved for each position along the droplet path. A program was developed to solve the non-linear, second order, ordinary differential equation that governs the model. A fourth order Runge-Kutta method was used to solve the equation. Experimental slip velocities from droplet breakup studies were used as input to the model which required slip velocity along the particle path. The center of mass displacement predictions were compared to the experimental measurements from the droplet breakup studies for droplets with radii in the range of 200 to 700 mm approaching the airfoil at 50 and 90 m/sec. The model predictions were good for the displacement of the center of mass for small and medium sized droplets. For larger droplets the model predictions did not agree with the experimental results.

Vargas, Mario↗
NASA NTRS2012

Droplet Deformation Prediction with the Droplet Deormation and Break Up Model (DDB)

The Droplet Deformation and Breakup Model was used to predict deformation of droplets approaching the leading edge stagnation line of an airfoil. The quasi-steady model was solved for each position along the droplet path. A program was developed to solve the non-linear, second order, ordinary differential equation that governs the model. A fourth order Runge-Kutta method was used to solve the equation. Experimental slip velocities from droplet breakup studies were used as input to the model which required slip velocity along the particle path. The center of mass displacement predictions were compared to the experimental measurements from the droplet breakup studies for droplets with radii in the range of 200 to 700 mm approaching the airfoil at 50 and 90 m/sec. The model predictions were good for the displacement of the center of mass for small and medium sized droplets. For larger droplets the model predictions did not agree with the experimental results.

Vargas, Mario↗
NASA NTRS1983

Transient responses of phosphoric acid fuel cell power plant system

An analytical and computerized study of the steady state and transient response of a phosphoric acid fuel cell (PAFC) system was completed. Parametric studies and sensitivity analyses of the PAFC system's operation were accomplished. Four non-linear dynamic models of the fuel cell stack, reformer, shift converters, and heat exchangers were developed based on nonhomogeneous non-linear partial differential equations, which include the material, component, energy balance, and electrochemical kinetic features. Due to a lack of experimental data for the dynamic response of the components only the steady state results were compared with data from other sources, indicating reasonably good agreement. A steady state simulation of the entire system was developed using, nonlinear ordinary differential equations. The finite difference method and trial-and-error procedures were used to obtain a solution. Using the model, a PAFC system, that was developed under NASA Grant, NCC3-17, was improved through the optimization of the heat exchanger network. Three types of cooling configurations for cell plates were evaluated to obtain the best current density and temperature distributions. The steady state solutions were used as the initial conditions in the dynamic model. The transient response of a simplified PAFC system, which included all of the major components, subjected to a load change was obtained. Due to the length of the computation time for the transient response calculations, analysis on a real-time computer was not possible. A simulation of the real-time calculations was developed on a batch type computer. The transient response characteristics are needed for the optimization of the design and control of the whole PAFC system. All of the models, procedures and simulations were programmed in Fortran and run on IBM 370 computers at Cleveland State University and the NASA Lewis Research Center.

Lu, Cheng-Yi↗
NASA NTRS2004

Variable Step Integration Coupled with the Method of Characteristics Solution for Water-Hammer Analysis, A Case Study

One-dimensional water-hammer modeling involves the solution of two coupled non-linear hyperbolic partial differential equations (PDEs). These equations result from applying the principles of conservation of mass and momentum to flow through a pipe, and usually the assumption that the speed at which pressure waves propagate through the pipe is constant. In order to solve these equations for the interested quantities (i.e. pressures and flow rates), they must first be converted to a system of ordinary differential equations (ODEs) by either approximating the spatial derivative terms with numerical techniques or using the Method of Characteristics (MOC). The MOC approach is ideal in that no numerical approximation errors are introduced in converting the original system of PDEs into an equivalent system of ODEs. Unfortunately this resulting system of ODEs is bound by a time step constraint so that when integrating the equations the solution can only be obtained at fixed time intervals. If the fluid system to be modeled also contains dynamic components (i.e. components that are best modeled by a system of ODEs), it may be necessary to take extremely small time steps during certain points of the model simulation in order to achieve stability and/or accuracy in the solution. Coupled together, the fixed time step constraint invoked by the MOC, and the occasional need for extremely small time steps in order to obtain stability and/or accuracy, can greatly increase simulation run times. As one solution to this problem, a method for combining variable step integration (VSI) algorithms with the MOC was developed for modeling water-hammer in systems with highly dynamic components. A case study is presented in which reverse flow through a dual-flapper check valve introduces a water-hammer event. The predicted pressure responses upstream of the check-valve are compared with test data.

Turpin, Jason B.↗
NASA NTRS1992

Combustion noise and combustion instabilities in propulsion systems

This paper is concerned with some aspects of non-linear behavior of unsteady motions in combustion chambers. The emphasis is on conditions under which organized oscillations having discrete frequencies may exist in the presence of random motions. In order to treat the two types of motions together, and particularly to investigate coupling between noise and combustion instabilities, the unsteady field is represented as a synthesis of acoustic modes having time-varying amplitudes. Each of the amplitudes are written as the sum of two parts, one associated with the random field and the remainder representing the organized oscillations. After spatial averaging, the general problem is reduced to solution of a set of second-order ordinary differential equations whose structure depends on the sorts of nonlinear processes accounted for. This formulation accommodates any physical process; in particular, terms are included to represent noise sources, although only limited modeling is discussed. Our results suggest that random sources of noise have only small effects on combustion instabilities and seem not to be a cause of unstable motions. However, the coupling between the two sorts of unsteady motions may be important as an essential process in a proposed scheme for noise control. It is now a familiar observation that many nonlinear deterministic systems are capable of exhibiting apparently random motions called 'chaos.' This is a particularly interesting possibility for systems which also executed non-deterministic random motions. In combustion chambers, a nonlinear deterministic system (acoustical motions) exists in the presence of noise produced by flow separation, turbulent motions, and energy released by combustion processes. The last part of the paper is directed to the matter of discovering whether or not chaotic motions exist in combustion systems. Analysis has not progressed sufficiently far to answer the question. We report here recent results of processing data taken in one combustor to determine the dimensions of any attractors in the motions. No evidence has been found for chaos in the strict sense, but the method seems to be an important means of investigating the nonlinear behavior of combustion systems.

F. E. C. Culick↗
NASA NTRS1978

Investigation of ODE integrators using interactive graphics

Two FORTRAN programs using an interactive graphic terminal to generate accuracy and stability plots for given multistep ordinary differential equation (ODE) integrators are described. The first treats the fixed stepsize linear case with complex variable solutions, and generates plots to show accuracy and error response to step driving function of a numerical solution, as well as the linear stability region. The second generates an analog to the stability region for classes of non-linear ODE's as well as accuracy plots. Both systems can compute method coefficients from a simple specification of the method. Example plots are given.

Brown, R. L.↗