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At least 469 records · Page 26
All sourcesNASA NTRSDOE OSTI
NASA NTRS1992

Stability of a class of interconnected evolution systems

Stability conditions for a class of interconnected systems modeled by linear abstract evolution equations and a memoryless nonlinearity are derived. These conditions are stated in terms of the passivity of each of the subsystems and can be considered as a partial generalization of the hyperstability theorem. A Liapunov function approach is used in the proof without requiring the positive definiteness of the Liapunov function. Application to the robustness analysis of the infinite-dimensional linear quadratic regulator is also discussed.

Wen, John T.↗
NASA NTRS1993

Euler solutions for blunt bodies using triangular meshes - Artificial viscosity forms and numerical boundary conditions

A finite volume method is used to calculate compressible inviscid flows over blunt bodies using, in general, unstructured grids. Artificial viscosity forms are derived based on a simplified least squares procedure. The extra second order terms are consistent with the governing equations, hence a systematic treatment of the numerical boundary conditions can be easily implemented. A special treatment of blunt bodies may be required. The discrete equations are linearized and the resulting system is solved by a relaxation method. Preliminary results indicate that the effect of the numerical dissipation is minimal. For subsonic flows over smooth bodies, the solution is practically vorticity-free and the total pressure loss is of the same order as the truncation error. Finally, some extensions of the present method are briefly discussed.

Winterstein, R.↗
NASA NTRS1994

Dynamic analysis of Space Shuttle/RMS configuration using continuum approach

The initial assembly of Space Station Freedom involves the Space Shuttle, its Remote Manipulation System (RMS) and the evolving Space Station Freedom. The dynamics of this coupled system involves both the structural and the control system dynamics of each of these components. The modeling and analysis of such an assembly is made even more formidable by kinematic and joint nonlinearities. The current practice of modeling such flexible structures is to use finite element modeling in which the mass and interior dynamics is ignored between thousands of nodes, for each major component. The model characteristics of only tens of modes are kept out of thousands which are calculated. The components are then connected by approximating the boundary conditions and inserting the control system dynamics. In this paper continuum models are used instead of finite element models because of the improved accuracy, reduced number of model parameters, the avoidance of model order reduction, and the ability to represent the structural and control system dynamics in the same system of equations. Dynamic analysis of linear versions of the model is performed and compared with finite element model results. Additionally, the transfer matrix to continuum modeling is presented.

Ramakrishnan, Jayant↗
NASA NTRS2003

Preliminary Planar Formation: Flight Dynamics Near Sun-Earth L2 Point

NASA's Goddard Space Flight Center is planning a series of missions in the vicinity of the Sun-Earth L2 libration point. Some of these projects will involve a distributed space system of telescope spacecraft acting together as a single telescope for high-resolution. The individual telescopes will be configured in a plane, surrounding a hub, where the telescope plane can be aimed toward various astronomical targets of interest. In preparation for these missions, it is necessary to develop an improved understanding of the dynamical behavior of objects in a planar configuration near L2. The classical circular restricted three body problem is taken as the basis for the analysis. At first order, the motion of such a telescope relative to the hub is described by a system of linear second order differential equations. These equations are identical to the circular restricted problem's linear equations describing the hub motion about L2. Therefore, the fundamental frequencies, both parallel to and normal to the ecliptic plane, are the same for the relative telescope motion as for the hub motion. To maintain the telescope plane for the duration necessary for the planned observations, a halo-type orbit of the telescopes about the hub is investigated. By using a halo orbit, the individual telescopes remain in approximately the same plane over the observation duration. For such an orbit, the fundamental periods parallel to and normal to the ecliptic plane are forced to be the same by careful selection of the initial conditions in order to adjust the higher order forces. The relative amplitudes of the resulting oscillations are associated with the orientation of the telescope plane relative to the ecliptic. As in the circular restricted problem, initial conditions for the linearized equations must be selected so as not to excite the convergent or divergent linear modes. In a higher order analysis, the telescope relative motion equations include the effects of the position of the hub relative to L2. In this paper, the differential equations are developed through second order in the distance of the hub from the libration point. A modified Lindstedt-Poincad perturbation method is employed to construct the solution of these differential equations through that same order of magnitude. In the course of the solution process, relationships are determined between the initial conditions of the telescopes, selected in order to avoid resonance excitation. As the differential equations include the hub position, it is necessary to simultaneously develop the solution for the hub. As has been done in past analyses of the circular restricted problem, the hub position is written in a power series formulation in terms of its distance from L2. Then, in order to be included in the telescope equations, the hub solution is cast in terms of the nonlinear frequency of the relative telescope motion. In the course of the analysis, it is determined that the hub should also maintain a halo orbit - about L2. Additionally, relationships are formed between the initial conditions of the telescopes and the hub. These relationships may be used to associate sets of initial conditions with particular orientations of the telescope plane. The accuracy of the analytical solution is verified through various simulations and comparison to numerical integration of the differential equations. The results of the simulations are presented, along with a graphical representation of the relationships between the initial conditions of the telescopes and hub.

Segerman, Alan M.↗
NASA NTRS1993

A least-squares finite element method for 3D incompressible Navier-Stokes equations

The least-squares finite element method (LSFEM) based on the velocity-pressure-vorticity formulation is applied to three-dimensional steady incompressible Navier-Stokes problems. This method can accommodate equal-order interpolations, and results in symmetric, positive definite algebraic system. An additional compatibility equation, i.e., the divergence of vorticity vector should be zero, is included to make the first-order system elliptic. The Newton's method is employed to linearize the partial differential equations, the LSFEM is used to obtain discretized equations, and the system of algebraic equations is solved using the Jacobi preconditioned conjugate gradient method which avoids formation of either element or global matrices (matrix-free) to achieve high efficiency. The flow in a half of 3D cubic cavity is calculated at Re = 100, 400, and 1,000 with 50 x 52 x 25 trilinear elements. The Taylor-Gortler-like vortices are observed at Re = 1,000.

Jiang, Bo-Nan↗
DOE OSTI2026

Block encoding of the three-dimensional heterogeneous Poisson equation with application to fracture flow

Quantum linear system (QLS) algorithms offer the potential to solve large-scale linear systems exponentially faster than classical methods. However, applying QLS algorithms to real-world problems remains challenging due to issues such as state preparation, data loading, and efficient information extraction. In this work, we study the feasibility of applying QLS algorithms to solve discretized three-dimensional (3D) heterogeneous Poisson equations, with specific examples relating to groundwater flow through geologic fracture networks. We explicitly construct a block encoding for the 3D heterogeneous Poisson matrix by leveraging the sparse local structure of the discretized operator. While classical solvers benefit from preconditioning, we show that block encoding the system matrix and preconditioner separately does not improve the effective condition number that dominates the QLS run-time. This differs from classical approaches where the preconditioner and the system matrix can often be implemented independently. Nevertheless, due to the structure of the problem in three dimensions, the quantum algorithm achieves a run-time of 𝑂⁡(𝑁 2/3 polylog 𝑁 ⋅log (1/𝜖)), outperforming the best classical methods (with run times of 𝑂⁡(𝑁⁢log 𝑁 ⋅log (1/𝜖))) and offering exponential memory savings. These results highlight both the promise and limitations of QLS algorithms for practical scientific computing, and point to effective condition-number reduction as a key barrier in achieving quantum advantages.

58 GEOSCIENCES↗
NASA NTRS1983

Identification and stochastic control of helicopter dynamic modes

A general treatment of parameter identification and stochastic control for use on helicopter dynamic systems is presented. Rotor dynamic models, including specific applications to rotor blade flapping and the helicopter ground resonance problem are emphasized. Dynamic systems which are governed by periodic coefficients as well as constant coefficient models are addressed. The dynamic systems are modeled by linear state variable equations which are used in the identification and stochastic control formulation. The pure identification problem as well as the stochastic control problem which includes combined identification and control for dynamic systems is addressed. The stochastic control problem includes the effect of parameter uncertainty on the solution and the concept of learning and how this is affected by the control's duel effect. The identification formulation requires algorithms suitable for on line use and thus recursive identification algorithms are considered. The applications presented use the recursive extended kalman filter for parameter identification which has excellent convergence for systems without process noise.

Molusis, J. A.↗
NASA NTRS1989

Recursive linearization of multibody dynamics equations of motion

The equations of motion of a multibody system are nonlinear in nature, and thus pose a difficult problem in linear control design. One approach is to have a first-order approximation through the numerical perturbations at a given configuration, and to design a control law based on the linearized model. Here, a linearized model is generated analytically by following the footsteps of the recursive derivation of the equations of motion. The equations of motion are first written in a Newton-Euler form, which is systematic and easy to construct; then, they are transformed into a relative coordinate representation, which is more efficient in computation. A new computational method for linearization is obtained by applying a series of first-order analytical approximations to the recursive kinematic relationships. The method has proved to be computationally more efficient because of its recursive nature. It has also turned out to be more accurate because of the fact that analytical perturbation circumvents numerical differentiation and other associated numerical operations that may accumulate computational error, thus requiring only analytical operations of matrices and vectors. The power of the proposed linearization algorithm is demonstrated, in comparison to a numerical perturbation method, with a two-link manipulator and a seven degrees of freedom robotic manipulator. Its application to control design is also demonstrated.

Lin, Tsung-Chieh↗
NASA NTRS1985

Variation in efficiency of parallel algorithms

The present study has the objective to investigate some iterative parallel-processor linear equation solving algorithms with respect to efficiency for analyses of typical linear engineering systems. Attention is given to a set of n linear equations, Ku = p, where K = an n x n positive definite, sparsely populated, symmetric matrix, u = an n x 1 vector of unknown responses, and p = an n x 1 vector of prescribed constants. This study is concerned with a hybrid method in which iteration is used to solve the problem, while a direct method is used on the local processor level. Variations in the efficiency of parallel algorithms are explored. Measures of the efficiency are based on computer experiments regarding the algorithms. For all the algorithms, the wall clock time is found to decrease as the number of processors increases.

Hayashi, A.↗
NASA NTRS1991

Least-squares solution of incompressible Navier-Stokes equations with the p-version of finite elements

A p-version of the least squares finite element method, based on the velocity-pressure-vorticity formulation, is developed for solving steady state incompressible viscous flow problems. The resulting system of symmetric and positive definite linear equations can be solved satisfactorily with the conjugate gradient method. In conjunction with the use of rapid operator application which avoids the formation of either element of global matrices, it is possible to achieve a highly compact and efficient solution scheme for the incompressible Navier-Stokes equations. Numerical results are presented for two-dimensional flow over a backward facing step. The effectiveness of simple outflow boundary conditions is also demonstrated.

Jiang, Bo-Nan↗
NASA NTRS2001

Multi-Tethered Space-Based Interferometers: Particle System Model

Dynamics models are presented for a class of space-based interferometers comprised of multiple component bodies, interconnected in various arrangements, by low-mass flexible tethers of variable length. The tethered constellations are to perform coordinated rotational scanning accompanied by baseline dimensional changes, as well as spin axis realignments and spin-up/spin-down maneuvers. The mechanical idealization is a system of N point masses interconnected by massless tethers of variable length. Both extensible and inextensible tethers are considered. Expressions for system angular and linear momenta are developed. The unrestricted nonlinear motion equations are derived via Lagranges equations. Rheonomic constraints are introduced to allow prescribed motion of any degrees of freedom, and the associated physical forces are determined. The linearized equations of motion are obtained for the steady rotation of a system with extensible tethers of constant unstrained length.

Gates, Stephen S.↗
NASA NTRS2007

Newton-Euler Dynamic Equations of Motion for a Multi-body Spacecraft

The Magnetospheric MultiScale (MMS) mission employs a formation of spinning spacecraft with several flexible appendages and thruster-based control. To understand the complex dynamic interaction of thruster actuation, appendage motion, and spin dynamics, each spacecraft is modeled as a tree of rigid bodies connected by spherical or gimballed joints. The method presented facilitates assembling by inspection the exact, nonlinear dynamic equations of motion for a multibody spacecraft suitable for solution by numerical integration. The building block equations are derived by applying Newton's and Euler's equations of motion to an "element" consisting of two bodies and one joint (spherical and gimballed joints are considered separately). Patterns in the "mass" and L'force" matrices guide assembly by inspection of a general N-body tree-topology system. Straightforward linear algebra operations are employed to eliminate extraneous constraint equations, resulting in a minimum-dimension system of equations to solve. This method thus combines a straightforward, easily-extendable, easily-mechanized formulation with an efficient computer implementation.

Stoneking, Eric↗
NASA NTRS1994

Modal decomposition of Hamiltonian variational equations

Over any finite arc of trajectory, the variational equations of a Hamiltonian system can be separated into 'normal' modes. This transformation is canonical, and the Lyapunov exponents over the trajectory arc occur as positive/negative pairs for conjugate modes, while the modal vectors remain unit vectors. This decomposition effectively solves the variational equations for any canonical, linear-dependent system. As an example, we study the Voyager I trajectory. In an interplanetary flyby, some of the modal variables increase by very large multiplicative factors, but this means that their conjugate modal variables decrease by those same very large multiplicative vectors. Maneuver strategies for this case are explored, and the minimum delta upsilon maneuver is found.

Wiesel, William E.↗
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 NTRS1990

Computational methods for probability of instability calculations

This paper summarizes the development of the methods and a computer program to compute the probability of instability of a dynamic system than can be represented by a system of second-order ordinary linear differential equations. Two instability criteria based upon the roots of the characteristics equation or Routh-Hurwitz test functions are investigated. Computational methods based on system reliability analysis methods and importance sampling concepts are proposed to perform efficient probabilistic analysis. Numerical examples are provided to demonstrate the methods.

Wu, Y.-T.↗
NASA NTRS1983

Proposed solution methodology for the dynamically coupled nonlinear geared rotor mechanics equations

The equations which describe the three-dimensional motion of an unbalanced rigid disk in a shaft system are nonlinear and contain dynamic-coupling terms. Traditionally, investigators have used an order analysis to justify ignoring the nonlinear terms in the equations of motion, producing a set of linear equations. This paper will show that, when gears are included in such a rotor system, the nonlinear dynamic-coupling terms are potentially as large as the linear terms. Because of this, one must attempt to solve the nonlinear rotor mechanics equations. A solution methodology is investigated to obtain approximate steady-state solutions to these equations. As an example of the use of the technique, a simpler set of equations is solved and the results compared to numerical simulations. These equations represent the forced, steady-state response of a spring-supported pendulum. These equations were chosen because they contain the type of nonlinear terms found in the dynamically-coupled nonlinear rotor equations. The numerical simulations indicate this method is reasonably accurate even when the nonlinearities are large.

Mitchell, L. D.↗
NASA NTRS1989

Implementation of an algorithm for cylindrical object identification using range data

One of the problems in 3-D object identification and localization is addressed. In robotic and navigation applications the vision system must be able to distinguish cylindrical or spherical objects as well as those of other geometric shapes. An algorithm was developed to identify cylindrical objects in an image when range data is used. The algorithm incorporates the Hough transform for line detection using edge points which emerge from a Sobel mask. Slices of the data are examined to locate arcs of circles using the normal equations of an over-determined linear system. Current efforts are devoted to testing the computer implementation of the algorithm. Refinements are expected to continue in order to accommodate cylinders in various positions. A technique is sought which is robust in the presence of noise and partial occlusions.

Bozeman, Sylvia T.↗
NASA NTRS1975

A lunar soil evolution model

Comminution, agglutination, and replenishment processes in a lunar soil are modeled by a system of time-dependent linear differential equations. In the model, a soil is subdivided into coarse-particle, fine-particle, and agglutinate fractions. The relative mass abundance of each component in a mature soil is found to be proportional to rates for the reworking processes. Evolution of the grain-size distribution from a fresh ejecta blanket to a mature soil is described quantitatively in terms of the changing proportions of the three soil constituents. If size data are available for an immature soil and a mature soil of the same system, rates for the various processes can be calculated under certain simplifying assumptions.

Mendell, W. W.↗