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NASA NTRS1988

Consistent linearization of the element-independent corotational formulation for the structural analysis of general shells

A consistent linearization is provided for the element-dependent corotational formulation, providing the proper first and second variation of the strain energy. As a result, the warping problem that has plagued flat elements has been overcome, with beneficial effects carried over to linear solutions. True Newton quadratic convergence has been restored to the Structural Analysis of General Shells (STAGS) code for conservative loading using the full corotational implementation. Some implications for general finite element analysis are discussed, including what effect the automatic frame invariance provided by this work might have on the development of new, improved elements.

Rankin, C. C.↗

NASA NTRS1988

Electrochemical growth of linear conducting crystals in microgravity

Much attention has been given to the synthesis of linear conducting materials. These inorganic, organic, and polymeric materials have some very interesting electrical and optical properties, including low temperature superconductivity. Because of the anisotropic nature of these compounds, impurities and defects strongly influences the unique physical properties of such crystals. Investigations have demonstrated that electrochemical growth has provided the most reproducible and purest crystals. Space, specifically microgravity, eliminates phenomena such as buoyancy driven convection, and could permit formation of crystals many times purer than the ones grown to date. Several different linear conductors were flown on Get Away Special G-007 on board the Space Shuttle Columbia, STS 61-C, the first of a series of Project Explorer payloads. These compounds were grown by electrochemical methods, and the growth was monitored by photographs taken throughout the mission. Due to some thermal problems, no crystals of appreciable size were grown. The experimental results will be incorporated into improvements for the next 2 missions of Project Explorer. The results and conclusions of the first mission are discussed.

Cronise, Raymond J., IV↗

NASA NTRS1988

Linear prediction of stationary vector sequences

The class of all linear predictors of minimal order for a stationary vector-valued process is specified in terms of linear transformations on the associated Hankel covariance matrix. Two particular transformations, yielding computationally efficient construction schemes, are proposed.

Baram, Yoram↗

NASA NTRS1988

A labeling procedure for linear finite-state codes

A method to define the labels of the state diagram of a linear finite-state code is presented and investigated. This method is particularly suitable for simple hardware implementation since it simplifies the encoder structure. The method can also be applied to the labeling of a state diagram that is not completely connected to obtain a linear finite state code with larger free distance.

Cheung, K.-M.↗

NASA NTRS1987

A hybrid-stress finite element approach for stress and vibration analysis in linear anisotropic elasticity

A hybrid stress finite element method is developed for accurate stress and vibration analysis of problems in linear anisotropic elasticity. A modified form of the Hellinger-Reissner principle is formulated for dynamic analysis and an algorithm for the determination of the anisotropic elastic and compliance constants from experimental data is developed. These schemes were implemented in a finite element program for static and dynamic analysis of linear anisotropic two dimensional elasticity problems. Specific numerical examples are considered to verify the accuracy of the hybrid stress approach and compare it with that of the standard displacement method, especially for highly anisotropic materials. It is that the hybrid stress approach gives much better results than the displacement method. Preliminary work on extensions of this method to three dimensional elasticity is discussed, and the stress shape functions necessary for this extension are included.

Oden, J. Tinsley↗

NASA NTRS1987

Design of reduced-order state estimators for linear time-varying multivariable systems

The design of reduced-order state estimators for linear time-varying multivariable systems is considered. Employing the concepts of matrix operators and the method of canonical transformations, this paper shows that there exists a reduced-order state estimator for linear time-varying systems that are 'lexicography-fixedly observable'. In addition, the eigenvalues of the estimator can be arbitrarily assigned. A simple algorithm is proposed for the design of the state estimator.

Nguyen, Charles C.↗

NASA NTRS1988

Modified non-linear Burgers' equations and cosmic ray shocks

A reductive perturbation scheme is used to derive a generalized non-linear Burgers' equation, which includes the effects of dispersion, in the long wavelength regime for the two-fluid hydrodynamical model used to describe cosmic ray acceleration by the first-order Fermi process in astrophysical shocks. The generalized Burger's equation is derived for both relativistic and non-relativistic cosmic ray shocks, and describes the time evolution of weak shocks in the theory of diffusive shock acceleration. The inclusion of dispersive effects modifies the phase velocity of the shock obtained from the lower order non-linear Burger's equation through the introduction of higher order terms from the long wavelength dispersion equation. The travelling wave solution of the generalized Burgers' equation for a single shock shows that larger cosmic ray pressures result in broader shock transitions. The results for relativistic shocks show a steepening of the shock as the shock speed approaches the relativistic cosmic ray sound speed. The dependence of the shock speed on the cosmic ray pressure is also discussed.

Zank, G. P.↗

NASA NTRS1987

Mean-square-error bounds for reduced-order linear state estimators

The mean-square error of reduced-order linear state estimators for continuous-time linear systems is investigated. Lower and upper bounds on the minimal mean-square error are presented. The bounds are readily computable at each time-point and at steady state from the solutions to the Ricatti and the Liapunov equations. The usefulness of the error bounds for the analysis and design of reduced-order estimators is illustrated by a practical numerical example.

Baram, Y.↗

NASA NTRS1987

Stability and robustness of slowly time-varying linear systems

A well-known result for finite-dimensional time-varying linear systems is that if each `frozen time' is stable, then the time-varying system is stable for sufficiently slow time-variations. These results are reviewed and extended to a class of Volterra integrodifferential equations, specifically, differential equations with a convolution operator in the right-hand-side. The results are interpreted in the context of robustness of time-varying linear systems with special emphasis on analysis of gain-scheduled control systems.

Shamma, Jeff S.↗

NASA NTRS1987

Estimability and regulability of linear systems

A linear state-space system will be said to be estimable if in estimating its state from its output the posterior error covariance matrix is strictly smaller than the prior covariance matrix. It will be said to be regulable if the quadratic cost of state feedback control is strictly smaller than the cost when no feedback is used. These properties, which are are shown to be dual, are different from the well known observability and controllability properties of linear systems. Necessary and sufficient conditions for estimability and regulability are derived for time variant and time invariant systems, in discrete and continuous time.

Baram, Yoram↗

NASA NTRS1988

Linear stability of the H(+) - O(+) two-stream interaction in a magnetized plasma

The Bergman and Lotko (1986) analysis of the linear stability of the H(+) - O(+) two-stream interactions for parallel propagating wave modes is extended to an analysis of the stability of wave modes that propagate at an angle to the terrestrial magnetic field. The analaysis provides an insight into the manner in which the oblique modes linearly couple and become unstable; the conditions under which the particular couplings are important; the qualitative behavior of the frequency, wave vector, and growth rate of the most unstable mode; and the changes to these introduced by finite ion temperatures. Consideration is also given to the relative importance of obliquely propagating and parallel propagating waves for the condition when the velocity difference between two ion beams is below the upper limit.

Bergmann, R.↗

NASA NTRS1988

Relations between broad-band linear polarization and Ca II H and K emission in late-type dwarf stars

Broadband UBV linear polarization data acquired for a sample of late-type dwarfs are compared with contemporaneous measurements of Ca II H and K line core emission. A weighted average of the largest values of the polarization degree is shown to be the best parameter for chromospheric activity diagnosis. The average maximum polarization in the UV is found to increase from late-F to late-G stars. It is noted that polarization in the U band is considerably more sensitive to activity variations than that in the B or V bands. The results indicate that stellar magnetic fields and the resulting saturation in the Zeeman-sensitive absorption lines are the most probably source of linear polarization in late-type main-sequence stars.

Huovelin, Juhani↗

NASA NTRS1988

Simulation and non-linear stage of the electrostatic waves observed during the AMPTE lithium release in the solar wind

During the AMPTE lithium releases in the solar wind intense electrostatic waves with frequencies between a few tens of Hz to several kHz were observed outside the diamagnetic cavity. The results of linear Vlasov theory have suggested that these waves may be generated through two types of instabilities. One is the ion-ion instability associated with the relative drift between the lithium ions and the solar wind protons, and the other is the ion-acoustic instability due to the relative drift between the electrons and the ions. In order to look at the non-linear behavior of the wave-particle interactions, and discern the effect of waves on the particles, full particle electrostatic simulations have been performed, and the results are presented here. It is shown that the ion-ion instability whose phase velocity is oblique to the solar wind velocity can cause considerable anisotropic 'heating' of both the lithium ions and the solar wind protons.

Omidi, N.↗

NASA NTRS1988

Sequential design of linear quadratic state regulators via the optimal root-locus techniques

The use of well-known root-locus techniques for sequentially finding the weighting matrices and the linear quadratic state regulators of multivariable control systems in the frequency domain is considered. This sequential design method permits the retention of some stable open-loop poles and the associated eigenvectors in the closed-loop system; it also allows some optimal closed-loop poles to be placed in a specific region of the complex plane. In addition, it provides a design procedure for determining the weighting matrices and linear quadratic state regulators for the optimal control of multivariable systems in the frequency domain.

Shieh, L. S.↗

NASA NTRS1988

A linearized model of helicopter dynamics including correlation with flight test

A linearized model for rotorcraft dynamics has been developed through the use of symbolic automatic equation generation techniques. The dynamic model has been formulated in a unique way such that it can be used to analyze a variety of rotor/body coupling problems including a rotor mounted on a flexible shaft with a number of modes as well as free flight stability and control characteristics. Direct comparison of the time response to longitudinaol, lateral and directional control inputs at various trim conditions shows that the linear model yields good to very good correlation with flight test. In particular it is also shown that a dynamic inflow model is essential to obtain good time response correlation especially for the hover trim condition and also that at translational flight trim conditions the main rotor wake interaction with the tail rotor and fixed tail surfaces is a significant contributor to the response. A relatively simple model for the downwash and sidewash at tail surfaces based on flat vortex wake theory is shown to produce good agreement. Finaly, the influence of rotor flap and lag dynamics on automatic control systems feedback gain limitations is investigated with the model.

Curtiss, H. C.↗

NASA NTRS1988

Coprime factorizations in stable linear systems

A block-diagonal linear (not necessarily time-invariant) map P with a right-coprime factorization ND-1 (or a left-coprime factorization D-1N) is considered. It is shown that the individual blocks in P have right-coprime factorizations (left-coprime factorizations, respectively) if and only if the denominator map D has a special block-triangular structure. This condition is applied to the stable linear feedback system S(P1,P2).

Desoer, Charles A.↗

NASA NTRS1987

Linear and nonlinear mechanical properties of a series of epoxy resins

The linear viscoelastic properties have been measured for a series of bisphenol-A-based epoxy resins cured with the diamine DDS. The linear viscoelastic master curves were constructed via time-temperature superposition of frequency dependent G-prime and G-double-prime isotherms. The G-double-prime master curves exhibited two sub-Tg transitions. Superposition of isotherms in the glass-to-rubber transition (i.e., alpha) and the beta transition at -60 C was achieved by simple horizontal shifts in the log frequency axis; however, in the region between alpha and beta, superposition could not be effected by simple horizontal shifts along the log frequency axis. The different temperature dependency of the alpha and beta relaxation mechanisms causes a complex response of G-double-prime in the so called alpha-prime region. A novel numerical procedure has been developed to extract the complete relaxation spectra and its temperature dependence from the G-prime and G-double-prime isothermal data in the alpha-prime region.

Curliss, D. B.↗

NASA NTRS1988

Approximation theory for LQG (Linear-Quadratic-Gaussian) optimal control of flexible structures

An approximation theory is presented for the LQG (Linear-Quadratic-Gaussian) optimal control problem for flexible structures whose distributed models have bounded input and output operators. The main purpose of the theory is to guide the design of finite dimensional compensators that approximate closely the optimal compensator. The optimal LQG problem separates into an optimal linear-quadratic regulator problem and an optimal state estimation problem. The solution of the former problem lies in the solution to an infinite dimensional Riccati operator equation. The approximation scheme approximates the infinite dimensional LQG problem with a sequence of finite dimensional LQG problems defined for a sequence of finite dimensional, usually finite element or modal, approximations of the distributed model of the structure. Two Riccati matrix equations determine the solution to each approximating problem. The finite dimensional equations for numerical approximation are developed, including formulas for converting matrix control and estimator gains to their functional representation to allow comparison of gains based on different orders of approximation. Convergence of the approximating control and estimator gains and of the corresponding finite dimensional compensators is studied. Also, convergence and stability of the closed-loop systems produced with the finite dimensional compensators are discussed. The convergence theory is based on the convergence of the solutions of the finite dimensional Riccati equations to the solutions of the infinite dimensional Riccati equations. A numerical example with a flexible beam, a rotating rigid body, and a lumped mass is given.

Gibson, J. S.↗