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NASA NTRS1988

Optimal post-experiment estimation of poorly modeled dynamic systems

Recently, a novel strategy for post-experiment state estimation of discretely-measured dynamic systems has been developed. The method accounts for errors in the system dynamic model equations in a more general and rigorous manner than do filter-smoother algorithms. The dynamic model error terms do not require the usual process noise assumptions of zero-mean, symmetrically distributed random disturbances. Instead, the model error terms require no prior assumptions other than piecewise continuity. The resulting state estimates are more accurate than filters for applications in which the dynamic model error clearly violates the typical process noise assumptions, and the available measurements are sparse and/or noisy. Estimates of the dynamic model error, in addition to the states, are obtained as part of the solution of a two-point boundary value problem, and may be exploited for numerous reasons. In this paper, the basic technique is explained, and several example applications are given. Included among the examples are both state estimation and exploitation of the model error estimates.

Mook, D. Joseph↗

NASA NTRS1988

The dynamics and control of large flexible space structures, part 11

A mathematical model is developed to predict the dynamics of the proposed Spacecraft Control Laboratory Experiment during the stationkeeping phase. The Shuttle and reflector are assumed to be rigid, while the mass connecting the Shuttle to the reflector is assumed to be flexible with elastic deformations small as compared with its length. It is seen that in the presence of gravity-gradient torques, the system assumes a new equilibrium position primarily due to the offset in the mass attachment point to the reflector from the reflector's mass center. Control is assumed to be provided through the Shuttle's three torquers and throught six actuators located by painrs at two points on the mass and at the reflector mass center. Numerical results confirm the robustness of an LQR derived control strategy during stationkeeping with maximum control efforts significantly below saturation levels. The linear regulator theory is also used to derive control laws for the linearized model of the rigidized SCOLE configuration where the mast flexibility is not included. It is seen that this same type of control strategy can be applied for the rapid single axis slewing of the SCOLE through amplitudes as large as 20 degrees. These results provide a definite trade-off between the slightly larger slewing times with the considerable reduction in over-all control effort as compared with the results of the two point boundary value problem application of Pontryagin's Maximum Principle.

Bainum, Peter M.↗

NASA NTRS1988

Boundary force method analyses of notched plates with cracks

The boundary-force method developed by Tan (1985) and Tan et al. (1986) is extended to two-dimensional mixed boundary-value problems in fracture mechanics. The formulation is explained in detail, and results are presented in tables and graphs for sample problems involving an edge crack emanating from a semielliptical and a semicircular notch and an edge crack emanating from a V notch. The accuracy of the approach is demonstrated by comparisons with collocation solutions for an edge-clamped center-cracked tension specimen.

Newman, J. C., Jr.↗

NASA NTRS1988

Methods for analysis of cracks in three-dimensional solids

Various methods used for determining stress-intensity factors for cracked three-dimensional bodies are reviewed. The review is limited to the determination of mode-I stress-intensity factors. Some exact solutions for cracks in infinite solids are presented, and various approximate methods that have been used for the solution of the boundary-value problem of finite solids with cracks are addressed. The techniques used to extract the stress-intensity factors from these solutions are considered, and various methods used to calculate the stress-intensity factors for through-the-thickness cracks, semielliptical surface cracks, and quarter-elliptical corner cracks at holes are compared for the case of remote tensile loading.

Raju, I. S.↗

NASA NTRS1989

Galerkin/Runge-Kutta discretizations for parabolic equations with time-dependent coefficients

A new class of fully discrete Galerkin/Runge-Kutta methods is constructed and analyzed for linear parabolic initial boundary value problems with time dependent coefficients. Unlike any classical counterpart, this class offers arbitrarily high order convergence while significantly avoiding what has been called order reduction. In support of this claim, error estimates are proved, and computational results are presented. Additionally, since the time stepping equations involve coefficient matrices changing at each time step, a preconditioned iterative technique is used to solve the linear systems only approximately. Nevertheless, the resulting algorithm is shown to preserve the original convergence rate while using only the order of work required by the base scheme applied to a linear parabolic problem with time independent coefficients. Furthermore, it is noted that special Runge-Kutta methods allow computations to be performed in parallel so that the final execution time can be reduced to that of a low order method.

Keeling, Stephen L.↗

NASA NTRS1989

Use of the quasilinearization algorithm for the simulation of LSS slewing

The use of the Maximum Principle for the large angle slewing of large space structures (LSS) usually results in the so-called two-point boundary-value problem, in which many requirements (e.g., minimum time, small amplitude, and limited control power, etc.) must be satisfied simultaneously. The successful solution of this problem depends largely on the use of an efficient numerical algorithm. There are many candidate algorithms available for this problem (e.g., quasilinearization, gradient, etc.). Here researchers discuss only the quasilinearization method which has been used for several cases of large angle slewing of LSS. The basic idea of this algorithm is to make a series of successive approximations of the solution from a particular solvable case (linear or nonlinear) to a more general practical case. For the rigid spacecraft slewing problem with no constraints on the controls, the solution procedure can be found in the literature. This procedure needs to be modified if a minimum time for the slewing problem is desired with control limits given. Recently, an indirect method for finding the minimum time was developed to meet all these requirements. For the general mixed (including both rigid and flexible parts) problem, an additional constraint of small vibrational amplitude on the flexible parts is imposed.

Li, Feiyue↗

NASA NTRS1989

Mechanics of the crack path formation

A detailed analysis of experimentally obtained curvilinear crack path trajectories formed in a heterogeneous stress field is presented. Experimental crack path trajectories were used as data for numerical simulations, recreating the actual stress field governing the development of the crack path. Thus, the current theories of crack curving and kinking could be examined by comparing them with the actual stress field parameters as they develop along the experimentally observed crack path. The experimental curvilinear crack path trajectories were formed in the tensile specimens with a hole positioned in the vicinity of a potential crack path. The numerical simulation, based on the solution of equivalent boundary value problems with the possible perturbations of the crack path, is presented here.

Rubinstein, Asher A.↗

NASA NTRS1989

An Eigenvalue Analysis of finite-difference approximations for hyperbolic IBVPs

The eigenvalue spectrum associated with a linear finite-difference approximation plays a crucial role in the stability analysis and in the actual computational performance of the discrete approximation. The eigenvalue spectrum associated with the Lax-Wendroff scheme applied to a model hyperbolic equation was investigated. For an initial-boundary-value problem (IBVP) on a finite domain, the eigenvalue or normal mode analysis is analytically intractable. A study of auxiliary problems (Dirichlet and quarter-plane) leads to asymptotic estimates of the eigenvalue spectrum and to an identification of individual modes as either benign or unstable. The asymptotic analysis establishes an intuitive as well as quantitative connection between the algebraic tests in the theory of Gustafsson, Kreiss, and Sundstrom and Lax-Richtmyer L(sub 2) stability on a finite domain.

Warming, Robert F.↗

NASA NTRS1989

The dynamics and control of large flexible space structures - 12, supplement 11

The rapid 2-D slewing and vibrational control of the unsymmetrical flexible SCOLE (Spacecraft Control Laboratory Experiment) with multi-bounded controls is considered. Pontryagin's Maximum Principle is applied to the nonlinear equations of the system to derive the necessary conditions for the optimal control. The resulting two point boundary value problem is then solved by using the quasilinearization technique, and the near minimum time is obtained by sequentially shortening the slewing time until the controls are near the bang-bang type. The tradeoff between the minimum time and the minimum flexible amplitude requirements is discussed. The numerical results show that the responses of the nonlinear system are significantly different from those of the linearized system for rapid slewing. The SCOLE station-keeping closed loop dynamics are re-examined by employing a slightly different method for developing the equations of motion in which higher order terms in the expressions for the mast modal shape functions are now included. A preliminary study on the effect of actuator mass on the closed loop dynamics of large space systems is conducted. A numerical example based on a coupled two-mass two-spring system illustrates the effect of changes caused in the mass and stiffness matrices on the closed loop system eigenvalues. In certain cases the need for redesigning control laws previously synthesized, but not accounting for actuator masses, is indicated.

Bainum, Peter M.↗

NASA NTRS1989

Modified Chebyshev pseudospectral method with O(N exp -1) time step restriction

The extreme eigenvalues of the Chebyshev pseudospectral differentiation operator are O(N exp 2) where N is the number of grid points. As a result of this, the allowable time step in an explicit time marching algorithm is O(N exp -2) which, in many cases, is much below the time step dictated by the physics of the partial differential equation. A new set of interpolating points is introduced such that the eigenvalues of the differentiation operator are O(N) and the allowable time step is O(N exp -1). The properties of the new algorithm are similar to those of the Fourier method. The new algorithm also provides a highly accurate solution for non-periodic boundary value problems.

Kosloff, Dan↗

NASA NTRS1989

The role of topography in geodetic gravity field modelling

Masses associated with the topography, bathymetry, and its isostatic compensation are a dominant source of gravity field variations, especially at shorter wavelengths. On global scales the topographic/isostatic effects are also significant, except for the lowest harmonics. In practice, though, global effects need not be taken into account as such effects are included in the coefficients of the geopotential reference fields. On local scales, the short-wavelength gravity variations due to the topography may, in rugged terrain, be an order of magnitude larger than other effects. In such cases, explicit or implicit terrain reduction procedures are mandatory in order to obtain good prediction results. Such effects may be computed by space-domain integration or by fast Fourier transformation (FFT) methods. Numerical examples are given for areas of the Canadian Rockies. In principle, good knowledge of the topographic densities is required to produce the smoothest residual field. Densities may be determined from sample measurements or by gravimetric means, but both are somewhat troublesome methods in practice. The use of a standard density, e.g., 2.67 g/cu cm, may often yield satisfactory results and may be put within a consistent theoretical framework. The independence of density assumptions is the key point of the classical Molodensky approach to the geodetic boundary value problem. The Molodensky solutions take into account that land gravity field observations are done on a non-level surface. Molodensky's problem may be solved by integral expansions or more effective FFT methods, but the solution should not be intermixed with the use of terrain reductions. The methods are actually complimentary and may both be required in order to obtain the smoothest possible signal, least prone to aliasing and other effects coming from sparse data coverage, typical of rugged topography.

Forsberg, R.↗

NASA NTRS1989

Compact finite volume methods for the diffusion equation

An approach to treating initial-boundary value problems by finite volume methods is described, in which the parallel between differential and difference arguments is closely maintained. By using intrinsic geometrical properties of the volume elements, it is possible to describe discrete versions of the div, curl, and grad operators which lead, using summation-by-parts techniques, to familiar energy equations as well as the div curl = 0 and curl grad = 0 identities. For the diffusion equation, these operators describe compact schemes whose convergence is assured by the energy equations and which yield both the potential and the flux vector with second order accuracy. A simplified potential form is especially useful for obtaining numerical results by multigrid and alternating direction implicit (ADI) methods. The treatment of general curvilinear coordinates is shown to result from a specialization of these general results.

Rose, Milton E.↗

NASA NTRS1989

Extension-torsion coupling behavior of advanced composite tilt-rotor blades

An analytic model was developed to study the extension-bend-twist coupling behavior of an advanced composite helicopter or tilt-rotor blade. The outer surface of the blade is defined by rotating an arbitrary cross section about an initial twist axis. The cross section can be nonhomogeneous and composed of generally anisotropic materials. The model is developed based upon a three dimensional elasticity approach that is recast as a coupled two-dimensional boundary value problem defined in a curvilinear coordinate system. Displacement solutions are written in terms of known functions that represent extension, bending, and twisting and unknown functions for local cross section deformations. The unknown local deformation functions are determined by applying the principle of minimum potential energy to the discretized two-dimensional cross section. This is an application of the Ritz method, where the trial function family is the displacement field associated with a finite element (8-node isoparametric quadrilaterals) representation of the section. A computer program was written where the cross section is discretized into 8-node quadrilateral subregions. Initially the program was verified using previously published results (both three-dimensional elasticity and technical beam theory) for pretwisted isotropic bars with an elliptical cross section. In addition, solid and thin-wall multi-cell NACA-0012 airfoil sections were analyzed to illustrate the pronounced effects that pretwist, initial twist axis location, and spar location has on coupled behavior. Currently, a series of advanced composite airfoils are being modeled in order to assess how the use of laminated composite materials interacts with pretwist to alter the coupling behavior of the blade. These studies will investigate the use of different ply angle orientations and the use of symmetric versus unsymmetric laminates.

Kosmatka, J. B.↗

NASA NTRS1990

Theory of circular dielectric waveguide with anisotropic sheet cover

A circular dielectric waveguide consisting of an isotropic core covered with a thin anisotropic sheet is considered. The sheet is represented as a jump immittance and Maxwell's equations are applied. Solution of the boundary value problem yields the characteristic equation, or dispersion relation, which is then solved numerically. The results are verified for the step-index fiber and circular, metallic waveguides. Finally, examples are included to investigate the effects of the anisotropic sheet.

Kish, Jerome Thomas↗

NASA NTRS1989

Recent advances in engineering science; Proceedings of the A. Cemal Eringen Symposium, University of California, Berkeley, June 20-22, 1988

Various papers on recent advances in engineering science are presented. Some individual topics addressed include: advances in adaptive methods in computational fluid mechanics, mixtures of two medicomorphic materials, computer tests of rubber elasticity, shear bands in isotropic micropolar elastic materials, nonlinear surface wave and resonator effects in magnetostrictive crystals, simulation of electrically enhanced fibrous filtration, plasticity theory of granular materials, dynamics of viscoelastic media with internal oscillators, postcritical behavior of a cantilever bar, boundary value problems in nonlocal elasticity, stability of flexible structures with random parameters, electromagnetic tornadoes in earth's ionosphere and magnetosphere, helicity fluctuations and the energy cascade in turbulence, mechanics of interfacial zones in bonded materials, propagation of a normal shock in a varying area duct, analytical mechanics of fracture and fatigue.

Koh, Severino L.↗

NASA NTRS1990

Parallel computation using boundary elements in solid mechanics

The inherent parallelism of the boundary element method is shown. The boundary element is formulated by assuming the linear variation of displacements and tractions within a line element. Moreover, MACSYMA symbolic program is employed to obtain the analytical results for influence coefficients. Three computational components are parallelized in this method to show the speedup and efficiency in computation. The global coefficient matrix is first formed concurrently. Then, the parallel Gaussian elimination solution scheme is applied to solve the resulting system of equations. Finally, and more importantly, the domain solutions of a given boundary value problem are calculated simultaneously. The linear speedups and high efficiencies are shown for solving a demonstrated problem on Sequent Symmetry S81 parallel computing system.

Chien, L. S.↗

NASA NTRS1990

Rapid in-plane maneuvering of the flexible orbiting SCOLE

The rapid two-dimensional slewing and vibrational control of the unsymmetrical flexible SCOLE (Spacecraft Control Laboratory Experiment) with multi-bounded controls has been considered. Pontryagin's Maximum Principle has been applied to the nonlinear equations of the system to derive the necessary conditions for the optimal control. The resulting two-point boundary-value problem is then solved by using the quasilinearization technique, and the near-minimum time is obtained by sequentially shortening the slewing time until the controls are near the bang-bang type. The trade-off between the minimum time and the minimum flexible amplitude requirements has been discussed. The numerical results show that the responses of the nonlinear system are significantly different from those of the linearized system for rapid slewing.

Bainum, Peter M.↗

NASA NTRS1991

Compressible flows with periodic vortical disturbances around lifting airfoils

A numerical method is developed for solving periodic, three-dimensional, vortical flows around lifting airfoils in subsonic flow. The first-order method that is presented fully accounts for the distortion effects of the nonuniform mean flow on the convected upstream vortical disturbances. The unsteady velocity is split into a vortical component which is a known function of the upstream flow conditions and the Lagrangian coordinates of the mean flow, and an irrotational field whose potential satisfies a nonconstant-coefficient, inhomogeneous, convective wave equation. Using an elliptic coordinate transformation, the unsteady boundary value problem is solved in the frequency domain on grids which are determined as a function of the Mach number and reduced frequency. The numerical scheme is validated through extensive comparisons with known solutions to unsteady vortical flow problems. In general, it is seen that the agreement between the numerical and analytical results is very good for reduced frequencies ranging from 0 to 4, and for Mach numbers ranging from .1 to .8. Numerical results are also presented for a wide variety of flow configurations for the purpose of determining the effects of airfoil thickness, angle of attack, camber, and Mach number on the unsteady lift and moment of airfoils subjected to periodic vortical gusts. It is seen that each of these parameters can have a significant effect on the unsteady airfoil response to the incident disturbances, and that the effect depends strongly upon the reduced frequency and the dimensionality of the gust. For a one-dimensional (transverse) or two-dimensional (transverse and longitudinal) gust, the results indicate that airfoil thickness increases the unsteady lift and moment at the low reduced frequencies but decreases it at the high reduced frequencies. The results show that an increase in airfoil Mach number leads to a significant increase in the unsteady lift and moment for the low reduced frequencies, but a significant decrease for the high reduced frequencies.

Scott, James R.↗