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The states of the three-dimensional quantum harmonic oscillator classify optical aberrations of axis-symmetric systems due to the isomorphism between the two mathematical structures. Cartesian quanta and angular momentum classifications have their corresponding aberration classifications. The operation of concatenation of optical elements introduces a new operation between harmonic oscillator states.
A formulation is given for a collection of phonons (sound) in a fluid at a non-zero temperature which uses the simple harmonic oscillator twice; one to give a stochastic thermal 'noise' process and the other which generates a coherent Glauber state of phonons. Simple thermodynamic observables are calculated and the acoustic two point function, 'contrast' is presented. The role of 'coherence' in an equilibrium system is clarified by these results and the simple harmonic oscillator is a key structure in both the formulation and the calculations.
Finite trigonometric series is fitted by harmonic analysis as an approximation function to the psi function of the Theodorsen arbitrary-airfoil potential theory. By harmonic synthesis, the corresponding conjugate trigonometric series is used as an approximation to the epsilon function. A set of coefficients of particularly simple form is obtained algebraically for direct calculation of the epsilon values from the corresponding set of psi values. Complete derivation of this process is presented.
A parametric study of vibratory rotor hub loads in a nonrotating system is presented. The study is based on a CAMRAD/JA model constructed for the GBH (Growth Version of Blackhawk Helicopter) Mach-scaled wind tunnel rotor model with high blade twist (-16 deg). The theoretical hub load predictions are validated by correlation with available measured data. Effects of various blade aeroelastic design changes on the harmonic nonrotating frame hub loads at both low and high forward flight speeds are investigated. The study aims to illustrate some of the physical mechanisms for change in the harmonic rotor hub loads due to blade design variations.
A new two-dimensional monochromatic method that computes the transfer of solar or thermal radiation through atmospheres with arbitrary optical properties is described. The model discretizes the radiative transfer equation by expanding the angular part of the radiance field in a spherical harmonic series and representing the spatial part with a discrete grid. The resulting sparse coupled system of equations is solved iteratively with the conjugate gradient method. A Monte Carlo model is used for extensive verification of outgoing flux and radiance values from both smooth and highly variable (multifractal) media. The spherical harmonic expansion naturally allows for different levels of approximation, but tests show that the 2D equivalent of the two-stream approximation is poor at approximating variations in the outgoing flux. The model developed here is shown to be highly efficient so that media with tens of thousands of grid points can be computed in minutes. The large improvement in efficiency will permit quick, accurate radiative transfer calculations of realistic cloud fields and improve our understanding of the effect of inhomogeneity on radiative transfer in cloudy atmospheres.
A four-parameter potential is analyzed, which contains the three-dimensional harmonic oscillator as a special case. This potential is exactly solvable and retains several characteristics of the harmonic oscillator, and also of the Coulomb problem. The possibility of similar generalizations of other potentials is also pointed out.
We discuss the semimicroscopic algebraic cluster model introduced recently, in which the internal structure of the nuclear clusters is described by the harmonic oscillator shell model, while their relative motion is accounted for by the Vibron model. The algebraic formulation of the model makes extensive use of techniques associated with harmonic oscillators and their symmetry group, SU(3). The model is applied to some cluster systems and is found to reproduce important characteristics of nuclei in the sd-shell region. An approximate SU(3) dynamical symmetry is also found to hold for the C-12 + C-12 system.
Given a magnetic field B = -del(vector differential operator)(sub gamma) with gamma expanded in spherical harmonics, it is shown that analytic Euler potentials may be derived for B if gamma is asymmetrical but contains only the contribution of a single index n. This work generalizes a result for sectorial harmonics with n = m, derived by Willis and Gardiner (1988).
Novel variation of split-waveguide mount for millimeter-and submillimeter-wavelength frequency multipliers and harmonic mixers developed. Designed to offer wide range of available matching impedances, while maintaining relatively simple fabrication sequence. Wide tuning range achieved with separate series and parallel elements, consisting of two pairs of noncontacting sliding backshorts, at fundamental and harmonic frequencies. Advantages include ease of fabrication, reliability, and tunability.
Half-daily global wind speeds in the east-west (u) and north-south (v) directions at the 10-meter height level were obtained from the European Centre for Medium Range Weather Forecasts (ECMWF) data set of global analyses. The data set covered the period 1985 January to 1995 January. A spherical harmonic expansion to degree and order 50 was used to perform harmonic analysis of the east-west (u) and north-south (v) velocity field components. The resulting wind field is displayed, as well as the residual of the fit, at a particular time. The contribution of particular coefficients is shown. The time variability of the coefficients up to degree and order 3 is presented. Corresponding power spectrum plots are given. Time series analyses were applied also to the power associated with degrees 0-10; the results are included.
This document details the methodology and use of the CAMRAD.Mod1/HIRES codes, which were developed at NASA Langley Research Center for the prediction of helicopter harmonic and Blade-Vortex Interaction (BVI) noise. CANMAD.Mod1 is a substantially modified version of the performance/trim/wake code CANMAD. High resolution blade loading is determined in post-processing by HIRES and an associated indicial aerodynamics code. Extensive capabilities of importance to noise prediction accuracy are documented, including a new multi-core tip vortex roll-up wake model, higher harmonic and individual blade control, tunnel and fuselage correction input, diagnostic blade motion input, and interfaces for acoustic and CFD aerodynamics codes. Modifications and new code capabilities are documented with examples. A users' job preparation guide and listings of variables and namelists are given.
Measurements of geomagnetic scalar intensity on a thin spherical shell alone are not enough to separate internal from external source fields; moreover, such scalar data are not enough for accurate modeling of the vector field from internal sources because of unmodeled fields and small data errors. Spherical harmonic models of the geomagnetic potential fitted to scalar data alone therefore suffer from well-understood Backus effect and perpendicular errors. Curiously, errors in some models of simulated 'data' are very much less than those in models of real data. We analyze select Magsat vector and scalar measurements separately to illustrate Backus effect and perpendicular errors in models of real scalar data. By using a model to synthesize 'data' at the observation points, and by adding various types of 'noise', we illustrate such errors in models of synthetic 'data'. Perpendicular errors prove quite sensitive to the maximum degree in the spherical harmonic expansion of the potential field model fitted to the scalar data. Small errors in models of synthetic 'data' are found to be an artifact of matched truncation levels. For example, consider scalar synthetic 'data' computed from a degree 14 model. A degree 14 model fitted to such synthetic 'data' yields negligible error, but amplifies 4 nT (rmss) added noise into a 60 nT error (rmss); however, a degree 12 model fitted to the noisy 'data' suffers a 492 nT error (rmms through degree 12). Geomagnetic measurements remain unaware of model truncation, so the small errors indicated by some simulations cannot be realized in practice. Errors in models fitted to scalar data alone approach 1000 nT (rmss) and several thousand nT (maximum).
A power divider divides an RF signal into two output signals having a phase difference of 180° or a multiple thereof. When the RF signal is a square wave or another harmonically rich signal. the phases of the fundamental and the harmonics have the proper relationship. The divider can be implemented in the form of rnicrostrips on a board, with one of the output microstrips having several bends to provide a different electrical length from the other.
The aeroelastic characteristics of a compressor stage were analyzed using a computational fluid dynamic (CFD) solver that uses the harmonic balance method to solve the governing equations. The three dimensional solver models the unsteady flow field due to blade vibration using the Reynolds-Averaged Navier-Stokes equations. The formulation enables the study of the effect of blade row interaction through the inclusion of coupling modes between blade rows. It also enables the study of nonlinear effects of high amplitude blade vibration by the inclusion of higher harmonics of the fundamental blade vibration frequency. In the present work, the solver is applied to study in detail the aeroelastic characteristics of a transonic compressor stage. Various parameters were included in the study: number of coupling modes, blade row axial spacing, and operating speeds. Only the first vibration mode is considered with amplitude of oscillation in the linear range. Both aeroelastic stability (flutter) of rotor blade and unsteady loading on the stator are calculated. The study showed that for the stage considered, the rotor aerodynamic damping is not influenced by the presence of the stator even when the axial spacing is reduced by nearly 25 percent. However, the study showed that blade row interaction effects become important for the unsteady loading on the stator when the axial spacing is reduced by the same amount.
This slide presentation reviews the efforts made at Goddard Space Flight Center in harmonizing the ground systems to assist in collaboration in space ventures. The key elements of this effort are: (1) Moving to a Common Framework (2) Use of Consultative Committee for Space Data Systems (CCSDS) Standards (3) Collaboration Across NASA Centers (4) Collaboration Across Industry and other Space Organizations. These efforts are working to bring into harmony the GSFC systems with CCSDS standards to allow for common software, use of Commercial Off the Shelf Software and low risk development and operations and also to work toward harmonization with other NASA centers
The results of a computational study examining the effects of non-harmonic active-twist control on blade-vortex interaction (BVI) noise for the Apache Active Twist Rotor are presented. Rotor aeroelastic behavior was modeled using the Comprehensive Analytical Model of Rotorcraft Aerodynamics and Dynamics code and the rotor noise was predicted using the acoustics code PSU-WOPWOP. The application of non-harmonic active-twist inputs to the main rotor blade system comprised three parameters: azimuthal location to start actuation, azimuthal duration of actuation, and magnitude of actuation. The acoustic analysis was conducted for a single low-speed flight condition of advance ratio mu=0.14 and shaft angle-of-attack, a(sub s)=+6deg. BVI noise levels were predicted on a flat plane of observers located 1.1 rotor diameters beneath the rotor. The results indicate significant reductions of up to 10dB in BVI noise using a starting azimuthal location for actuation of 90?, an azimuthal duration of actuation of 90deg, and an actuation magnitude of +1.5 ft-lb.
We present here the design, development and test of an integrated sub-millimeter front-end featuring a 520-600 GHz sub-harmonic mixer and a 260-300 GHz frequency tripler in a single cavity. Both devices used GaAs MMIC membrane planar Schottky diode technology. The sub-harmonic mixer/tripler circuit has been tested using conventional machined as well as silicon micro-machined blocks. Measurement results on the metal block give best DSB mixer noise temperature of 2360 K and conversion losses of 7.7 dB at 520 GHz. Preliminary results on the silicon micro-machined blocks give a DSB mixer noise temperature of 4860 K and conversion losses of 12.16 dB at 540 GHz. The LO input power required to pump the integrated tripler/sub-harmonic mixer for both packages is between 30 and 50 mW.
We present here the design, development and test of an integrated sub-millimeter front-end featuring a 520-600 GHz sub-harmonic mixer and a 260-300 GHz frequency tripler in a single cavity. Both devices used GaAs MMIC membrane planar Schottky diode technology. The sub-harmonic mixer/tripler circuit has been tested using conventional machined as well as silicon micro-machined blocks. Measurement results on the metal block give best DSB mixer noise temperature of 2360 K and conversion losses of 7.7 dB at 520 GHz. Preliminary results on the silicon micro-machined blocks give a DSB mixer noise temperature of 4860 K and conversion losses of 12.16 dB at 540 GHz. The LO input power required to pump the integrated tripler/sub-harmonic mixer for both packages is between 30 and 50 mW