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NASA NTRS1975

Necessary and sufficient conditions for the complete controllability and observability of systems in series using the coprime factorization of a rational matrix

The series connection of two linear time-invariant systems that have minimal state space system descriptions is considered. From these descriptions, strict-system-equivalent polynomial matrix system descriptions in the manner of Rosenbrock are derived. They are based on the factorization of the transfer matrix of the subsystems as a ratio of two right or left coprime polynomial matrices. They give rise to a simple polynomial matrix system description of the tandem connection. Theorem 1 states that for the complete controllability and observability of the state space system description of the series connection, it is necessary and sufficient that certain 'denominator' and 'numerator' groups are coprime. Consequences for feedback systems are drawn in Corollary 1. The role of pole-zero cancellations is explained by Lemma 3 and Corollaires 2 and 3.

Callier, F. M.↗

NASA NTRS1975

Applications of matrix derivatives to optimization problems in statistical pattern recognition

A necessary condition for a real valued Frechet differentiable function of a vector variable have an extremum at a vector x sub 0 is that the Frechet derivative vanishes at x sub 0. A relationship between Frechet differentials and matrix derivatives was established that obtains a necessary condition on the matrix derivative at an extrema. These results are applied to various scalar functions of matrix variables which occur in statistical pattern recognition.

Morrell, J. S.↗

NASA NTRS1976

A comparison of matrix methods for calculating eigenvalues in acoustically lined ducts

Three approximate methods - finite differences, weighted residuals, and finite elements - were used to solve the eigenvalue problem which arises in finding the acoustic modes and propagation constants in an absorptively lined two-dimensional duct without airflow. The matrix equations derived for each of these methods were solved for the eigenvalues corresponding to various values of wall impedance. Two matrix orders, 20 x 20 and 40 x 40, were used. The cases considered included values of wall admittance for which exact eigenvalues were known and for which several nearly equal roots were present. Ten of the lower order eigenvalues obtained from the three approximate methods were compared with solutions calculated from the exact characteristic equation in order to make an assessment of the relative accuracy and reliability of the three methods. The best results were given by the finite element method using a cubic polynomial. Excellent accuracy was consistently obtained, even for nearly equal eigenvalues, by using a 20 x 20 order matrix.

Watson, W.↗

NASA NTRS1976

The use of the Winograd matrix multiplication algorithm in digital multispectral processing

The Winograd procedure for matrix multiplication provides a method whereby general matrix products may be computed more efficiently than the normal method. The algorithm and the time savings that can be effected are described. A FORTRAN program is provided which performs a general matrix multiply according to this algorithm. A variation of this procedure that may be used to calculate Gaussian probability density functions is also described. It is shown how a time savings can be effected in this calculation. The extension of this method to other similar calculations should yield similar savings.

Vanrooy, D. L.↗

NASA NTRS1975

Numerical solution of quadratic matrix equations for free vibration analysis of structures

This paper is concerned with the efficient and accurate solution of the eigenvalue problem represented by quadratic matrix equations. Such matrix forms are obtained in connection with the free vibration analysis of structures, discretized by finite 'dynamic' elements, resulting in frequency-dependent stiffness and inertia matrices. The paper presents a new numerical solution procedure of the quadratic matrix equations, based on a combined Sturm sequence and inverse iteration technique enabling economical and accurate determination of a few required eigenvalues and associated vectors. An alternative procedure based on a simultaneous iteration procedure is also described when only the first few modes are the usual requirement. The employment of finite dynamic elements in conjunction with the presently developed eigenvalue routines results in a most significant economy in the dynamic analysis of structures.

Gupta, K. K.↗

NASA NTRS1977

Application of symbolic/numeric matrix solution techniques to the NASTRAN program

The matrix solving algorithm of any finite element algorithm is extremely important since solution of the matrix equations requires a large amount of elapse time due to null calculations and excessive input/output operations. An alternate method of solving the matrix equations is presented. A symbolic processing step followed by numeric solution yields the solution very rapidly and is especially useful for nonlinear problems.

Buturla, E. M.↗

NASA NTRS1977

Matrix glass vs. intruded glass in lunar breccia 15286

The viscous flow and crystallization behavior of the matrix composition of breccia 15286 have been determined in the 1159-1307 C and 644-770 C temperature ranges. The Shaw (1972) model and, to a slightly lesser extent, the semiempirical Bottinga and Weill (1972) model describe the variation of viscosity with temperature. The crystal growth rate has been determined over the temperature range 821-1185 C. Time-temperature-transformation, logarithmic cooling, and continuous cooling curves were constructed; a nucleation barrier of 60 kT at a relative undercooling of 0.2 is assumed. It is estimated that the matrix of 15286 breccia cooled at a rate greater than 80 K/min through the glass transition region. The increase in density that occurs when the matrix glass is annealed suggests that the breccia was formed by cooling from the molten state rather than by direct shock-induced transition from crystal to glass. Viscous sintering under continuous cooling and logarithmic cooling conditions is analyzed.

Handwerker, C. A.↗

NASA NTRS1979

A new approach to the solution of large, full matrix equations: A two-dimensional potential flow feasibility study

An approach to the solution of matrix problems resulting from integral equations of mathematical physics is presented. Based on the inherent smoothness in such equations, the problem is reformulated using a set of orthogonal basis vectors, leading to an equivalent coefficient problem which can be of lower order without significantly impairing the accuracy of the solution. This approach was evaluated using a two-dimensional Neumann problem describing the inviscid, incompressible flow over an airfoil. Two different kinds of mode functions were investigated, namely eigenfunction series and Fourier series. The method using Fourier series was found preferable. It uses all of the coefficients from a Fast Fourier Transform algorithm in an approximate method which exploits the known structure of the transformed coefficient matrix and very promising results for the flow over a realistic airfoil are obtained. On the basis of the results presented here, an order of magnitude reduction in this computer time can be expected for such problems as compared with the time for a direct matrix solution.

James, R. M.↗

NASA NTRS1979

Space Shuttle communications RF switch matrix

The Shuttle Orbiter communications equipment includes phase modulation (PM) and frequency modulation (FM) channels. The PM section has the capability of routing high levels of energy (175 W) from any one of four transmitters to any one of four antennas, mutually exclusive. The FM channel uses a maximum of 15-W power routed from either of two transmitters to one of two antennas, mutually exclusive. The paper describes the design and the theory of a logic-controlled RF switch matrix devised for the purposes cited. Both PM and FM channels are computer-controlled with manual overrides. The logic interface is realized with CMOS logic for low power consumption and high noise immunity. The interior of the switch matrix is maintained at a pressure of 15 psi (90% nitrogen, 10% helium) by an electron beam-welded encapsulation. The computational results confirm the viability of the RF switch matrix concept.

Winch, R.↗

NASA NTRS1979

Analysis of a unidirectional composite containing broken fibers and matrix damage

An analytical solution is developed for the determination of the stresses and displacements in a unidirectional fiber-reinforced composite containing an arbitrary number of broken fibers as well as longitudinal yielding and splitting of the matrix. The solution is developed using a materials-modeling approach which is based on a shear-lag stress transfer mechanism. The equilibrium equation in the axial direction gives a pair of integral equations which are solved numerically. Excellent agreement is shown to exist between the solution and experimental results for notched unidirectional boron/aluminum laminates without splitting. For brittle matrix composites (i.e. epoxy) equally good results are indicated for both matrix yielding and splitting. For yielding without splitting the fracture strength depends on crack length while for large splitting it is crack length independent.

George, J. G.↗

NASA NTRS1980

Research on graphite reinforced glass matrix composites

High levels of mechanical performance in tension, flexure, fatigue, and creep loading situations of graphite fiber reinforced glass matrix composites are discussed. At test temperatures of up to 813 K it was found that the major limiting factor was the oxidative instability of the reinforcing graphite fibers. Particular points to note include the following: (1) a wide variety of graphite fibers were found to be comparable with the glass matrix composite fabrication process; (2) choice of fiber, to a large extent, controlled resultant composite performance; (3) composite fatigue performance was found to be excellent at both 300 K and 703 K; (4) composite creep and stress rupture at temperatures of up to 813 K was limited by the oxidative stability of the fiber; (5) exceptionally low values of composite thermal expansion coefficient were attributable to the dimensional stability of both matrix and fiber; and (6) component fabricability was demonstrated through the hot pressing of hot sections and brazing using glass and metal joining phases.

Prewo, K. M.↗

NASA NTRS1980

Analysis of a unidirectional composite containing broken fibers and matrix damage

An analytical solution is developed for the determination of the stresses and displacements in a unidirectional fiber-reinforced composite containing an arbitrary number of broken fibers as well as longitudinal yielding and splitting of the matrix. The solution is developed using a 'materials-modeling' approach which is based on a shear-lag stress transfer mechanism. The equilibrium equation in the axial direction gives a pair of integral equations which are solved numerically. Excellent agreement is shown to exist between the solution and experimental results for notched unidirectional boron/aluminum laminates without splitting. For brittle matrix composites (i.e., epoxy) equally good results are indicated for both matrix yielding and splitting. For yielding without splitting the fracture strength is found to depend on crack length while for large splitting it is crack length independent.

Goree, J. G.↗

NASA NTRS1981

New approach to the solution of large, full matrix equations

A new approach to the solution of matrix equations resulting from integral equations is presented and applied to the solution of two-dimensional Neumann problems describing the inviscid, incompressible flow past an airfoil. The problem is reformulated in terms of a preselected set of mode functions giving an equivalent matrix equation to be solved for the mode-function expansion coefficients. Because of the inherent smoothness of the original problem, the coefficient problem can be solved approximately without significantly affecting the accuracy of the final solution. Very promising two-dimensional results are obtained and the extension of the method to three-dimensional problems is investigated. On the basis of these results it is shown that the computing time for the matrix solution for a large three-dimensional panel method calculation could be reduced by an order of magnitude compared with that required for a direct solution.

Clark, R. W.↗

NASA NTRS1981

Degradation Mechanisms in Aluminum Matrix Composites: Alumina/Aluminum and Boron/Aluminum

The effects of fabrication and long term thermal exposure (up to 10,000 hours at 590 K) on two types of aluminum matrix composites were examined. An alumina/aluminum composite, was made of continuous alpha Al2O3 fibers in a matrix of commercially pure aluminum alloyed with 2.8% lithium. The mechanical properties of the material, the effect of isothermal exposure, cyclic thermal exposure, and fatigue are presented. Two degradation mechanisms are identified. One was caused by formation of a nonstoichiometric alumina during fabrication, the other by a loss of lithium to a surface reaction during long term thermal exposure. The other composite, boron/aluminum, made of boron fibers in an aluminum matrix, was investigated using five different aluminum alloys for the matrices. The mechanical properties of each material and the effect of isothermal and cyclic thermal exposure are presented. The effects of each alloy constituent on the degradation mechanisms are discussed. The effects of several reactions between alloy constituents and boron fibers on the composite properties are discussed.

Olsen, G. C.↗

NASA NTRS1982

Integrating matrix formulations for vibrations of rotating beams including the effects of concentrated masses

By expressing partial differential equations of motion in matrix notation, utilizing the integrating matrix as a spatial operator, and applying the boundary conditions, the resulting ordinary differential equations can be cast into standard eigenvalue form upon assumption of the usual time dependence. As originally developed, the technique was limited to beams having continuous mass and stiffness properties along their lengths. Integrating matrix methods are extended to treat the differential equations governing the flap, lag, or axial vibrations of rotating beams having concentrated masses. Inclusion of concentrated masses is shown to lead to the same kind of standard eigenvalue problem as before, but with slightly modified matrices.

Lakin, W. D.↗

NASA NTRS1983

Matrix differentiation formulas

A compact differentiation technique (without using indexes) is developed for scalar functions that depend on complex matrix arguments which are combined by operations of complex conjugation, transposition, addition, multiplication, matrix inversion and taking the direct product. The differentiation apparatus is developed in order to simplify the solution of extremum problems of scalar functions of matrix arguments.

Usikov, D. A.↗

NASA NTRS1983

Characterization of the matrix glass transition in carbon-epoxy laminates using the CSD test geometry

A new test geometry, referred to as centro-symmetric deformation (CSD), is proposed for characterizing the viscoelastic behavior of the matrix of carbon-epoxy laminates. The sample consists of a thin disk, typically 6-14 plies thick, having a nominal diameter of 30 mm. The disk is freely supported on a circular anvil; the load is applied to the center of the disk using an 8-mm-diameter ball bearing nosepiece. The CSD test geometry provides viscoelastic dispersion data which are independent of the angular orientation of the sample. The test geometry is sufficiently sensitive to matrix changes to allow its use for postcuring, humidity, crosslink density, and other matrix change studies. Test results are presented for a carbon-epoxy laminate.

Sternstein, S. S.↗

NASA NTRS1984

Analysis of stress-strain, fracture and ductility behavior of aluminum matrix composites containing discontinuous silicon carbide reinforcement

Mechanical properties and stress-strain behavior for several types of commercially fabricated aluminum matrix composites, containing up to 40 vol % discontinuous SiC whisker, nodule, or particulate reinforcement were evaluated. It was found that the elastic modulus of the composites was isotropic, to be independent of type of reinforcement, and to be controlled solely by the volume percentage of SiC reinforcement present. The yield/tensile strengths and ductility were controlled primarily by the matrix alloy and temper condition. Ductility decreased with increasing reinforcement content, however, the fracture strains observed were higher than those reported in the literature for this type of composite. This increase in fracture strain is attributed to cleaner matrix powder and increased mechanical working during fabrication. Conventional aluminum and titanium structural alloys were compared and have shown that the properties of these low cost, lightweight composites have good potential for application to aerospace structures.

Mcdanels, D. L.↗