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Methods are given for computing the covariance matrix for the solution vector of an equality-constrained least squares problem. The methods are matched to the solution algorithms given in the book, 'Solving Least Squares Problems.'
Experimental results of the mechanical behavior of two metal matrix composite systems at room temperature are presented. Ultimate stress, ultimate strain, Poisson's ratio, and initial Young's Modulus are documented for BORSIC/Aluminum in uniaxial tension and Boron/Aluminum in uniaxial tension and compression. Poisson's ratio is used for nonlinear stress-strain behavior. A comparison of compression results for B/Al as obtained from sandwich beam compression specimens and IITRI coupon compression specimens is presented.
The three turbulent models currently available in the JANNAF version of the Aerotherm Boundary Layer Integral Matrix Procedure (BLIMP-J) code were studied. The BLIMP-J program is the standard prediction method for boundary layer effects in liquid rocket engine thrust chambers. Experimental data from flow fields with large edge-to-wall temperature ratios are compared to the predictions of the three turbulence models contained in BLIMP-J. In addition, test conditions necessary to generate additional data on a flat plate or in a nozzle are given. It is concluded that the Cebeci-Smith turbulence model be the recommended model for the prediction of boundary layer effects in liquid rocket engines. In addition, the effects of homogeneous chemical reaction kinetics were examined for a hydrogen/oxygen system. Results show that for most flows, kinetics are probably only significant for stoichiometric mixture ratios.
A study is done of solution methods for Linear Matrix Equations including Lyapunov's equation, using methods of modern algebra. The emphasis is on the use of finite algebraic procedures which are easily implemented on a digital computer and which lead to an explicit solution to the problem. The action f sub BA is introduced a Basic Lemma is proven. The equation PA + BP = -C as well as the Lyapunov equation are analyzed. Algorithms are given for the solution of the Lyapunov and comment is given on its arithmetic complexity. The equation P - A'PA = Q is studied and numerical examples are given.
A computer algorithm for extracting a quaternion from a direction-cosine matrix (DCM) is described. The quaternion provides a four-parameter representation of rotation, as against the nine-parameter representation afforded by a DCM. Commanded attitude in space shuttle steering is conveniently computed by DCM, while actual attitude is computed most compactly as a quaternion, as is attitude error. The unit length of the rotation quaternion, and interchangeable of a quaternion and its negative, are used to advantage in the extraction algorithm. Protection of the algorithm against square root failure and division overflow are considered. Necessary and sufficient conditions for handling the rotation vector element of largest magnitude are discussed
The electric dipole moment function of the ground electronic state of carbon monoxide has been determined by combining numerical solutions of the radial Schrodinger equation with absolute intensity data of vibration-rotation bands. The derived dipole moment function is used to calculate matrix elements of interest to stellar astronomy and of importance in the carbon monoxide laser.
Webeler's (1976) experimental results for the generation of atomic hydrogen in a hydrogen matrix by tritium decay are reexamined with a variant of Rosen's (1976) mathematical treatment. The analysis retains Rosen's equations for the number densities of trapped and mobile hydrogen atoms, but replaces his enthalpy equation with an equation for the directly measured temperature. Theoretical expressions are derived for the dependence of storage time, recombination time, and maximum density of trapped hydrogen atoms as a function of temperature for a given tritium concentration. A comparison of predictions for the maximum trapped atomic hydrogen number density as a function of storage time reveals that Rosen's estimate for the maximum number density of hydrogen atoms for the zero magnetic field case is a little more optimistic than the estimate obtained in the paper.
A scanning electron microscope was used in a petrographic investigation of the matrix textures of 41 lunar breccias ranging from very friable soil clods through coherent microbreccias and tough vitric breccias to tough, fine-grained crystalline breccias. It was found that as their coherence increases, the matrices display a gradual increase in the content of glass from 1 or 2% as filaments less than 1 micron across through 5-50% as irregularly shaped patches up to 200 microns across to over 50% as continuous networks.
A directed matrix abstraction procedure is described which allows a static solution to be obtained in phase 1 of a three phase NASTRAN substructuring analysis while at the same time storing all information needed to complete the substructuring analysis.
Nonlinear matrix inversion operators have been developed which, applied to observed radiances, infer maximal information regarding atmospheric scattering parameters and vertical distribution of radiant sources and sinks. The algorithm has the attractive feature of noise discrimination, attributing instrumental errors to extra-atmospheric sources.
Detailed computer resource measurements of the NASTRAN matrix decomposition spill logic were made using a software input/output monitor. These measurements showed that, in general, job cost can be reduced by avoiding spill. The results indicated that job cost can be minimized by using dynamic memory management. A prototype memory management system is being implemented and evaluated for the CDC Cyber computer.
The classical LQG stochastic control law is reformulated using the matrix factorization S equals UDU super T. This method yields a statistical guidance analysis algorithm that is numerically superior to the classical solution yet requires negligible additional computation and storage. Moreover, experience with U-D algorithms has shown them to be adaptable and easy to implement on a variety of problems.
A description is presented of the preliminary results of interdisciplinary studies of matrix samples. The significance of the 73215 studies is considered and the relationship of 73215 to other Apollo 17 highland breccias is discussed. According to a tentative hypothesis 73215 is an aggregate of fragments plus melt generated in a very large impact, possibly the Serenitatis basin-forming event. If this is correct, studies of this sample will make it possible to date the Serenitatis event and will provide an insight into the breccia-forming processes associated with such events. Studies of the clasts in the breccia will provide an opportunity to formulate a partial characterization of the preimpact source terrain. An investigation of the deep lunar crust might perhaps also be possible.
The definition of Poisson's ratio for nonlinear behavior of metal matrix composite laminates is discussed and experimental results for tensile and compressive loading of five different boron-aluminum laminates are presented. It is shown that there may be considerable difference in the value of Poisson's ratio as defined by a total strain or an incremental strain definition. It is argued that the incremental definition is more appropriate for nonlinear material behavior. Results from a (0) laminate indicate that the incremental definition provides a precursor to failure which is not evident if the total strain definition is used.
The characterization of polynomials whose zeros lie in certain algebraic domains (and the unification of the ideas of Hermite and Lyapunov) is the basis for developing finite algorithms for the solution of linear matrix equations. Particular attention is given to equations PA + A'P = Q (the Lyapunov equation) and P - A'PA = Q the (discrete Lyapunov equation). The Lyapunov equation appears in several areas of control theory such as stability theory, optimal control (evaluation of quadratic integrals), stochastic control (evaluation of covariance matrices) and in the solution of the algebraic Riccati equation using Newton's method.
The synthesis of a class of bis(imide-amine) curing agents for epoxy matrix resins is discussed. Glass transition temperatures and char yield data of an epoxy cured with various bis(imide-amines) are presented. The room temperature and 350 F mechanical properties, and char yields of unidirectional graphite fiber laminates prepared with conventional epoxy and imide-modified epoxy resins are presented.
An estimate of a state vector for a physical system when the weight matrix in the method of least squares is a function of this vector is considered. An iterative procedure is proposed for calculating the desired estimate. Conditions for the existence and uniqueness of the limit of this procedure are obtained, and a domain is found which contains the limit estimate. A second method for calculating the desired estimate which reduces to the solution of a system of algebraic equations is proposed. The question of applying Newton's method of tangents to solving the given system of algebraic equations is considered and conditions for the convergence of the modified Newton's method are obtained. Certain properties of the estimate obtained are presented together with an example.
The Fisher Information Matrix provides the nucleus of a design procedure for obtaining parameter-insensitive feedback gains in Linear-Quadratic-Gaussian problems. The procedure minimizes a sum of performance and closed-loop sensitivity costs, the latter being related to the information content of the system response. Analytical expressions for the appended cost functional and its gradient with respect to the feedback gains are derived. These derivatives serve as the basis of a computationally efficient iterative algorithm that finds the optimal gains. Application of the technique is made to determine low sensitivity feedback gains for a C-5A wing loading alleviation system that has 15 states and three uncertain parameters.