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Integrating matrices are derived for arbitrarily spaced grid points using either interpolating or least squares fit orthogonal polynomials. Several features of the equally spaced grid case are also discussed.
The matrices of 16 unequilibrated chondrites were examined by optical microscopy, an electron microprobe, and a scanning electron microscope. The fine-grained, opaque, silicate matrix of type 3 unequilibrated chondrites was compositionally, mineralogically, and texturally different from the chondrules and their fragments; it may be the low temperature condensate proposed by Larimer and Anders (1967, 1970). Each meteorite has been metamorphosed by a combination of processes including thermal metamorphism and the passage of shock waves; the appearance of each chondrite results from the temperature and pressure conditions which formed it, and subsequent metamorphic alterations.
Matrix isolation techniques are used to investigate the spectra of lead molecules and, in particular, to obtain resonance Raman spectra of lead vapors isolated in solid xenon matrices. The presence of Pb2 is confirmed by the visible adsorption, and Raman spectra yield a vibrational frequency for the ground state of 108 per cm and a dissociation energy of 8200 per cm. A second resonance Raman progression indicates a Pb3 species of D3h symmetry. Finally, two additional Raman features at approximately 111 per cm spacing are evidence for a third species, tentatively identified as Pb4.
Two off-line schemes are proposed for the identification of unknown noise covariance matrices Q and R of a discrete-time dynamic system. The first scheme is based on a maximum a posteriori cost function utilizing smoothed state estimates, while the second is based on a maximum likelihood cost function utilizing filtered state estimates. Sensitivity of the cost functions to Q and R is analyzed for the following cases: (1) single-input single-output systems; (2) multiinput single-output systems; and (3) single-input multioutput systems with a diagonal R. Identifiability criteria are presented for the cases considered and demonstrated by examples.
Degradation mechanisms were investigated for graphite/polysulfone and graphite/epoxy laminates exposed to ultraviolet and high-energy electron radiations in vacuum up to 960 equivalent sun hours and 10 to the ninth power rads respectively. Based on GC and combined GC/MS analysis of volatile by-products evolved during irradiation, several free radical mechanisms of composite degradation were identified. The radiation resistance of different matrices was compared in terms of G values and quantum yields for gas formation. All the composite materials evaluated show high electron radiation stability and relatively low ultraviolet stability as indicated by low G values and high quantum for gas formation. Mechanical property measurements of irradiated samples did not reveal significant changes, with the possible exception of UV exposed polysulfone laminates. Hydrogen and methane were identified as the main by-products of irradiation, along with unexpectedly high levels of CO and CO2.
The objective of the Rotor Force Test Facility at the California Institute of Technology is to artificially orbit the center of rotation of an impeller enclosed within a volute over a range of frequencies from zero to synchronous and to measure the resulting forces on the impeller. Preliminary data from the first stage experiments in which the shaft is orbited at low frequency is reported. Steady volute forces along with stiffness matrices due to the change in position of the rotor center are measured. Static pressure taps around the volute are used to obtain volute pressure distributions for various fixed positions of the impeller center and for various flow rates. Static pressure forces are calculated from these pressure distributions allowing a more complete analysis of the components of the impeller forces. Comparison is made with various existing theoretical and experimental results.
Research was conducted on reinforcing nickel based matrices with alumina trichites by using powder metallurgy. Alumina trichites previously coated with nickel are magnetically aligned. The felt obtained is then sintered under a light pressure at a temperature just below the melting point of nickel. The halogenated atmosphere technique makes it possible to incorporate a large number of additive elements such as chromium, titanium, zirconium, tantalum, niobium, aluminum, etc. It does not appear that going from laboratory scale to a semi-industrial scale in production would create any major problems.
Estimates made in the plane to improve on noisy unbiased estimates were combined. Only a small fraction of points in a giant grid were used to do this, those that are most like a given point. A component of this process defining an affinity matrix of values, indicating which points are relevant to others. Minimax rules are shown to be based on affinity matrices.
An algorithm is presented which can be used to develop compliance matrices for cracked bodies. The method relies on the numerical solution of singular integral equations with Cauchy-type kernels and provides an efficient and accurate procedure for relating applied loadings to crack opening displacements. The algorithm should be of interest to those performing repetitive calculations in the analysis of experimental results obtained from fracture specimens.
Aramid-reinforced composite materials of equal fiber volume and varied polymer thermoset matrices were pultruded and flexurally tested to failure. The objective was to improve the flexural properties of aramid-reinforced pultrusions. Pultrusions of both sized and unsized aramid fiber with four different resin systems were compared to determine the effects of sizing compounds and postcuring on flexural strength, fiber wettability, and fiber-to-resin interface bonding. Improvements in flexural strength resulting from pretreatments with the sizing solutions used were marginal. The most significant improvements in flexural properties resulted from postcuring. Flexural strengths ranged from a low of 39,647 psi (273MPa) to a high of 80,390 psi (554 MPa), an overall increase of 103 percent. The fact that postcuring improved the flexural properties of the pultrusions of the four resin systems indicates that a full cure did not occur in any of the resin systems during the pultrusion process. The increased flexural strengths of the polyester and vinyl ester pultrusions were the most surprising. The four resin systems examined were Interplastic Corporation VE 8300 vinyl ester, Ashland Chemical Company Aropol 7430 Polyester, and Shell Chemical Company Epon 9302 and Epon 9310 epoxides.
During the process of solving a mathematical model numerically, there is often a need to operate on a vector v by an operator which can be expressed as f(A) while A is NxN matrix (ex: exp(A), sin(A), A sup -1). Except for very simple matrices, it is impractical to construct the matrix f(A) explicitly. Usually an approximation to it is used. In the present research, an algorithm is developed which uses a polynomial approximation to f(A). It is reduced to a problem of approximating f(z) by a polynomial in z while z belongs to the domain D in the complex plane which includes all the eigenvalues of A. This problem of approximation is approached by interpolating the function f(z) in a certain set of points which is known to have some maximal properties. The approximation thus achieved is almost best. Implementing the algorithm to some practical problem is described. Since a solution to a linear system Ax = b is x= A sup -1 b, an iterative solution to it can be regarded as a polynomial approximation to f(A) = A sup -1. Implementing the algorithm in this case is also described.
Aramid reinforcement composite materials of equal fiber volume having varied polymer thermoset matrices have been pultruded and flexurally tested to failure. The objective was to improve flexural properties of aramid reinforced pultrusions. Pultrusions of both sized and unsized aramid fiber with four different resin systems were compared to determine the effects of sizing compounds and post-thermal treatments on the flexural strength as an indication of fiber wettability and fiber-to-resin interface bonding. Improvements in flexural strength as the result of pretreatments with sizing solutions used in this study were marginal. The most significant improvements in flexural properties were the results of postcuring. Overall improvements ranged from a low 39,647 Psi(273 Mpa) to a high of 80,390 Psi(554 Mpa), or 103 percent. The fact that post-thermal treatments improved the flexural properties of the four pultrusions indicates that a full cure did not occur in either resin system during the pultrusion process. The increased flexural strengths of the polyester and vinyl ester pultrusions were the most surprising. Based on data presented, the most promising resin system of the four examined (VE 8300, Aropol 7430, Epon 9302, and Epon 9310) for Kevlar reinforced pultrusion is Epon 9310 epoxy.
First, the Cholesky factorization is extended to cover uniformly partitioned banded positive definite matrices of rank n which may be real symmetric or Hermitian. Then, two stratagems are given for the use of the algorithm in concurrent machines where the number of processing elements is less than required to factor the matrix in as few serial steps as possible, and where uniformly high efficiency is expected from all processing elements. Expressions are given for the efficiency factor e appearing in the speed-up expression q = eN, and these are specialized for the N node hypercube machine as a function of partition size s, the number N of processing elements of the hypercube machine, and the cost mu of interelement transmission relative to computation. It is shown that the efficiency factor e is inversely proportional to mu/s, and that e is almost independent of N when N is large and mu/s = 0. The task is completed in n/s serial steps with no limit on n. The half bandwidth b of the matrix is 2 Ns.
An algorithm is developed for finding the inverse of polynomial matrices in the irreducible form. The computational method involves the use of the left (right) matrix division method and the determination of linearly dependent vectors of the remainders. The obtained transfer function matrix has no nontrivial common factor between the elements of the numerator polynomial matrix and the denominator polynomial.
Inviscid flux Jacobian matrices and their properties used in numerical solutions of conservation laws are extended to general, equilibrium gas laws. Exact and approximate generalizations of the Roe average are presented. Results are given for one-dimensional flow, and then extended to three-dimensional flow with time-varying grids.
The principles and operation of MMAX, a symbolic-computation program which automates the process of generating property matrices for structural materials, are briefly described and illustrated with sample analyses of a rubberlike material and an elastoplastic material. MMAX is written in LISP under the symbolic finite-element generator FINGER and the general symbolic manipulator MACSYMA; it first derives the formulas required by mathematical manipulation, and then translates the formulas into FORTRAN code, adapted to the particular type of machine to be used for the numerical calculations. This approach is shown to combine efficiently the advantages of symbolic and numerical computation for engineering applications.
Load and stress recovery from transient dynamic studies are improved upon using an extended acceleration vector in the modal acceleration technique applied to structural analysis. Extension of the normal LTM (load transformation matrices) stress recovery to automatically compute margins of safety is presented with an application to the Hubble space telescope.
The Bunch-Kaufman algorithm is the method of choice for factoring symmetric indefinite matrices in many applications. However, the Bunch-Kaufman algorithm does not take advantage of high-performance architectures such as the Cray Y-MP. Three new algorithms, based on Bunch-Kaufman factorization, that take advantage of such architectures are described. Results from an implementation of the third algorithm are presented.