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A dual extremum principle for the Verhulst-Pearl population equation is constructed using a complementary variational technique. The dual formulation utilizes a minimum principle recently developed by Leitmann to convert the functional optimization problem into a parameter optimization problem.
A variational formulation is given for the equations of motion for an unconstrained elastic body, and Hamilton's principle is used to derive the equations of motion and deformation of the body. Finite-element approximations are developed for these dynamical equations with respect to a body axis system satisfying the mean axis conditions. The free-body influence matrix for the body then is developed in terms of the finite-element model parameters.
The phenomena of dissipative destabilization of external Rossby waves and the acceleration of the zonal mean jet during baroclinic life cycles are described in terms of potential vorticity. The main principle of the potential temperature variations at rigid boundaries have the same effect on the interior flow as do sheets of potential vorticity located just within the boundaries. It is noted that the potential vorticity theory is useful for understanding the dynamical behavior of meterological phenomena.
The concepts of early-prompt delay tracking, multipath correction of early-prompt delay tracking from correlation shape, and carrier phase multipath correction are addressed. In early-prompt delay tracking, since multipath is always delayed with respect to the direct signals, the system derives phase and pseudorange observables from earlier correlation lags. In multipath correction of early-prompt delay tracking from correlation shape, the system looks for relative variations of amplitude across the code correlation function that do not match the predicted multipath-free code cross-correlation shape. The system then uses deviations from the multipath-free shape to infer the magnitude of multipath, and to generate corrections pseudorange observables. In carrier phase multipath correction, the system looks for variations of phase among plural early and prompt lags. The system uses the measured phase variations, along with the general principle that the multipath errors are larger for later lags, to infer the presence of multipath, and to generate corrections for carrier-phase observables.
Selection of the best shape of a fillet in a tension bar such that no yielding occurs has long attracted the attention of engineers. Dimensions and notations for the bar and fillet are shown. With symmetry, only the upper half of the bar is considered. The optimal design problem is to find a boundary shape to minimize the total area of the fillet such that no yielding occurs. The classical boundary value problem is reduced to a variational or energy related problem which not only has excellent properties of existence and uniqueness but also provides the mathematical foundation for finite element analysis. The variational formulation may be viewed as the principle of virtual work and the finite element method as a application of the Galerkin method to the variational equation for approximate solution of the boundary value problem.
The aerodynamic phenomena that may profitably be employed by the designer at subsonic speeds seem now to be well understood. At supersonic speeds such phenomena show a greater and more interesting variety. Search for the minimum number of guiding principles of design thus becomes more difficult and more dangerous. Studies which can cover an adequate range of geometrical form are at present limited to the linearized version of aerodynamic theory. Such studies, especially those by variational methods, have disclosed certain basic principles of design for aerodynamic efficiency. In present-day experiments, however, the indicated trends are rather quickly confronted with effects of viscosity and nonlinearities. While the theory indicates that good values of aerodynamic efficiency are possible at supersonic speeds it is not yet clear how closely these expectations may be approached in practice. In the present paper several arrangements of supporting surfaces and bodies are discussed and in some cases comparisons of theory and experiment are made. Finally, certain phenomena connected with lift and drag in a rarefied medium are considered briefly.
In order for Kepler to achieve its required less than 20 PPM photometric precision for magnitude 12 and brighter stars, instrument-induced variations in the CCD readout bias pattern (our "2D black image"), which are either fixed or slowly varying in time, must be identified and the corresponding pixels either corrected or removed from further data processing. The two principle sources of these readout bias variations are crosstalk between the 84 science CCDs and the 4 fine guidance sensor (FGS) CCDs and a high frequency amplifier oscillation on less than 40% of the CCD readout channels. The crosstalk produces a synchronous pattern in the 2D black image with time-variation observed in less than 10% of individual pixel bias histories. We will describe a method of removing the crosstalk signal using continuously-collected data from masked and over-clocked image regions (our "collateral data"), and occasionally-collected full-frame images and reverse-clocked readout signals. We use this same set to detect regions affected by the oscillating amplifiers. The oscillations manifest as time-varying moir pattern and rolling bands in the affected channels. Because this effect reduces the performance in only a small fraction of the array at any given time, we have developed an approach for flagging suspect data. The flags will provide the necessary means to resolve any potential ambiguity between instrument-induced variations and real photometric variations in a target time series. We will also evaluate the effectiveness of these techniques using flight data from background and selected target pixels.
Solution of trajectory optimization problems by calculus of variations, dynamic programming, and Pontryagin's Principle
The problem of predicting the minimum fuel trajectory for a six degree of freedom vehicle which has a motion characterized by the first order differential equations of translational and rotational dynamics is considered. The thrust direction and center of gravity of the vehicle are assumed to be fixed with respect to the vehicle. Thrust magnitude and the control moment are used as control variables and appear linearly in the equation of motion. Pontryagin's Maximum Principle is used to solve the variational problem. With this formulation, the extremal controls are bang-bang with the exception of the singular case.
The effect of periodic surface roughness on the radar cross section (RCS) was studied. The surface roughness was formed by a small sinusoidal variation in a planar surface. RCS measurements were obtained for two different sinusoidal variations near grazing incidence for both principle polarizations. Significant grating lobes were observed in the measurements which directly correspond to the roughness characteristics. A physical optics solution was generated and compared to the measurements with reasonable agreement.
Pontryagin maximum principle used to study stability of periodic equations in time optimal control problems
A variational integral is formulated from Hamilton's Principle and is proved to be equivalent to the usual differential equations of low-gravity sloshing in ellipsoidal tanks. It is shown that for a zero-degree contact angle the contact line boundary condition corresponds to the stuck condition, a result that is due to the linearization of the equations and the ambiguity in the definition of the wave height at the wall. The variational integral is solved by a Rayleigh-Ritz technique. Results for slosh frequency when the free surface is not bent-over compare well with previous numerical solutions. When the free surface is bent over, however, the results for slosh frequency are considerably larger than those predicted by previous finite-difference, numerical approaches: the difference may be caused by the use of a zero degree contact angle in the present theory in contrast to the nonzero contact angle used in the numerical approaches.
An analytical model is proposed for calculating the rotor performance and aerodynamic blade forces for Darrieus wind turbines with curved blades. The method of analysis uses a multiple-streamtube model, divided into two parts: one modeling the upstream half-cycle of the rotor and the other, the downstream half-cycle. The upwind and downwind components of the induced velocities at each level of the rotor were obtained using the principle of two actuator disks in tandem. Variation of the induced velocities in the two parts of the rotor produces larger forces in the upstream zone and smaller forces in the downstream zone. Comparisons of the overall rotor performance with previous methods and field test data show the important improvement obtained with the present model. The calculations were made using the computer code CARDAA developed at IREQ. The double-multiple streamtube model presented has two major advantages: it requires a much shorter computer time than the three-dimensional vortex model and is more accurate than multiple-streamtube model in predicting the aerodynamic blade loads.
Progress and plans are reported for investigations of: (1) the mechanical properties of high performance carbon fibers; (2) fatigue in composite materials; (3) moisture and temperature effects on the mechanical properties of graphite-epoxy laminates; (4) the theory of inhomogeneous swelling in epoxy resin; (5) numerical studies of the micromechanics of composite fracture; (6) free edge failures of composite laminates; (7) analysis of unbalanced laminates; (8) compact lug design; (9) quantification of Saint-Venant's principles for a general prismatic member; (10) variation of resin properties through the thickness of cured samples; and (11) the wing fuselage ensemble of the RP-1 and RP-2 sailplanes.
Two versions of the LOWTRAN5 radiance code are used in a study of the earth's clear sky infrared radiation budget in the interval 30 per cm (333.3 microns) to 3530 per cm (2.8 microns). One version uses 5 per cm resolution and temperature dependent molecular absorption coefficients, and the second uses 20 per cm resolution and temperature independent molecular absorption coefficients. Both versions compare well with Nimbus 3 IRIS spectra, with some discrepancies at particular wavenumber intervals. Up and downgoing fluxes, calculated as functions of latitude, are displayed for wavenumbers at which the principle absorbers are active. Most of the variation of the fluxes with latitude is found in the higher wavenumber intervals for both clear and cloudy skies. The main features of the wavenumber integrated cooling rates are explained with reference to calculations in more restricted wavenumber intervals. A tropical lower tropospheric cooling maximum is produced by water vapor continuum effects in the 760-1240 per cm window. A secondary upper tropospheric cooling maximum, with wide meridional extent, is produced by water vapor rotational lines between 30-430 per cm. Water vapor lines throughout the terrestrial infrared spectrum prevent the upflux maximum from coinciding with the surface temperature maximum.
A formulation of the stability behavior of a finite-length pinch is presented. A general initial perturbation is expressed as a uniformly convergent sum over a complete discrete k set. A variational calculation is then performed, based on the energy principle, in which the end-boundary conditions appear as constraints. The requisite Lagrange multipliers mutually couple the elemental periodic excitations. The resulting extended form of delta-W still admits a proper second-variation treatment so that the minimization and stability considerations of Newcomb remain applicable. Comparison theorems are discussed as is the relevance of this end-effect model to the stability of solar coronal loops.
An investigation was conducted of maximum principle solutions for an initial 0.6 Mach number and 15,000-ft altitude. The authors generate these solutions for a family of prescribed final times tf, starting with tf = 0.5 s. Using a nonlinear wind-tunnel model they construct maximum principle solutions. Above tf = 1.2 s some small nonlinear variations in the aerodynamic pitching moment coefficient presented difficulty with respect to numerical convergence. This was circumvented by fitting analytical models to the aerodynamic coefficients of the wind-tunnel model at Mach 0.4. Maximum principle solutions of the analytical model are shown to compare well with those obtained for tf of less than 1.2 s. Using the analytical model the authors extended the prescribed final time to a value of 13.65 s at which time the aircraft completes the half-loop maneuver. This is 0.53 s longer than that obtained using the singular perturbation feedback control law.
Researchers in the Astrobiology Technology Branch at Ames Research Center have begun investigating alternate concepts for the detection of extraterrestrial life. We suggest searching for extraterrestrial ecology, exoecology, as well as for extraterrestrial biology, exobiology. Ecology describes the interactions of living things with their environment. All ecosystems are highly constrained by their environment and are constrained by well-known system design principles. Ecology could exist wherever there is an energy source and living I things have discovered some means to capture, store, and use the available energy. Terrestrial ecosystems use as energy sources, light, organic molecules, and in thermal vents and elsewhere, simple inorganic molecules. Ecosystem behavior is controlled by matter and energy conservation laws and can be described by linear and nonlinear dynamic systems theory. Typically in an ecosystem different molecules are not in chemical equilibrium and scarce material is conserved, stored, or recycled. Temporal cycles and spatial variations are often observed. These and other -eneral principles of exoecology can help guide the search for extraterrestrial life. The chemical structure observed in terrestrial biology may be highly contingent on evolutionary accidents. Oxygen was not always abundant on Earth. Primitive sulfur bacteria use hydrogen sulfide and sulfur to perform photosynthesis instead of water and oxygen. Astrobiologists have assumed, for the sake of narrowing and focusing our life detection strategies, that extraterrestrial life will have detailed chemical similarities with terrestrial life. Such assumptions appear very reasonable and they allow us to design specific and highly sensitive life detection experiments. But the fewer assumptions we make, the less chance we have of being entirely wrong The best strategy for the detection of extraterrestrial life could be a mixed strategy. We should use detailed assumptions based on terrestrial biology to guide some but not all future searches for alien life. The systems principles of exoecology seem much more fundamental and inescapable than the terrestrial biology analogies of exobiology. We should search for exoecology as well as exobiology.