Search NASA⌕ Search

SEARCH · Search NASA

Results for “Closed form solution”

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 73 records · Page 4

Collapse of composite tubes under end moments

Cylindrical tubes of moderate wall thickness such as those proposed for the original space station truss, may fail due to the gradual collapse of the tube cross section as it distorts under load. Sometimes referred to as the Brazier instability, it is a nonlinear phenomenon. This paper presents an extension of an approximate closed form solution of the collapse of isotropic tubes subject to end moments developed by Reissner in 1959 to include specially orthotropic material. The closed form solution was verified by an extensive nonlinear finite element analysis of the collapse of long tubes under applied end moments for radius to thickness ratios and composite layups in the range proposed for recent space station truss framework designs. The finite element analysis validated the assumption of inextensional deformation of the cylindrical cross section and the approximation of the material as specially orthotropic.

Stockwell, Alan E.↗

Automatic Generation of Algorithms for the Statistical Analysis of Planetary Nebulae Images

Analyzing data sets collected in experiments or by observations is a Core scientific activity. Typically, experimentd and observational data are &aught with uncertainty, and the analysis is based on a statistical model of the conjectured underlying processes, The large data volumes collected by modern instruments make computer support indispensible for this. Consequently, scientists spend significant amounts of their time with the development and refinement of the data analysis programs. AutoBayes [GF+02, FS03] is a fully automatic synthesis system for generating statistical data analysis programs. Externally, it looks like a compiler: it takes an abstract problem specification and translates it into executable code. Its input is a concise description of a data analysis problem in the form of a statistical model as shown in Figure 1; its output is optimized and fully documented C/C++ code which can be linked dynamically into the Matlab and Octave environments. Internally, however, it is quite different: AutoBayes derives a customized algorithm implementing the given model using a schema-based process, and then further refines and optimizes the algorithm into code. A schema is a parameterized code template with associated semantic constraints which define and restrict the template s applicability. The schema parameters are instantiated in a problem-specific way during synthesis as AutoBayes checks the constraints against the original model or, recursively, against emerging sub-problems. AutoBayes schema library contains problem decomposition operators (which are justified by theorems in a formal logic in the domain of Bayesian networks) as well as machine learning algorithms (e.g., EM, k-Means) and nu- meric optimization methods (e.g., Nelder-Mead simplex, conjugate gradient). AutoBayes augments this schema-based approach by symbolic computation to derive closed-form solutions whenever possible. This is a major advantage over other statistical data analysis systems which use numerical approximations even in cases where closed-form solutions exist. AutoBayes is implemented in Prolog and comprises approximately 75.000 lines of code. In this paper, we take one typical scientific data analysis problem-analyzing planetary nebulae images taken by the Hubble Space Telescope-and show how AutoBayes can be used to automate the implementation of the necessary anal- ysis programs. We initially follow the analysis described by Knuth and Hajian [KHO2] and use AutoBayes to derive code for the published models. We show the details of the code derivation process, including the symbolic computations and automatic integration of library procedures, and compare the results of the automatically generated and manually implemented code. We then go beyond the original analysis and use AutoBayes to derive code for a simple image segmentation procedure based on a mixture model which can be used to automate a manual preproceesing step. Finally, we combine the original approach with the simple segmentation which yields a more detailed analysis. This also demonstrates that AutoBayes makes it easy to combine different aspects of data analysis.

Fischer, Bernd↗

An Approximate Analytical Method for Studying Atmosphere Entry of Vehicles with Modulated Aerodynamic Forces

The dimensionless, transformed, nonlinear differential equation developed in NASA TR R-11 for describing the approximate motion and heating during entry into planetary atmospheres for constant aerodynamic coefficients and vehicle shape has been modified to include entries during which the aerodynamic coefficients and the vehicle shape are varied. The generality of the application of the original equation to vehicles of arbitrary weight, size, and shape and to arbitrary atmospheres is retained. A closed-form solution for the motion, heating, and the variation of drag loading parameter m/C(D)A has been obtained for the case of constant maximum resultant deceleration during nonlifting entries. This solution requires certain simplifying assumptions which do not compromise the accuracy of the results. The closed-form solution has been used to determine the variation of m/C(D)A required to reduce peak decelerations and to broaden the corridor for nonlifting entry into the earth's atmosphere at escape velocity. The attendant heating penalty is also studied.

Levy, Lionel L., Jr.↗

Vibration suppression using a constrained rate-feedback-threshold control strategy

A finite time, minimum force rate-feedback-threshold controller is developed to bring a system with or without known external disturbances back into an 'allowable' state bound in finite time. The disturbances are assumed to be expandable in terms of Fourier series. The optimal control is defined by a two-point boundary value problem coupled to a set of definite integral constraints. Quasi-closed form solutions are derived which replace the solution of the two-point boundary value problem and definite integral constraints with the solution of algebraic equations and the calculation of matrix exponentials. Examples are provided which demonstrate the threshold control technique and compare the quasi-closed form solutions with numerical and MACSYMA generated exact solutions.

Zimmerman, D. C.↗

Elastic-Plastic J-Integral Solutions or Surface Cracks in Tension Using an Interpolation Methodology. Appendix C -- Finite Element Models Solution Database File, Appendix D -- Benchmark Finite Element Models Solution Database File

No closed form solutions exist for the elastic-plastic J-integral for surface cracks due to the nonlinear, three-dimensional nature of the problem. Traditionally, each surface crack must be analyzed with a unique and time-consuming nonlinear finite element analysis. To overcome this shortcoming, the authors have developed and analyzed an array of 600 3D nonlinear finite element models for surface cracks in flat plates under tension loading. The solution space covers a wide range of crack shapes and depths (shape: 0.2 less than or equal to a/c less than or equal to 1, depth: 0.2 less than or equal to a/B less than or equal to 0.8) and material flow properties (elastic modulus-to-yield ratio: 100 less than or equal to E/ys less than or equal to 1,000, and hardening: 3 less than or equal to n less than or equal to 20). The authors have developed a methodology for interpolating between the goemetric and material property variables that allows the user to reliably evaluate the full elastic-plastic J-integral and force versus crack mouth opening displacement solution; thus, a solution can be obtained very rapidly by users without elastic-plastic fracture mechanics modeling experience. Complete solutions for the 600 models and 25 additional benchmark models are provided in tabular format.

Allen, Phillip A.↗

Composite transport wing technology development

The design, fabrication, testing, and analysis of stiffened wing cover panels to assess damage tolerance criteria are discussed. The damage tolerance improvements were demonstrated in a test program using full-sized cover panel subcomponents. The panels utilized a hard skin concept with identical laminates of 44-percent 0-degree, 44-percent plus or minus 45-degree, and 12-percent 90-degree plies in the skins and stiffeners. The panel skins were impacted at midbay between the stiffeners, directly over the stiffener, and over the stiffener flange edge. The stiffener blades were impacted laterally. Impact energy levels of 100 ft-lb and 200 ft-lb were used. NASTRAN finite-element analyses were performed to simulate the nonvisible damage that was detected in the panels by nondestructive inspection. A closed-form solution for generalized loading was developed to evaluate the peel stresses in the bonded structure. Two-dimensional delamination growth analysis was developed using the principle of minimum potential energy in terms of closed-form solution for critical strain. An analysis was conducted to determine the residual compressive stress in the panels after impact damage, and the analytical predictions were verified by compression testing of the damaged panels.

Madan, Ram C.↗

Stability of flow over axisymmetric bodies with porous suction strips

Linear triple deck, closed form solutions for mean-flow quantities are developed for axisymmetric incompressible flow past a body with porous strips. The solutions account for upstream influence and are linear superpositions of the flow past the body without suction plus the perturbations due to the suction strips. Flow past the suctionless body is calculated using the Transition Analysis Program System, and a simple linear optimization scheme to determine number, spacing, and mass flow rate through the strips on an axisymmetric body is developed using the linear, triple-deck, closed-form solutions. The theory is demonstrated by predicting optimal strip distributions, and the effect of various adverse pressure-gradient situations on stability is studied.

Nayfeh, A. H.↗

Influence of Material Distribution on Impact Resistance of Hybrid Composites

Impact events occur in a wide variety of circumstances. A typical example is a bullet impacting a target made of composite material. These impact events produce time-varying loads on a structure that can result in damage. As a first step to understanding the damage resistance issue in composite laminates, an accurate prediction of the transient response during an impact event is necessary. The analysis of dynamic loadings on laminated composite plates has undergone considerable development recently. Rayleigh-Ritz energy method was used to determine the impact response of laminated plates. The impact response of composite plates using shear deformation plate theory was analyzed. In recent work a closed-form solution was obtained for a rectangular plate with four edges simply supported subjected to a center impact load using classical plate theory. The problem was further investigated and the analysis results compared of both classical plate theory and shear deformation theory, and found that classical plate theory predicts very accurate results for the range of small deformations considered. In this study, the influence of cross sectional material distribution on the comparative impact responses of hybrid metal laminates subjected to low and medium velocity impacts is investigated. A simple linear model to evaluate the magnitude of the impact load is proposed first, and it establishes a relation between the impact velocity and the impact force. Then a closed-form solution for impact problem is presented. The results were compared with the finite element analysis results. For an 11 layer-hybrid laminate, the impact response as a function of material distribution in cross-section is presented. With equal areal weight, the effect of the number of laminate layers on the impact resistance is also investigated. Finally, the significance of the presented results is discussed.

Abatan, Ayu↗

Combined structures-controls optimization of lattice trusses

The role that distributed parameter model can play in CSI is demonstrated, in particular in combined structures controls optimization problems of importance in preliminary design. Closed form solutions can be obtained for performance criteria such as rms attitude error, making possible analytical solutions of the optimization problem. This is in contrast to the need for numerical computer solution involving the inversion of large matrices in traditional finite element model (FEM) use. Another advantage of the analytic solution is that it can provide much needed insight into phenomena that can otherwise be obscured or difficult to discern from numerical computer results. As a compromise in level of complexity between a toy lab model and a real space structure, the lattice truss used in the EPS (Earth Pointing Satellite) was chosen. The optimization problem chosen is a generic one: of minimizing the structure mass subject to a specified stability margin and to a specified upper bond on the rms attitude error, using a co-located controller and sensors. Standard FEM treating each bar as a truss element is used, while the continuum model is anisotropic Timoshenko beam model. Performance criteria are derived for each model, except that for the distributed parameter model, explicit closed form solutions was obtained. Numerical results obtained by the two model show complete agreement.

Balakrishnan, A. V.↗

Approximate optimal atmospheric entry trajectories.

For a reentry glider, approximate solutions are found in closed form for the problem of maximizing a function of the terminal velocity, altitude, flight-path angle and heading angle subject to, at most, three terminal nonlinear constraints. The results given here extend the previous results in two important ways: (1) the second-order approximation of the entry dynamics of Loh is used instead of the first-order approximation of Allen and Eggers. This approximation is found to compare extremely well with an exact numerical optimal path. (2) A three-dimensional optimization problem is solved which includes Loh's second-order approximation where the roll angle as well as the lift coefficient are determined subject to constraints which include the terminal heading angle. Furthermore, closed-form solutions can also be obtained subject to in-flight constraints on the lift coefficient and roll angle.

Speyer, J. L.↗

A closed-form trim solution yielding minimum trim drag for airplanes with multiple longitudinal-control effectors

Airplane designs are currently being proposed with a multitude of lifting and control devices. Because of the redundancy in ways to generate moments and forces, there are a variety of strategies for trimming each airplane. A linear optimum trim solution (LOTS) is derived using a Lagrange formulation. LOTS enables the rapid calculation of the longitudinal load distribution resulting in the minimum trim drag in level, steady-state flight for airplanes with a mixture of three or more aerodynamic surfaces and propulsive control effectors. Comparisons of the trim drags obtained using LOTS, a direct constrained optimization method, and several ad hoc methods are presented for vortex-lattice representations of a three-surface airplane and two-surface airplane with thrust vectoring. These comparisons show that LOTS accurately predicts the results obtained from the nonlinear optimization and that the optimum methods result in trim drag reductions of up to 80 percent compared to the ad hoc methods.

Goodrich, Kenneth H.↗

An analytic solution to the steady-state double adiabatic equations

A set of 20 generalized moment-transport equations applicable to low-beta (low random energy density/magnetic energy density) plasmas of thermal origin in rotating planetary magnetospheres. An analytic solution is obtained for a set of simplified equations characterizing the steady-state transport of collisionless plasma; the novel element of this analytic solution is a closed-form solution for the parallel-flow velocity variation along magnetic field lines. It is shown that only T(perpendicular)/B remains constant along the field line, while density, parallel temperature, and parallel Mach number vary.

Gombosi, Tamas I.↗

The double universal joint wrist on a manipulator: Solution of inverse position kinematics and singularity analysis

This paper presents three methods to solve the inverse position kinematics position problem of the double universal joint attached to a manipulator: (1) an analytical solution for two specific cases; (2) an approximate closed form solution based on ignoring the wrist offset; and (3) an iterative method which repeats closed form position and orientation calculations until the solution is achieved. Several manipulators are used to demonstrate the solution methods: cartesian, cylindrical, spherical, and an anthropomorphic articulated arm, based on the Flight Telerobotic Servicer (FTS) arm. A singularity analysis is presented for the double universal joint wrist attached to the above manipulator arms. While the double universal joint wrist standing alone is singularity-free in orientation, the singularity analysis indicates the presence of coupled position/orientation singularities of the spherical and articulated manipulators with the wrist. The cartesian and cylindrical manipulators with the double universal joint wrist were found to be singularity-free. The methods of this paper can be implemented in a real-time controller for manipulators with the double universal joint wrist. Such mechanically dextrous systems could be used in telerobotic and industrial applications, but further work is required to avoid the singularities.

Williams, Robert L., III↗

Distributed parameter estimation for NASA Mini-Mast truss through displacement measurements

Most methods of system identification of large flexible structures by far are based on the lumped parameter approach. Because of the considerable computational burden due to the large number of unknown parameters, distributed parameter approach, which greatly decreases the number of unknowns, has being investigated. In this paper a distributed parameter model for the estimation of modal characteristics of NASA Mini-Mast truss has been formulated. Both Bernoulli-Euler beam and Timoshenko beam equations are used to characterize the lateral bending vibrations of the truss. The measurement of the lateral displacement at the tip of the truss is provided to the maximum likelihood estimator. Closed-form solutions of the partial differential equations and closed-form expressions of the sensitivity functions are derived so that the estimation algorithm is highly efficient. The resulting estimates from test data by using Timoshenko beam model are found to be comparable to those derived from finite element analysis.

Huang, Jen-Kuang↗

Structure of steady state accretion shocks with several cooling functions: Closed integral-form solution

We present, for the first time, a closed integral-form solution to the accretion shock structures for the case where the cooling is due to optically thin bremsstrahlung emission and a series of power-law cooling functions of density and temperature. Our results can provide useful checks on numerical calculations and simple accurate estimates for valuable parameters such as the shock height. For the case where the cooling rate j = (2/3)Arho(exp 2)(P/rho)(exp 1/2)(1 + epsilon (sub s)(P/P(sub s)(exp alpha)(rho(sub s)/rho)(exp beta)), we find that a substantial amount of the accretion energy is released at the base of the accretion shock in the form of bremsstrahlung radiation. This implies that for a cyclotron-dominated shock (qualitatively given by alpha = 2.0, beta = 3.85, and epsilon(sub s) is much greater than 1), bremsstrahlung cooling still plays a crucial role in determining the shock structure. Our results are shown to be consistent with detailed numerical calculations.

Wu, Kinwah↗

Solution to the backward-Kolmogorov equation for a nonstationary oscillation problem

The transition probability density function of a Markovian approximation of the response amplitude of an oscillator under nonstationary excitation is determined in an analytical form. A solution is presented for the associated backward-Kolmogorov equation by transforming the equation into a form amenable to solution by the method of the separation of variables. This procedure results in a boundary value problem which is then solved by using an infinite series of Laguerre polynomials. It is found that the infinite series solution is equivalent to a closed-form solution involving a Bessel function.

Solomos, G. P.↗

First-order corrections to approximate solutions to two-point boundary-condition problems

A method, applicable to real time guidance, is developed for accurate solution to exo-atmospheric space flight optimization problems. In principle the method is applicable to many other two-point boundary-condition (TPBC) problems. The first step of the method is the iterative solution (using a shooting method) of a TPBC problem with differential equations simplified so that they may be solved analytically by means of a single closed-form solution over each stage of the flight. The second step is the addition of a closed-form correction to the solution to the TPBC problem obtained in the first step. The correction accounts (to first-order accuracy) for the errors due to the aforementioned simplifications. Numerical results are given for several orbital injection problems.

Andrus, J. F.↗