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NASA NTRS1987

Nonlinear beam kinematics by decomposition of the rotation tensor

A simple matrix expression is obtained for the strain components of a beam in which the displacements and rotations are large. The only restrictions are on the magnitudes of the strain and of the local rotation, a newly-identified kinematical quantity. The local rotation is defined as the change of orientation of material elements relative to the change of orientation of the beam reference triad. The vectors and tensors in the theory are resolved along orthogonal triads of base vectors centered along the undeformed and deformed beam reference axes, so Cartesian tensor notation is used. Although a curvilinear coordinate system is natural to the beam problem, the complications usually associated with its use are circumvented. Local rotations appear explicitly in the resulting strain expressions, facilitating the treatment of beams with both open and closed cross sections in applications of the theory. The theory is used to obtain the kinematical relations for coupled bending, torsion, extension, shear deformation, and warping of an initially curved and twisted beam.

Danielson, D. A.↗

NASA NTRS1987

The effect of wave breaking on wave spectrum in water of finite depth

An approximate method is devised to compute the energy-containing portion of the spectrum of waves in water of finite depth, taking into account the effect of wave breaking. It is assumed that there exists a linear and Gaussian ideal wave train whose spectrum is first calculated using the wave energy flux balance equation without considering wave breaking. The Miche wave-breaking criterion for waves in water of finite depth is then applied to limit the wave elevation and establish an expression for the breaking wave elevation in terms of the elevation and elevation's second time derivative of the ideal waves. Simple expressions for the mean value, the mean square value, and the spectrum of the breaking waves are then obtained, and numerical results are presented graphically.

Tung, C. C.↗

NASA NTRS1987

Magnetic field-aligned electric field acceleration and the characteristics of the optical aurora

The long-recognized association of brighter aurora with more deeply penetrating, and hence more energetic, electrons is examined. Using the Knight (1973) relation between the magnetic-field-aligned current density and potential drop (derived from the theory of single-particle motion in the presence of a magnetic-field-aligned electric field), an approximate expression relating the energy flux of the precipitating electrons over discrete aurora and the mean particle energy is derived. This expression is used in conjunction with an auroral optical excitation and emission model to specify the dependence of the red/blue ratio of auroral optical emissions on the brightness of the aurora. It is shown that the quantitative predictions of the discrete auroral theory are in accord with observations of the aurora.

Christensen, A. B.↗

NASA NTRS1987

S-matrix calculation of the triple-alpha reaction

An S-matrix formalism is developed which can be applied to reactions in which electron screening is important, including three-body reactions and reactions involving weak interactions. The various regimes of the triple-alpha reactions are systematically discussed, and a new nonresonant regime at high densities is identified. Using the S-matrix formalism, an analytic expression is obtained for the screened triple-alpha reactions which is accurate for all temperatures and densities. The results are compared with those of Cameron (1959) and Nomoto et al. (1985), and the latter's expression for the unscreened reaction rate is verified. However, it is shown that the reaction rate in the pycnonuclear regime cannot be obtained from the unscreened rate using a screening factor, and that the results of Nomoto et al. therefore cannot be used in this regime.

Fushiki, Ikko↗

NASA NTRS1987

Flexible system model reduction and control system design based upon actuator and sensor influence functions

A model reduction technique based on aggregation with respect to sensor and actuator influence functions rather than modes is presented for large systems of coupled second-order differential equations. Perturbation expressions which can predict the effects of spillover on both the reduced-order plant model and the neglected plant model are derived. For the special case of collocated actuators and sensors, these expressions lead to the derivation of constraints on the controller gains that are, given the validity of the perturbation technique, sufficient to guarantee the stability of the closed-loop system. A case study demonstrates the derivation of stabilizing controllers based on the present technique. The use of control and observation synthesis in modifying the dimension of the reduced-order plant model is also discussed. A numerical example is provided for illustration.

Yam, Yeung↗

NASA NTRS1987

Determination of the scaled optical thickness of clouds from reflected solar radiation measurements

A method for determination of scaled optical thickness from reflected solar radiation measurements is described in which measurements of the reflected function are compared with asymptotic expressions for the reflection function of optically thick layers. Analytic formulas are derived which show the dependence of the reflection and transmission functions of nonabsorbing atmospheres on cloud optical thickness, ground albedo, and the asymmetry factor. An expression is derived which shows that the ground albedo produces a constant bias in the derived optical thickness, regardless of the value of the measured reflection function. Results from an analysis of high-resolution images, obtained in Oklahoma, of the reflection function of clouds shows the impact of details of the single scattering phase function on the derived optical thickness.

King, Michael D.↗

NASA NTRS1988

Stress-and-Strain Analysis Of Hot Metal/Fiber Composites

Macroscopic mechanical properties derived from micromechanics. Stress-and-strain equations developed to express microscopic and macroscopic mechanical properties of metals reinforced by unidirectional fibers, over range of temperatures. New equations reduce computational load by providing approximate, closed-form expressions for microscopic and pseudohomogeneous anisotropic properties of single ply reinforced by unidirectional fibers. Typical application is calculation of residual stress in newly manufactured article.

Hopkins, Dale A.↗

NASA NTRS1988

Regressed relations for forced convection heat transfer in a direct injection stratified charge rotary engine

Currently, the heat transfer equation used in the rotary combustion engine (RCE) simulation model is taken from piston engine studies. These relations have been empirically developed by the experimental input coming from piston engines whose geometry differs considerably from that of the RCE. The objective of this work was to derive equations to estimate heat transfer coefficients in the combustion chamber of an RCE. This was accomplished by making detailed temperature and pressure measurements in a direct injection stratified charge (DISC) RCE under a range of conditions. For each specific measurement point, the local gas velocity was assumed equal to the local rotor tip speed. Local physical properties of the fluids were then calculated. Two types of correlation equations were derived and are described in this paper. The first correlation expresses the Nusselt number as a function of the Prandtl number, Reynolds number, and characteristic temperature ratio; the second correlation expresses the forced convection heat transfer coefficient as a function of fluid temperature, pressure and velocity.

Lee, Chi M.↗

NASA NTRS1987

Investigation, development, and application of optimal output feedback theory. Volume 3: The relationship between dynamic compensators and observers and Kalman filters

Relationships between observers, Kalman Filters and dynamic compensators using feedforward control theory are investigated. In particular, the relationship, if any, between the dynamic compensator state and linear functions of a discrete plane state are investigated. It is shown that, in steady state, a dynamic compensator driven by the plant output can be expressed as the sum of two terms. The first term is a linear combination of the plant state. The second term depends on plant and measurement noise, and the plant control. Thus, the state of the dynamic compensator can be expressed as an estimator of the first term with additive error given by the second term. Conditions under which a dynamic compensator is a Kalman filter are presented, and reduced-order optimal estimaters are investigated.

Broussard, John R.↗

NASA NTRS1988

Calculation of symmetric and asymmetric vortex seperation on cones and tangent ogives based on discrete vortex models

An inviscid discrete vortex model, with newly derived expressions for the tangential velocity imposed at the separation points, is used to investigate the symmetric and asymmetric vortex separation on cones and tangent ogives. The circumferential locations of separation are taken from experimental data. Based on a slender body theory, the resulting simultaneous nonlinear algebraic equations in a cross-flow plane are solved with Broyden's modified Newton-Raphson method. Total force coefficients are obtained through momentum principle with new expressions for nonconical flow. It is shown through the method of function deflation that multiple solutions exist at large enough angles of attack, even with symmetric separation points. These additional solutions are asymmetric in vortex separation and produce side force coefficients which agree well with data for cones and tangent ogives.

Chin, S.↗

NASA NTRS1987

Relativistic electromagnetic waves in an electron-ion plasma

High power laser beams can drive plasma particles to relativistic energies. An accurate description of strong waves requires the inclusion of ion dynamics in the analysis. The equations governing the propagation of relativistic electromagnetic waves in a cold electron-ion plasma can be reduced to two equations expressing conservation of energy-momentum of the system. The two conservation constants are functions of the plasma stream velocity, the wave velocity, the wave amplitude, and the electron-ion mass ratio. The dynamic parameter, expressing electron-ion momentum conversation in the laboratory frame, can be regarded as an adjustable quantity, a suitable choice of which will yield self-consistent solutions when other plasma parameters were specified. Circularly polarized electromagnetic waves and electrostatic plasma waves are used as illustrations.

Chian, Abraham C.-L.↗

NASA NTRS1988

Toward a comprehensive theory for the sweeping of trapped radiation by inert orbiting matter

There is a need to calculate loss rates when trapped Van Allen radiation encounters inert orbiting material such as planetary rings and satellites. An analytic expression for the probability of a hit in a bounce encounter is available for all cases where the absorber is spherical and the particles are gyrotropically distributed on a cylindrical flux tube. The hit probability is a function of the particle's pitch angle, the size of the absorber, and the distance between flux tube and absorber, when distances are scaled to the gyroradius of a particle moving perpendicular to the magnetic field. Using this expression, hit probabilities have been computed in drift encounters for all regimes of particle energies and absorber sizes. This technique generalizes the approach to sweeping lifetimes, and is particularly suitable for attacking the inverse problem, where one is given a sweeping signature and wants to deduce the properties of the absorber(s).

Fillius, Walker↗

NASA NTRS1988

Optimal placement of tuning masses for vibration reduction in helicopter rotor blades

Described are methods for reducing vibration in helicopter rotor blades by determining optimum sizes and locations of tuning masses through formal mathematical optimization techniques. An optimization procedure is developed which employs the tuning masses and corresponding locations as design variables which are systematically changed to achieve low values of shear without a large mass penalty. The finite-element structural analysis of the blade and the optimization formulation require development of discretized expressions for two performance parameters: modal shaping parameter and modal shear amplitude. Matrix expressions for both quantities and their sensitivity derivatives are developed. Three optimization strategies are developed and tested. The first is based on minimizing the modal shaping parameter which indirectly reduces the modal shear amplitudes corresponding to each harmonic of airload. The second strategy reduces these amplitudes directly, and the third strategy reduces the shear as a function of time during a revolution of the blade. The first strategy works well for reducing the shear for one mode responding to a single harmonic of the airload, but has been found in some cases to be ineffective for more than one mode. The second and third strategies give similar results and show excellent reduction of the shear with a low mass penalty.

Pritchard, Jocelyn I.↗

NASA NTRS1988

A physically consistent model for artificial dissipation in transonic potential flow computations

The effect that artificial dissipation has on numerical solutions of the transonic Full Potential Equation (FPE) are investigated by comparing the artificially dissipative FPE to a Physically Dissipative Potential (PDP) equation. Analytic expressions were derived from the variables C and M sub c that are used in the artificial density formulation. It was shown that these new values generate artificial dissipation which is equivalent to the physical dissipation existing in the PDP equation. The new expression for the variables C and M sub c can easily be incorporated into the existing full potential codes which are based either on the artificial density or on the artificial viscosity formulation. A comparison of Physically Dissipative Potential (PDP), Artificial Density or Viscosity (ADV), Artificial Mass Flux (AMF), and ADV with variable C and M sub c formulation (MCC) is also presented.

Dulikravich, George S.↗

NASA NTRS1986

Review of convectively cooled structures for hypersonic flight

Resurgent interest in development of Aerospace Plane and Orient Express type vehicles promises to stretch structural technology for hypersonic flight vehicles to the uppermost limits. Significant portions of the structure may require active cooling of some type to survive hostile environments. Despite a lack of recent research activity for cooled structures, a significant body of unclassified knowledge exists concerning such structures. Contractual and in-house research conducted mainly by NASA's Langley Research Center during the 60's and 70's on vehicles very similar to the proposed Orient Express has provided a substantial data base for convectively cooled hypersonic flight structures. Specifically, results are presented for regeneratively cooled structural concepts which have a relatively high heat flux capability and use the hydrogen fuel directly as a coolant; and for structural concepts which use a secondary coolant loop to absorb incident heating and then transfer the absorbed heat to the liquid hydrogen fuel as it flows to the engines. Results are presented to indicate application regions in terms of heat flux capability for various concepts and benefits for each concept. Experience gained and costs are discussed in terms of heat flux capability for various concepts and benefits for each concept. Additionally, experience gained and cost involved with design, fabrication, and testing of full-scale convectively cooled structures are discussed.

Shore, Charles P.↗

NASA NTRS1987

Delay time for the onset of beam plasma discharge

The interaction of a nonrelativistic electron beam with a neutral gas in a large chamber is considered, and the time interval before ignition of beam plasma discharge (BPD) is studied. A new theoretical expression for the time delay before BPD ignition is found as a function of the critical current necessary for BPD to be established. There are two parameters in the theoretical expression, and both are derived from two different experiments. These parameters are used to write the time evolution equation for plasma density as a function of time.

Parish, J. L.↗

NASA NTRS1987

Current to the ionosphere following a lightning stroke

A simple analytical expression for calculating the total current waveform to the ionosphere after a lightning stroke is derived. The validity of this expression is demonstrated by comparison with a more rigorous computer solution of Maxwell's equations. The analytic model demonstrates that the temporal variation of the current induced in the ionosphere and global circuit and the corresponding return current in the earth depends on the conductivity profile at intervening altitudes in the middle atmosphere. A conclusion is that capacitative coupling may provide tighter coupling between the lower atmosphere and the ionosphere than usually considered, in both directions, which may help to explain observations which seem to indicate that magnetospheric phenomena may in some instances trigger lightning.

Hale, L. C.↗

NASA NTRS1987

Measuring and modeling the backscattering cross section of a leaf

Leaves are a significant feature of any vegetation canopy, and for remote sensing purposes it is important to develop an effective model for predicting the scattering from a leaf. From measurements of the X band backscattering cross section of a coleus leaf in varying stages of dryness, it is shown that a uniform resistive sheet constitutes such a model for a planar leaf. The scattering is determined by the (complex) resistivity which is, in turn, entirely specified by the gravimetric moisture content of the leaf. Using an available asymptotic expression for the scattering from a rectangular resistive plate which includes, as a special case, a metallic plate whose resistivity is zero, the computed backscattering cross sections for both principal polarizations are found to be in excellent agreement with data measured for rectangular sections of leaves with different moisture contents. If the resistivity is sufficiently large, the asymptotic expressions do not differ significantly from the physical optics ones, and for naturally shaped leaves as well as rectangular sections, the physical optics approximation in conjunction with the resistive sheet model faithfully reproduces the dominant feataures of the scattering patterns under all moisture conditions.

Senior, T. B. A.↗