The generation and application of first and second order sensitivity coefficients
Linear ordinary differential equation systems first order output sensitivity coefficients generation and application to second order cross sensitivities
Kulas, C. E.↗
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Spin stability of dual-spin spacecrafts.
The spin-axis attitude stability in torque-free environment of dual-spin spacecrafts with energy losses in both bodies is studied by analytical means. A new mathematical model, which assumes the high speed rotor to be symmetrical, is introduced and analyzed. The linearized variational system of equations for the model with periodic coefficients is transformed to a kinematically similar autonomous system (according to Floquet and Liapunov) which can be analyzed routinely using established procedure. As an example, this technique is used on a recent mathematical model of the SAS-A spacecraft to establish steady state stability over a wide range of speeds.
Guha, A. K.↗
A velocity-command controller for a VTOL aircraft
A technique is presented for calculating feedback and feedforward gain matrices that enable a VTOL aircraft to track input commands of forward and vertical velocity while maintaining acceptable responses to pilot inputs. Leverrier's algorithm is used for determining a set of state-variable, feedback gains that force the closed-loop poles and zeros of one pilot-input transfer function to pre-selected positions in the s-plane. This set of feedback gains is then used to calculate the feedback and feedforward gains for the velocity-command controller. The method is computationally attractive since the gains are determined by solving systems of linear, simultaneous equations. The method is used in a digital simulation of the CH-47 helicopter to control longitudinal dynamics.
Reid, G. F.↗
Digital adaptive control of a VTOL aircraft
A technique has been developed for calculating feedback and feedforward gain matrices that stabilize a VTOL aircraft while enabling it to track input commands of forward and vertical velocity. Leverrier's algorithm is used in a procedure for determining a set of state variable, feedback gains that force the closed loop poles and zeroes of one pilot input transfer function to be at preselected positions in the s plane. This set of feedback gains is then used to calculate the feedback and feedforward gains for the velocity command controller. The method is computationally attractive since the gains are determined by solving systems of linear, simultaneous equations. Responses obtained using a digital simulation of the longitudinal dynamics of the CH-47 helicopter are presented.
Reid, G. F.↗
Recursive partitioned inversion of large (1500 x 1500) symmetric matrices
A recursive algorithm was designed to invert large, dense, symmetric, positive definite matrices using small amounts of computer core, i.e., a small fraction of the core needed to store the complete matrix. The described algorithm is a generalized Gaussian elimination technique. Other algorithms are also discussed for the Cholesky decomposition and step inversion techniques. The purpose of the inversion algorithm is to solve large linear systems of normal equations generated by working geodetic problems. The algorithm was incorporated into a computer program called SOLVE. In the past the SOLVE program has been used in obtaining solutions published as the Goddard earth models.
Putney, B. H.↗
A technique for pole-zero placement for dual-input control systems
A technique is presented for determining state variable feedback gains that will place both the poles and zeros of a selected transfer function of a dual-input control system at pre-determined locations in the s-plane. Leverrier's algorithm is used to determine the numerator and denominator coefficients of the closed-loop transfer function as functions of the feedback gains. The values of gain that match these coefficients to those of a pre-selected model are found by solving two systems of linear simultaneous equations. The algorithm has been used in a computer simulation of the CH-47 helicopter to control longitudinal dynamics.
Reid, G. F.↗
A velocity-command controller for a VTOL aircraft
A technique is presented for calculating feedback and feedforward gain matrices that enable a VTOL aircraft to track input commands of forward and vertical velocity while maintaining acceptable responses to pilot inputs. Leverrier's algorithm is used for determining a set of state-variable, feedback gains that force the closed-loop poles and zeros of one pilot-input transfer function to pre-selected positions in the s-plane. This set of feedback gains is then used to calculate the feedback and feedforward gains for the velocity-command controller. The method is computationally attractive since the gains are determined by solving systems of linear, simultaneous equations. The method has been used in a digital simulation of the CH-47 helicopter to control longitudinal dynamics.
Reid, G. F.↗
Numerical methods for problems involving the Drazin inverse
The objective was to try to develop a useful numerical algorithm for the Drazin inverse and to analyze the numerical aspects of the applications of the Drazin inverse relating to the study of homogeneous Markov chains and systems of linear differential equations with singular coefficient matrices. It is felt that all objectives were accomplished with a measurable degree of success.
Meyer, C. D., Jr.↗
A mixed-mode crack analysis of rectilinear anisotropic solids using conservation laws of elasticity
A very simple and convenient method of analysis for studying two-dimensional mixed-mode crack problems in rectilinear anisotropic solids is presented. The analysis is formulated on the basis of conservation laws of anisotropic elasticity and of fundamental relationships in anisotropic fracture mechanics. The problem is reduced to a system of linear algebraic equations in mixed-mode stress intensity factors. One of the salient features of the present approach is that it can determine directly the mixed-mode stress intensity solutions from the conservation integrals evaluated along a path removed from the crack-tip region without the need of solving the corresponding complex near-field boundary value problem. Several examples with solutions available in the literature are solved to ensure the accuracy of the current analysis. This method is further demonstrated to be superior to other approaches in its numerical simplicity and computational efficiency. Solutions of more complicated and practical engineering problems dealing with the crack emanating from a circular hole in composites are presented also to illustrate the capacity of this method.
Wang, S. S.↗
Time-temperature effect in adhesively bonded joints
The viscoelastic analysis of an adhesively bonded lap joint was reconsidered. The adherends are approximated by essentially Reissner plates and the adhesive is linearly viscoelastic. The hereditary integrals are used to model the adhesive. A linear integral differential equations system for the shear and the tensile stress in the adhesive is applied. The equations have constant coefficients and are solved by using Laplace transforms. It is shown that if the temperature variation in time can be approximated by a piecewise constant function, then the method of Laplace transforms can be used to solve the problem. A numerical example is given for a single lap joint under various loading conditions.
Delale, F.↗
Analysis of cracks emanating from a circular hole in unidirectional fiber reinforced composites, part 2
An analytical method is developed for cracks emanating from a circular hole in an off-axis unidirectional fiber-reinforced composite. The method which is formulated by using conservation laws of elasticity and fundamental relationships in anisotropic fracture mechanics, provides a convenient and accurate means to examine the complicated crack behavior, when used in conjunction with a suitable numerical scheme such as the finite element method. The formulation is eventually reduced to a system of linear algebraic equations of mixed-mode stress intensity factors. Fracture parameters, describing crack-tip deformation and fracture in the composite, are obtained explicitly. Effects of material anisotropy and crack/hole geometry are examined also. Of particular interest are the energy release rates associated with crack extension; their values are evaluated for various cases. Results show that mixed-mode stress intensity factors and energy release rates associated with the cracks emanating from a hole change very appreciably with fiber orientation in the composite. K sub 1 and G increase monotonically with increasing theta; but K sub 2 reaches its maximum at theta = 45 deg, and then decreases gradually as theta increases further.
Wang, S. S.↗
Subsonic 3-D surface panel method for rapid analysis of multiple geometry perturbations
For any baseline aircraft configuration, a matrix of partial derivatives of surface velocity potential with respect to geometry coordinates is calculated. Linear extrapolation is then used to analyze the subcritical potential flow corresponding to a series of arbitrary small geometry perturbations. Each perturbation analysis is more than an order of magnitude more efficient than a conventional panel method solution because no influence coefficients are calculated and no large system of linear algebraic equations is solved. Wing and wing-fuselage examples are presented to demonstrate that the predicted pressure distributions are nearly exact for large changes to wing camber, thickness, and leading edge radius.
Bristow, D. R.↗
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.↗
Acoustooptic linear algebra processors - Architectures, algorithms, and applications
Architectures, algorithms, and applications for systolic processors are described with attention to the realization of parallel algorithms on various optical systolic array processors. Systolic processors for matrices with special structure and matrices of general structure, and the realization of matrix-vector, matrix-matrix, and triple-matrix products and such architectures are described. Parallel algorithms for direct and indirect solutions to systems of linear algebraic equations and their implementation on optical systolic processors are detailed with attention to the pipelining and flow of data and operations. Parallel algorithms and their optical realization for LU and QR matrix decomposition are specifically detailed. These represent the fundamental operations necessary in the implementation of least squares, eigenvalue, and SVD solutions. Specific applications (e.g., the solution of partial differential equations, adaptive noise cancellation, and optimal control) are described to typify the use of matrix processors in modern advanced signal processing.
Casasent, D.↗
An approximate buckling analysis for rectangular orthotropic plates with centrally located cutouts
An approximate analysis for predicting buckling of rectangular orthotropic composite plates with centrally located cutouts is presented. In this analysis, prebuckling and buckling problems are converted from a two-dimensional to a one-dimensional system of linear differential equations with variable coefficients. The conversion is accomplished by expressing the displacements as series with each element containing a trigonometric function of one coordinate and a coefficient that is an arbitrary function of the other coordinate. Ordinary differential equations are then obtained from a variational principle. Analytical results obtained from the approximate analysis are compared with finite element analyses for isotropic plates and for specially orthotropic plates with central circular cutouts of various sizes. Experimental results for the specially orthotropic plates are also presented. In nearly all cases, the approximate analysis predicts the buckling mode shapes correctly and predicts the buckling loads to within a few percent of the finite element and experimental results.
Nemeth, M. P.↗
The neutral curve for stationary disturbances in rotating-disk flow
The neutral curve for stationary vortex disturbances in rotating-disk flow is computed up to a Reynolds number of 10 to the 7th using the sixth-order system of linear stability equations which includes the effects of streamline curvature and Coriolis force. It is found that the neutral curve has two minima: one at R = 285.36 (upper branch) and the other at R = 440.88 (lower branch). At large Reynolds numbers, the upper branch tends to Stuart's asymptotic solution while the lower branch tends to a solution that is associated with the wave angle corresponding to the direction of zero mean wall shear.
Malik, M. R.↗
Discontinuous solutions to hyperbolic systems under operator splitting
Two-dimensional systems of linear hyperbolic equations are studied with regard to their behavior under a solution strategy that in alternate time-steps solves exactly the component one-dimensional operators. The initial data is a step function across an oblique discontinuity. The manner in which this discontinuity breaks up under repeated applications of the split operator is analyzed, and it is shown that the split solution will fail to match the true solution in any case where the two operators do not share all their eigenvectors. The special case of the fluid flow equations is analyzed in more detail, and it is shown that arbitrary initial data gives rise to pseudo acoustic waves and a non-physical stationary wave. The implications of these findings for the design of high-resolution computing schemes are discussed.
Roe, P. L.↗
Direct finite element solution on an optical laboratory matrix-vector processor
The first optical laboratory system results employing a direct LU decomposition solution of a system of linear algebraic equations are presented for a finite element problem solution. This also represents the first laboratory demonstration of the use of sign-magnitude negative number representation as well as new bit partitioning techniques to increase the accuracy of an optical encoded processor beyond the number of bit channels available.
Casasent, David↗