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A perturbative method, using Lie transforms, is given for calculating the Hannay angle for slow, cyclic evolutions of a classical system, taking into account the finite rate of change of the Hamiltonian. The method is applied to the generalized harmonic oscillator. The classical Aharonov-Anandan angle is also calculated. The interpretational ambiguity in the definitions of geometrical angles is discussed.
A finite-volume unstructured-grid FEM scheme with multistage Runge-Kutta time stepping is applied to the three-dimensional time-dependent Euler equations for inviscid flows on complex aircraft configurations undergoing structural deformation. The derivation of the model, the solution procedure, and the computer implementation are described, and results are presented graphically for a NASA Langley supersonic fighter aircraft model in steady and unsteady (harmonic oscillation in complete-vehicle bending mode) flow regimes. Good agreement between FEM predictions and experimental data is demonstrated.
A study is made of the evolution of strongly three-dimensional disturbances which are generated by a point force in a parallel mixing layer. When the input force is a pulse, a wave packet develops whose wavefronts are approximately parallel to the spanwise direction. This is in sharp contrast to a wave packet in a wall boundary layer for which the wavefronts are strongly curved. On the other hand, when the input disturbance is oscillating harmonically in time, a spatially growing instability wave develops in a downstream wedge of (x, z)-space. The size of this wedge, as a function of excitation frequency and velocity ratio, is determined. The receptivity of the shear layer to pulse-type and harmonic excitation is also studied. It is found that the shear layer is especially sensitive to relatively high-frequency forcing on its centreline.
It is demonstrated that the Bargmann representation of quantum mechanics is ideally suited for semiclassical analysis, using as an example the WKB method applied to the bound-state problem in a single well of one degree of freedom. For the harmonic oscillator, this WKB method trivially gives the exact eigenfunctions in addition to the exact eigenvalues. For an anharmonic well, a self-consistent variational choice of the representation greatly improves the accuracy of the semiclassical ground state. Also, a simple change of scale illuminates the relationship of semiclassical versus linear perturbative expansions, allowing a variety of multidimensional extensions.
The purpose is to describe the development of unstructured grid methods which have several advantages when compared to methods which make use of structured grids. Unstructured grids, for example, easily allow the treatment of complex geometries, allow for general mesh movement for realistic motions and structural deformations of complete aircraft configurations which is important for aeroelastic analysis, and enable adaptive mesh refinement to more accurately resolve the physics of the flow. Steady Euler calculations for a supersonic fighter configuration to demonstrate the complex geometry capability; unsteady Euler calculations for the supersonic fighter undergoing harmonic oscillations in a complete-vehicle bending mode to demonstrate the general mesh movement capability; and vortex-dominated conical-flow calculations for highly-swept delta wings to demonstrate the adaptive mesh refinement capability are discussed. The basic solution algorithm is a multi-stage Runge-Kutta time-stepping scheme with a finite-volume spatial discretization based on an unstructured grid of triangles in 2D or tetrahedra in 3D. The moving mesh capability is a general procedure which models each edge of each triangle (2D) or tetrahedra (3D) with a spring. The resulting static equilibrium equations which result from a summation of forces are then used to move the mesh to allow it to continuously conform to the instantaneous position or shape of the aircraft. The adaptive mesh refinement procedure enriches the unstructured mesh locally to more accurately resolve the vortical flow features. These capabilities are described in detail along with representative results which demonstrate several advantages of unstructured grid methods. The applicability of the unstructured grid methodology to steady and unsteady aerodynamic problems and directions for future work are discussed.
A technique based on the Minimum Model Error optimal estimation approach is employed for robust identification of a nonlinear dynamic system. A simple harmonic oscillator with quadratic position feedback was simulated on an analog computer. With the aid of analog measurements and an assumed linear model, the Minimum Model Error Algorithm accurately identifies the quadratic nonlinearity. The tests demonstrate that the method is robust with respect to prior ignorance of the nonlinear system model and with respect to measurement record length, regardless of initial conditions.
For differential equations with one fast variable, a perturbation method is introduced that transforms a solution valid over only a short time interval to a new solution composed of averaged variables plus a periodic function of the averaged variables. The averaged variables are governed by a set of differential equations where the fast variable has been removed and thus can be numerically integrated quickly or solved directly. This method is applied to a perturbed harmonic oscillator with a cubic perturbation, van der Pol's equation, coorbital motion in the restricted three-body problem, and to nearly circular motion of a particle near one of the primaries in the restricted three-body problem.
A new model is proposed for energy transfer between the vibrational and translational modes for use in the direct simulation Monte Carlo method (DSMC). The model modifies the Landau-Teller theory for a harmonic oscillator and the rate transition is related to an experimental correlation for the vibrational relaxation time. Assessment of the model is made with respect to three different computations: relaxation in a heat bath, a one-dimensional shock wave, and hypersonic flow over a two-dimensional wedge. These studies verify that the model achieves detailed balance, and excellent agreement with experimental data is obtained in the shock wave calculation. The wedge flow computation reveals that the usual phenomenological method for simulating vibrational nonequilibrium in the DSMC technique predicts much higher vibrational temperatures in the wake region.
The plasma physical analog model (an extension of the damped, harmonic-oscillator dripping faucet model) is employed to consider explicitly the effect of time-varying the inputs. This work is equivalent to considering the effects of northward and southward turnings of the interplanetary magnetic field for various periods of time. It is found that relatively extended episodes (not less than 2 hours) of turned-on input with shorter (about 1 hour) periods of turned-off input lead to model behavior much like the continuously driven case. Going to short input intervals with longer periods of zero input leads to highly irregular and dramatically fluctuating episodes of magnetotail unloading. These results give an insight into the diversity of apparent magnetospheric responses during relatively isolated substorm conditions. This work shows the absolutely critical interdependence (in a nonlinear dynamical system) of input phasing and internal magnetospheric response cycles.
The effect of temperature on the optical properties of GaAs was investigated using spectroscopic ellipsometry measurements, between room temperature and about 610 C in increments of 50 C, of pseudodielectric functions and related optical constants of GaAs. A quantitative analysis of the pseudodielectric function spectrum was carried out using a harmonic-oscillator approximation (HOA) to fit the measured dielectric functions. Good fits were obtained with this model, which provides a convenient means of reproducing the GaAs dielectric function at any temperature, by using the temperature-dependent oscillator parameters. The HOA analysis also provides information about band-gap variation with temperature. Using the measured optical constants at a number of fixed temperatures, an algorithm was developed for computing the dielectric function spectrum at an arbitrary temperature in the range 22-610 C.
The receptivity to freestream sound of the laminar boundary layer over a semi-infinite flat plate with an elliptic leading edge is simulated numerically. The incompressible flow past the flat plate is computed by solving the full Navier-Stokes equations in general curvilinear coordinates. A finite-difference method which is second-order accurate in space and time is used. Spatial and temporal developments of the Tollmien-Schlichting wave in the boundary layer, due to small-amplitude time-harmonic oscillations of the freestream velocity that closely simulate a sound wave travelling parallel to the plate, are observed. The effect of leading-edge curvature is studied by varying the aspect ratio of the ellipse. The boundary layer over the flat plate with a sharper leading edge is found to be less receptive. The relative contribution of the discontinuity in curvature at the ellipse-flat-plate juncture to receptivity is investigated by smoothing the juncture with a polynomial. Continuous curvature leads to less receptivity. A new geometry of the leading edge, a modified super ellipse, which provides continuous curvature at the juncture with the flat plate, is used to study the effect of continuous curvature and inherent pressure gradient on receptivity.
Continued here is the development of a model of the electron (HYDRA), which includes rotational and magnetic terms. The atomic electron state is discussed and a comparison is made with a simple harmonic oscillator. Experimental data is reviewed that supports the possibility of a new lepton.
A method is developed for obtaining coherent states of a system admitting a supersymmetry. These states are called supercoherent states. The presented approach is based on an extension to supergroups of the usual group-theoretic approach. The example of the supersymmetric harmonic oscillator is discussed, thereby illustrating some of the attractive features of the method. Supercoherent states of an electron moving in a constant magnetic field are also described.
The following paper is presented in viewgraph format and topics include: (1) exact solutions are difficult to obtain for higher order systems; (2) impulsive forces cannot be implemented directly; (3) exact solution provide the basis for judging the optimality of approximate techniques; and (4) properties of the exact solution can be used to develop improved approximations. Furthermore, the author discusses a floating harmonic oscillator whose model possesses rigid and flexible body motion characteristics of the proposed spacecraft.
A theory which gives statistical predictions for how often sonic booms propagating through the earth's turbulent boundary layer will encounter caustics, given the spectral properties of the atmospheric turbulence, is outlined. The theory is simple but approximately accounts for the variation of ray tube areas along ray paths. This theory predicts that the variation of ray tube areas is determined by the product of two similar area factors, psi (x) and phi (x), each satisfying a generic harmonic oscillator equation. If an area factor increases the peak acoustic pressure decreases, and if the factor decreases the peak acoustic pressure increases. Additionally, if an area factor decreases to zero and becomes negative, the ray has propagated through a caustic, which contributes a phase change of 90 degrees to the wave. Thus, it is clear that the number of times that a sonic boom wave passes through a caustic should be related to the distorted boom waveform received on the ground. Examples are given based on a characterization of atmospheric turbulence due to the structure function of Tatarski as modified by Crow.
The two-dimensional, elliptic Navier-Stokes equations are used to investigate supersonic flows with nonequilibrium chemistry and thermodynamics, coupled with radiation, for hydrogen-air systems. The chemistry source term in the species equation is treated implicitly to alleviate the stiffness associated with fast reactions. The explicit, unsplit MacCormack finite-difference scheme is used to advance the governing equations in time, until convergence is achieved. The specific problem considered is the premixed, expanding flow in a supersonic nozzle. The reacting flow consists of seven species, one of which is the inert N2 molecule. The thermal state of the gas is modeled with one translational-rotational temperature and five vibrational temperatures. The harmonic oscillator model is used in the formulation for vibrational relaxation. The tangent slab approximation is used in the radiative flux formulation. A pseudo-gray model is used to represent the absorption-emission characteristics of the participating species. Results obtained for specific conditions indicate the presence of nonequilibrium in the expansion region. This reduces the radiative interactions and can have a significant influence on the flowfield.
This paper presents for the first time an exact solution to the fuel optimal propulsive reboost problem for flexible spacecraft. The spacecraft undergoes rigid-body motion and flexible-body motion and reboost is achieved propulsively through the use of reaction control jets. The exact fuel optimal solution to the associated minimization problem is found numerically based on an adaptive grid bisection search. The reboost of the floating harmonic oscillator reveals properties of the fuel optimal solution. Nondimensional plots of minimum fuel versus maneuver time expose the nature of the solution classes depending on the maneuver time. Some very interesting properties are observed, among them, the shifting of impulses to the smaller oscillator mass, and comparisons are made with near fuel optimal solutions obtained by other investigators.
Magnetic field measurements made with the AMPTE CCE spacecraft are used to investigate the distribution of ULF energy in the inner magnetosphere. The data base is employed to examine the spatial distribution of ULF energy. The spatial distribution of wave power and spectral structures are used to identify several pulsation types, including multiharmonic toroidal oscillations; equatorial compressional Pc 3 oscillations; second harmonic poloidal oscillations; and nightside compressional oscillations. The frequencies of the toroidal oscillations are applied to determine the statistical radial profile of the plasma mass density and Alfven velocity. A clear signature of the plasma pause in the profiles of these average parameters is found.