Theory of a high-intensity gas laser
Single mode gas laser theory for arbitrary field intensities in terms of ensemble averaged form of density-matrix equations of motion
Feld, M. S.↗
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Results for “Matrix equations”
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Numerical solution of potential flow problems in terms of flux components
Numerical determination of flux components in vapor flow considers split boundary conditions in both directions to form grid lines for flow continuity equations. Matrix of coefficients obtains direct solution for potential flow problem without iteration.
Vonrosenberg, D. U.↗
An approach for generation of second order RC-active filters.
It is shown that node (or loop) matrix equations can be written which yield second order low-pass, band-pass, and high-pass network functions. Networks corresponding to the equations can in turn be formulated by inspection of the equations. The networks employ ideal VCT or CVT sources which can be physically realized using transistors and/or operational amplifiers. The technique yields several structures which are believed to be new.
Dunn, W. R., Jr.↗
Error analysis of earth physics satellite systems
Error analysis of distant-satellite-to-close-satellite range-rate, satellite-to-sea altimetry, and ground station to satellite range are made by simulations in which observational variances are assumed, observation equations are formed, and normal equations incremented. The final normal equation matrix is inverted to obtain standard deviations and correlation coefficients. The natural parameters solved for are the broad variations of the gravity field, represented by harmonic coefficients; local variations of gravity, represented by point masses; and the departure of the sea level from the geoid, represented by area means. A standard case of a low (263 km) polar close satellite, three equatorial geosynchronous satellites, and eight ground tracking stations is set up.
Kaula, W. M.↗
Computation of optimal output-feedback compensators for linear time-invariant systems
The control of linear time-invariant systems with respect to a quadratic performance criterion was considered, subject to the constraint that the control vector be a constant linear transformation of the output vector. The optimal feedback matrix, f*, was selected to optimize the expected performance, given the covariance of the initial state. It is first shown that the expected performance criterion can be expressed as the ratio of two multinomials in the element of f. This expression provides the basis for a feasible method of determining f* in the case of single-input single-output systems. A number of iterative algorithms are then proposed for the calculation of f* for multiple input-output systems. For two of these, monotone convergence is proved, but they involve the solution of nonlinear matrix equations at each iteration. Another is proposed involving the solution of Lyapunov equations at each iteration, and the gradual increase of the magnitude of a penalty function. Experience with this algorithm will be needed to determine whether or not it does, indeed, possess desirable convergence properties, and whether it can be used to determine the globally optimal f*.
Platzman, L. K.↗
An analysis of the accuracy of a parameter optimization
The numerical operations involved in a currently used optimization technique are discussed and analyzed with special attention to the numerical accuracy. Alternative methods for deriving linear system transfer functions, finding the relationships between the transfer function coefficients and the design parameters, and solving a matrix equation are presented for more accurate and cost effective solutions.
Baram, Y.↗
Hybrid-coordinate spacecraft dynamics using large-deformation modal coordinates
For a vehicle amenable to idealization as a set of n + 1 rigid bodies interconnected by n line hinges (implying tree topology), a set of minimum-dimension matrix equations of motion is recorded in terms of variables including the relative rotations of contiguous bodies. When a substructure or portion of the vehicle is subjected to large gross deformations but is restricted to 'small' strains or local deformations, the n-body model is (for sufficiently large n) assumed to experience 'small' relative rotations of contiguous bodies within the substructure, and the equations of motion are linearized in these variables. The application of coordinate transformations and truncations to the linearized subset of the system equations of motion then produces a hybrid-coordinate formulation of the problem involving both discrete and distributed coordinates.
Likins, P. W.↗
Recent advances in numerical analysis of structural eigenvalue problems
A wide range of eigenvalue problems encountered in practical structural engineering analyses is defined, in which the structures are assumed to be discretized by any suitable technique such as the finite-element method. A review of the usual numerical procedures for the solution of such eigenvalue problems is presented and is followed by an extensive account of recently developed eigenproblem solution procedures. Particular emphasis is placed on the new numerical algorithms and associated computer programs based on the Sturm sequence method. Eigenvalue algorithms developed for efficient solution of natural frequency and buckling problems of structures are presented, as well as some eigenvalue procedures formulated in connection with the solution of quadratic matrix equations associated with free vibration analysis of structures. A new algorithm is described for natural frequency analysis of damped structural systems.
Gupta, K. K.↗
Application of modern control theory to the design of optimum aircraft controllers
A procedure is described for synthesis of optimal aircraft control systems by application of the concepts of optimal control theory to time-invariant linear systems with quadratic performance criteria. Essential in this synthesis procedure is the solution of the Riccati matrix equation which results in a constant linear feedback control law for an output regulator which maintains a plant in an equilibrium in the presence of impulse disturbances. An algorithm is derived for designing maneuverable output regulators with selected state variables for feedback.
Power, L. J.↗
Optimal control of a low wing-loading STOL aircraft
Linear optimal quadratic control theory is applied to a low wing-loading STOL aircraft for ride quality and flight path following. Design criteria include minimum rms response to wind turbulence and desired transient response characteristics. Design techniques include proper choosing of design versus evaluation models, choosing appropriate performance index responses, and use of classical evaluation techniques. Results are obtained through a combination of frequency response shaping and gust observation. Effects of control rate and authority saturation are examined with a new rapid calculation of random input describing functions. Parameter sensitivity is also evaluated using a Liapunov type matrix equation.
Cunningham, T. B.↗
On a finite dynamic element method for free vibration analysis of structures
This paper explores the concept of finite dynamic elements involving higher order dynamic correction terms in the associated stiffness and mass matrices. Such matrices are then developed for a rectangular prestressed membrane element. Next, efficient analysis techniques for the eigenproblem solution of the resulting quadratic matrix equations are described in detail. These are followed by suitable numerical examples which indicate that employment of such dynamic elements in conjunction with an efficient quadratic matric solution technique will result in a most significant economy in the free vibration analysis of structures.
Gupta, K. K.↗
A finite element technique for non-deterministic thermal deformation analyses including temperature dependent material properties
A technique utilizing the finite element displacement method is developed for the static analysis of structures subjected to non-deterministic thermal loading in which the material properties, assumed isotropic, are temperature dependent. Matrix equations are developed for the first two statistical moments of the displacements using a third order series expansion for the displacements in terms of the random temperatures. Sample problems are included to demonstrate the range of applicability of the third order series solutions. These solutions are compared with results from Monte Carlo analyses and also, for some problems, with solutions obtained by numerically integrating equations for the statistical properties of the displacements. In general, it is shown that the effect of temperature dependent material properties can have a significant effect on the covariances of the displacements.
Case, W. R., Jr.↗
Control logic to track the outputs of a command generator or randomly forced target
A procedure is presented for synthesizing time-invariant control logic to cause the outputs of a linear plant to track the outputs of an unforced (or randomly forced) linear dynamic system. The control logic uses feed-forward of the reference system state variables and feedback of the plant state variables. The feed-forward gains are obtained from the solution of a linear algebraic matrix equation of the Liapunov type. The feedback gains are the usual regulator gains, determined to stabilize (or augment the stability of) the plant, possibly including integral control. The method is applied here to the design of control logic for a second-order servomechanism to follow a linearly increasing (ramp) signal, an unstable third-order system with two controls to track two separate ramp signals, and a sixth-order system with two controls to track a constant signal and an exponentially decreasing signal (aircraft landing-flare or glide-slope-capture with constant velocity).
Trankle, T. L.↗
Elastic lithosphere thickness on the moon from mare tectonic features - A formal inversion
The thickness (T) of the lunar elastic lithosphere at the time (3.6 to 3.8 billion years ago) of the earliest preserved basalt flows in circular mare basins can be estimated by inverting the observed locations of extensional tectonic features in and surrounding the maria. In performing the inversion, the lithosphere is modeled as an elastic shell with a liquid interior, and the basalt load for each mare is approximated by a set of concentric cylinders. To permit solving the forward problem of placing radial limits on the positions of the rilles around a given mare, an additional parameter F, the ratio of the radial stress at the radial limits to the maximum radial stress, is introduced. T and F are chosen to give the best weighted-squares fit of the radial limits to the observations, and are used as the initial values in a linearized matrix inversion to check the resolution and estimate errors. The application of the procedure to three maria with prominent extensional features, Humorum, Orientale, and Serenitatis, gives values of T from about 40 + or - 10 to 50 + or - 10 km, and in each case the linearized matrix equation has an exact inverse.
Comer, R. P.↗
Design for active flutter suppression and gust alleviation using state-space aeroelastic modeling
An analytical design technique for an active flutter-suppression and gust-alleviation control system is presented. It is based on a rational approximation of the unsteady aerodynamic loads in the entire Laplace domain, which yields matrix equations of motion with constant coefficients. Some existing rational approximation schemes are reviewed, and a new technique which yields a minimal number of augmented states for a desired accuracy is presented. The state-space aeroelastic model is used to design a constant gain, partial-feedback control system, which simultaneously assures stability and optimizes any desired combination of gust response parameters throughout the entire flight envelope.
Karpel, M.↗
Spectral methods for problems in complex geometries
The properties of spectral methods are surveyed and their extension to solve problems in complex geometries is developed. A new iteration procedure is introduced to solve efficiently the full matrix equations resulting from spectral approximations to nonconstant coefficient boundary-value problems in complex geometries. It is shown that the work required to solve these spectral equations exceeds that of solving the lowest-order finite-difference approximation to the same problem by only O(N log N).
Orszag, S. A.↗
A symmetric modal formulation of fluid-structure interaction, including a static approximation to higher order fluid modes
A method of fluid-structure coupling which provides symmetrical matrix equations of standard form solved by existing finite element computer programs is presented. The method postulates that the uncoupled vibration modes of the fluid or the structure be calculated before the coupled analysis. A numerical solution of vibration modes in an axisymmetric container demonstrated that a static approximation to higher order fluid modes can improve the accuracy of dynamic response computations using modal methods.
Macneal, R. H.↗
Closed form evaluation of symmetric two-sided complex integrals
Evaluation of two-sided complex integrals is often required when analyzing linear systems to determine signal variances resulting from stochastic inputs and system noise bandwidths. Algebraic solutions of integrals in a closed matrix equation form, using coefficients of the numerator and denominator polynomials, are presented. The closed forms provide the possibility of obtaining some insight into parameter sensitivity in addition to greatly reducing the computational complexity required by the normal method of evaluation by residues.
Winkelstein, R.↗