Kinematical equations for bodies whose rotation is described by the Euler-Rodrigues parameter.
Matrix form of kinematical differential equations for rotating bodies described by Euler-Rodrigues parameter, deriving time derivatives
Roberson, R. E.↗
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Special computational procedures for differential equations
Analytical study of ordinary and partial differential equations, illustrating initial value problem and large matrix inversion
INITIAL VALUE PROBLEM↗
Coherent laser excitation of Ba-137 and Ba-138
Computations are carried out for the 1S(6s2)-1P(6s,6p) coherent laser excitation of Ba-137 and Ba-138 in a magnetic field. Results are presented for both the steady-state and time-dependent excited-state populations of the Zeeman-split magnetic sublevels. The quantum-statistical Liouville-equation approach (for the reduced density matrix) is compared to the rate-equations approach. Significant differences are found between these, due to the interference between strongly overlapping lines (especially for Ba-137). The time-evolution profiles indicate that the Ba-137 transient time is much longer than that of Ba-138.
Lam, Kai-Shue↗
Acoustic transmission matrix of a variable area duct or nozzle carrying a compressible subsonic flow
The differential equations governing the propagation of sound in a variable area duct or nozzle carrying a one dimensional subsonic compressible fluid flow are derived and put in state variable form using acoustic pressure and particle velocity as the state variables. The duct or nozzle is divided into a number of regions. The region size is selected so that in each region the Mach number can be assumed constant and the area variation can be approximated by an exponential area variation. Consequently, the state variable equation in each region has constant coefficients. The transmission matrix for each region is obtained by solving the constant coefficient acoustic state variable differential equation. The transmission matrix for the duct or nozzle is the product of the individual transmission matrices of each region. Solutions are presented for several geometries with and without mean flow.
Miles, J. H.↗
Acoustic transmission matrix of a variable area duct or nozzle carrying a compressible subsonic flow
The differential equations governing the propagation of sound in a variable area duct or nozzle carrying a one-dimensional subsonic compressible fluid flow are derived and put in state variable form using acoustic pressure and particle velocity as the state variables. The duct or nozzle is divided into a number of regions. The region size is selected so that in each region the Mach number can be assumed constant and the area variation can be approximated by an exponential area variation. Consequently, the state variable equation in each region has constant coefficients. The transmission matrix for each region is obtained by solving the constant coefficient acoustic state variable differential equation. The transmission matrix for the duct or nozzle is the product of the individual transmission matrices of each region. Solutions are presented for several geometries with and without mean flow.
Miles, J. H.↗
Progressive Failure And Life Prediction of Ceramic and Textile Composites
An engineering approach to predict the fatigue life and progressive failure of multilayered composite and textile laminates is presented. Analytical models which account for matrix cracking, statistical fiber failures and nonlinear stress-strain behavior have been developed for both composites and textiles. The analysis method is based on a combined micromechanics, fracture mechanics and failure statistics analysis. Experimentally derived empirical coefficients are used to account for the interface of fiber and matrix, fiber strength, and fiber-matrix stiffness reductions. Similar approaches were applied to textiles using Repeating Unit Cells. In composite fatigue analysis, Walker's equation is applied for matrix fatigue cracking and Heywood's formulation is used for fiber strength fatigue degradation. The analysis has been compared with experiment with good agreement. Comparisons were made with Graphite-Epoxy, C/SiC and Nicalon/CAS composite materials. For textile materials, comparisons were made with triaxial braided and plain weave materials under biaxial or uniaxial tension. Fatigue predictions were compared with test data obtained from plain weave C/SiC materials tested at AS&M. Computer codes were developed to perform the analysis. Composite Progressive Failure Analysis for Laminates is contained in the code CPFail. Micromechanics Analysis for Textile Composites is contained in the code MicroTex. Both codes were adapted to run as subroutines for the finite element code ABAQUS and CPFail-ABAQUS and MicroTex-ABAQUS. Graphic user interface (GUI) was developed to connect CPFail and MicroTex with ABAQUS.
Xue, David Y.↗
Analysis and Improvement of Iteration Methods for Solving Automatic Control Equations Semiannual Status Report, Feb. - Aug. 1966
Approximation and matrix iteration techniques for solving automatic control equations
ITERATIVE SOLUTION↗
Micromechanics for ceramic matrix composites
The fiber substructuring concepts and the micromechanics equations that are embedded in the Ceramic Matrix Composite Analyzer (CEMCAN) computer code are described as well as the code itself, its current features and capabilities, and some examples to demonstrate the code's versatility. The methodology is equally applicable to metal matrix and polymer matrix composites. The prediction of ply mechanical and thermal properties agree very well with the existing models in the Integrated Composite Analyzer and the Ceramic Matrix Composite Analyzer, lending credence to the fiber substructuring approach. Fiber substructuring can capture greater local detail than conventional unit-cell-based micromechanical theories. It offers promise in simulating complex aspects of micromechanics in ceramic matrix composites.
Murthy, P. L. N.↗
Posterior Covariance Matrix Approximations
Here, the Davis equation of state (EOS) is commonly used to model thermodynamic relationships for high explosive (HE) reactants. Typically, the parameters in the EOS are calibrated, with uncertainty, using a Bayesian framework and Markov Chain Monte Carlo (MCMC) methods. However, MCMC methods are computationally expensive, especially for complex models with many parameters. This paper provides a comparison between MCMC and less computationally expensive Variational methods (Variational Bayesian and Hessian Variational Bayesian) for computing the posterior distribution and approximating the posterior covariance matrix based on heterogeneous experimental data. All three methods recover similar posterior distributions and posterior covariance matrices. This study demonstrates that for this EOS parameter calibration application, the assumptions made in the two Variational methods significantly reduce the computational cost but do not substantially change the results compared to MCMC.
97 MATHEMATICS AND COMPUTING↗
Connective stability of competitive equilibrium
The purpose of this paper is to derive necessary and sufficient conditions for the connective stability of nonlinear matrix systems described by the equation x-dot = A(t, x) x, where the matrix A(t, x) has time-varying nonlinear elements. The results obtained can be used to study the stability of competitive equilibrium in fields as diverse as economics and engineering, model ecosystems, and the arms race.-
Siljak, D. D.↗
Design and Development of a Model to Simulate 0-G Treadmill Running Using the European Space Agency's Subject Loading System
Develop a model that simulates a human running in 0 G using the European Space Agency s (ESA) Subject Loading System (SLS). The model provides ground reaction forces (GRF) based on speed and pull-down forces (PDF). DESIGN The theoretical basis for the Running Model was based on a simple spring-mass model. The dynamic properties of the spring-mass model express theoretical vertical GRF (GRFv) and shear GRF in the posterior-anterior direction (GRFsh) during running gait. ADAMs VIEW software was used to build the model, which has a pelvis, thigh segment, shank segment, and a spring foot (see Figure 1).the model s movement simulates the joint kinematics of a human running at Earth gravity with the aim of generating GRF data. DEVELOPMENT & VERIFICATION ESA provided parabolic flight data of subjects running while using the SLS, for further characterization of the model s GRF. Peak GRF data were fit to a linear regression line dependent on PDF and speed. Interpolation and extrapolation of the regression equation provided a theoretical data matrix, which is used to drive the model s motion equations. Verification of the model was conducted by running the model at 4 different speeds, with each speed accounting for 3 different PDF. The model s GRF data fell within a 1-standard-deviation boundary derived from the empirical ESA data. CONCLUSION The Running Model aids in conducting various simulations (potential scenarios include a fatigued runner or a powerful runner generating high loads at a fast cadence) to determine limitations for the T2 vibration isolation system (VIS) aboard the International Space Station. This model can predict how running with the ESA SLS affects the T2 VIS and may be used for other exercise analyses in the future.
Caldwell, E. C.↗
Geometry and entanglement in the scattering matrix
A formulation of nucleon–nucleon scattering is developed in which the S-matrix, rather than an effective-field theory (EFT) action, is the fundamental object. Spacetime plays no role in this description: the S-matrix is a trajectory that moves between RG fixed points in a compact theory space defined by unitarity. This theory space has a natural operator definition, and a geometric embedding of the unitarity constraints in four-dimensional Euclidean space yields a flat torus, which serves as the stage on which the S-matrix propagates. Trajectories with vanishing entanglement are special geodesics between RG fixed points on the flat torus, while entanglement is driven by an external potential. The system of equations describing S-matrix trajectories is in general complicated, however the very-low-energy S-matrix –that appears at leading-order in the EFT description– possesses a UV/IR conformal invariance which renders the system of equations integrable, and completely determines the potential. In this geometric viewpoint, inelasticity is in correspondence with the radius of a three-dimensional hyperbolic space whose two-dimensional boundary is the flat torus. This space has a singularity at vanishing radius, corresponding to maximal violation of unitarity. The trajectory on the flat torus boundary can be explicitly constructed from a bulk trajectory with a quantifiable error, providing a simple example of a holographic quantum error correcting code.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
State-space formulations for flutter analysis
Two state-space methods for true-damping flutter analysis are presented here. The aerodynamic matrix is assumed to be function of the complex reduced frequency p (off-the-imaginary axis aerodynamics) as the ones provided for instance by the computer code SOUSSA (Steady, Oscillatory and Unsteady, Subsonic and Supersonic Aerodynamics). This matrix is approximated by a third order polynomial or by a Pade's matrix approximant. Then the equations are cast into a suitable state space format including the Laplace transform of the space vector, a frequency-independent matrix, and the flight speed. The solution is then obtained by standard root-locus methods. Results are in good agreement with existing ones in a suitable range of the flutter reduced frequency. A discussion on spurious roots is also included.
Weiss, S. J.↗
Strain Rate Dependent Deformation and Strength Modeling of a Polymer Matrix Composite Utilizing a Micromechanics Approach
Potential gas turbine applications will expose polymer matrix composites to very high strain rate loading conditions, requiring an ability to understand and predict the material behavior under extreme conditions. Specifically, analytical methods designed for these applications must have the capability of properly capturing the strain rate sensitivities and nonlinearities that are present in the material response. The Ramaswamy-Stouffer constitutive equations, originally developed to analyze the viscoplastic deformation of metals, have been modified to simulate the nonlinear deformation response of ductile, crystalline polymers. The constitutive model is characterized and correlated for two representative ductile polymers. Fiberite 977-2 and PEEK, and the computed results correlate well with experimental values. The polymer constitutive equations are implemented in a mechanics of materials based composite micromechanics model to predict the nonlinear, rate dependent deformation response of a composite ply. Uniform stress and uniform strain assumptions are applied to compute the effective stresses of a composite unit cell from the applied strains. The micromechanics equations are successfully verified for two polymer matrix composites. IM7/977-2 and AS4/PEEK. The ultimate strength of a composite ply is predicted with the Hashin failure criteria that were implemented in the composite micromechanics model. The failure stresses of the two composite material systems are accurately predicted for a variety of fiber orientations and strain rates. The composite deformation model is implemented in LS-DYNA, a commercially available transient dynamic explicit finite element code. The matrix constitutive equations are converted into an incremental form, and the model is implemented into LS-DYNA through the use of a user defined material subroutine. The deformation response of a bulk polymer and a polymer matrix composite are predicted by finite element analyses. The results compare reasonably well to experimental values, with some discrepancies. The discrepancies are at least partially caused by the method used to integrate the rate equations in the polymer constitutive model.
Goldberg, Robert K.↗
Numerical computation of exponential matrices using the Cayley-Hamilton theorem
A method for computing exponential matrices, which often arise naturally in the solution of systems of linear differential equations, is developed. An exponential matrix is generated as a linear combination of a finite number (equal to the matrix order) of matrices, the coefficients of which are scalar infinite sums. The method can be generalized to apply to any formal power series of matrices. Attention is focused upon the exponential function, and the matrix exponent is assumed tri-diagonal in form. In such cases, the terms in the coefficient infinite sums can be extracted, as recursion relations, from the characteristic polynomial of the matrix exponent. Two numerical examples are presented in some detail: (1) the three dimensional infinitesimal rotation rate matrix, which is skew symmetric, and (2) an N-dimensional tri-diagonal and symmetric finite difference matrix which arises in the numerical solution of the heat conduction partial differential equation. In the second example, the known eigenvalues and eigenvectors of the finite difference matrix permit an analytical solution for the exponential matrix, through the theory of diagonalization and similarity transformations, which is used for independent verification. The convergence properties of the scalar infinite summations are investigated for finite difference matrices of various orders up to ten, and it is found that the number of terms required for convergence increases slowly with the order of the matrix.
Walden, H.↗
Introduction to matrix generalized inverses and their applications
Inverse matrices applied to analysis and minimization of systems of linear equations
MATRIX ALGEBRA↗
Phenomenological R -Matrix parameterization of direct, doorway, and compound nuclear reactions [Abstract]
Although formal expressions for scattering matrix accounting for direct, doorway, and compound nuclear (CN) resonant reactions have been derived several decades ago in both the transition ( T -)matrix formalism and the reactance ( K -)matrix formalism, the absence of corresponding expressions in phenomenological R -matrix formalism has limited the application of the latter to CN resonant reactions only. We remove this limitation by parameterizing direct, doorway, and CN resonant reactions in a phenomenological R -matrix scattering matrix, and provide a parameterization for a corresponding Reich-Moore approximation of eliminated capture channels. Direct reactions induce (previously neglected) mixing among the incoming or outgoing R -matrix channel wave functions, parameterized by real and orthonormal channel-rotation matrix, M , whereby the original scattering matrix U is transformed into M T UM . Any real and orthonormal matrix, M , can be equivalently expressed as e η , where η is a real and skew-symmetric 2 rotation-generating matrix that subsequently yields a more intuitive parameterization of eliminated direct capture reactions in Reich-Moore approximation. A phenomenological R -matrix parameterization of doorway reactions is inferred by equating the expression for reactance ( K -)matrix, given in terms of Brune’s alternative R -matrix parameterization, to a corresponding expression derived using Feshbach’s projection operator formalism. Assuming that all doorway states, just like CN states, are confined within spheres defined by R -matrix channel radii, a new R -matrix-like term induced by doorway states is gleaned, wherein each doorway state is parameterized by its energy, width, and the strength of its coupling to each CN state. Since a Reich-Moore approximation for retained-channel scattering matrix ought to approximate the effect of eliminated capture channels taking place via direct, doorway, or CN reactions, each of the three kinds of reactions contributing to the capture entails a corresponding Reich-Moore parameterization in a first-order approximation: direct contribution is parameterized by introducing finite diagonal elements of a retained-channel rotation-generating matrix, doorway contribution is parameterized by doorway capture widths, while CN contribution is parameterized by conventional Reich-Moore capture widths. We will present evidence of direct and doorway reactions observed in recent measurements of resolved resonance cross sections at the Gaerttner LINAC Center at Rensselaer Polytechnic Institute, and will outline a path for implementing this new R -matrix parameterization into the SAMMY nuclear data evaluation code.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Development of MCAERO wing design panel method with interactive graphics module
A reliable and efficient iterative method has been developed for designing wing section contours corresponding to a prescribed subcritical pressure distribution. The design process is initialized by using MCAERO (MCAIR 3-D Subsonic Potential Flow Analysis Code) to analyze a baseline configuration. A second program DMCAERO is then used to calculate a matrix containing the partial derivative of potential at each control point with respect to each unknown geometry parameter by applying a first-order expansion to the baseline equations in MCAERO. This matrix is calculated only once but is used in each iteration cycle to calculate the geometry perturbation and to analyze the perturbed geometry. The potential on the new geometry is calculated by linear extrapolation from the baseline solution. This extrapolated potential is converted to velocity by numerical differentiation, and velocity is converted to pressure by using Bernoulli's equation. There is an interactive graphics option which allows the user to graphically display the results of the design process and to interactively change either the geometry or the prescribed pressure distribution.
Hawk, J. D.↗