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All sourcesNASA NTRSDOE OSTI
NASA NTRS1986

Formationkeeping of Spacecraft via Differential Drag

This paper examines the use of differential drag in the formationkeeping of spacecraft. In many future space missions one satellite will be required to fly in a specific position with respect to another satellite; this action is referred to as formationkeeping. In this study, differential drag is the difference in drag between the two satellites. Reasons to use differential drag as an actuator for formationkeeping include the avoidance of jet plume impingement effects on closely spaced satellites and possible fuel savings. The equations of relative motion between the two satellites are derived and a mathematical transformation is made to reduce the formationkeeping problem to the simultaneous solution of a double integrator and a harmonic oscillator. A two-part control law is developed that simultaneously and dependently solves the double oscillator and harmonic integrator. Results are presented of eight test cases being driven to a target position; two different simulations are used. The validity of assumptions made in the derivation of the control law is examined in the comparison of similar test cases run through different simulations. The control law developed can drive a satellite from an initial position to a target position and maintain the satellite at that location.

Leonard, Carolina Lee↗
NASA NTRS1996

Two Different Squeeze Transformations

Lorentz boosts are squeeze transformations. While these transformations are similar to those in squeezed states of light, they are fundamentally different from both physical and mathematical points of view. The difference is illustrated in terms of two coupled harmonic oscillators, and in terms of the covariant harmonic oscillator formalism.

Han, D.↗
NASA NTRS1991

Dissociation of diatomic gases

The Landau-Zener theory of reactive cross sections has been applied to diatomic molecules dissociating from a ladder of rotational and vibrational states. Although the preexponential factor of the Arrhenius rate expression is shown to be a complex function of the dimensionless activation energy, the average over all states in the ladder is well represented by a single factor that varies about as T exp (-n), where the coefficient n is the order of unity. This relation agrees very well with experimental data for dissociation of O2 and N2, for example. The results validate previous empirical assignment of a single preexponential factor in the Arrhenius expression and justify the extrapolation of the expression well beyond the range of data. The theory is then used to calculate the effect of vibrational nonequilibrium on dissociation rate. For Morse oscillators the results are about the same as for harmonic oscillators, and the dissociation from a ladder of equilibrium rotational and nonequilibrium vibrational states is close to an analytic approximation provided by Hammerling, Kivel, and Teare for harmonic oscillators all dissociating from the ground rotational state.

Hansen, C. F.↗
NASA NTRS1993

Point form relativistic quantum mechanics and relativistic SU(6)

The point form is used as a framework for formulating a relativistic quantum mechanics, with the mass operator carrying the interactions of underlying constituents. A symplectic Lie algebra of mass operators is introduced from which a relativistic harmonic oscillator mass operator is formed. Mass splittings within the degenerate harmonic oscillator levels arise from relativistically invariant spin-spin, spin-orbit, and tensor mass operators. Internal flavor (and color) symmetries are introduced which make it possible to formulate a relativistic SU(6) model of baryons (and mesons). Careful attention is paid to the permutation symmetry properties of the hadronic wave functions, which are written as polynomials in Bargmann spaces.

Klink, W. H.↗
NASA NTRS1987

Resonance absorption of magnetohydrodynamic surface waves Physical discussion

It is shown how the phenomenon of MHD surface wave resonance absorption can be described in simple terms, both physically and mathematically, by applying the 'thin flux tube equations' to the finite-thickness transition layer which supports the surface wave. The thin flux tubes support incompressible slow-mode waves that are driven by fluctuations in the total pressure which exist due to the presence of the surface wave. It is shown that the equations for the slow-mode waves can be reduced to a simple equation, equivalent to a driven harmonic oscillator. Certain field lines within the transition layer are equivalent to a harmonic oscillator driven at resonance, and neighboring field lines are effectively driven at resonance as long as a given condition is satisfied. Thus, a layer which secularly extracts energy from the surface wave develops. The analysis indicates that nonlinear effects may destroy the resonance which is crucial to the whole effect.

Hollweg, Joseph V.↗
NASA NTRS1995

The Duffin-Kemmer-Petiau oscillator

In view of current interest in relativistic spin-one systems and the recent work on the Dirac Oscillator, we introduce the Duffin-Kemmer-Petiau (DKP) equation obtained by using an external potential linear in r. Since, in the non-relativistic limit, the spin 1 representation leads to a harmonic oscillator with a spin-orbit coupling of the Thomas form, we call the equation the DKP oscillator. This oscillator is a relativistic generalization of the quantum harmonic oscillator for scalar and vector bosons. We show that it conserves total angular momentum and that it is exactly solvable. We calculate and discuss the eigenspectrum of the DKP oscillator in the spin 1 representation.

Nedjadi, Youcef↗
NASA NTRS1988

Linear canonical transformations of coherent and squeezed states in the Wigner phase space

It is shown that classical linear canonical transformations are possible in the Wigner phase space. Coherent and squeezed states are shown to be linear canonical transforms of the ground-state harmonic oscillator. It is therefore possible to evaluate the Wigner functions for coherent and squeezed states from that for the harmonic oscillator. Since the group of linear canonical transformations has a subgroup whose algebraic property is the same as that of the (2+1)-dimensional Lorentz group, it may be possible to test certain properties of the Lorentz group using optical devices. A possible experiment to measure the Wigner rotation angle is discussed.

Han, D.↗
NASA NTRS1995

User's Guide for ECAP2D: an Euler Unsteady Aerodynamic and Aeroelastic Analysis Program for Two Dimensional Oscillating Cascades, Version 1.0

This guide describes the input data required for using ECAP2D (Euler Cascade Aeroelastic Program-Two Dimensional). ECAP2D can be used for steady or unsteady aerodynamic and aeroelastic analysis of two dimensional cascades. Euler equations are used to obtain aerodynamic forces. The structural dynamic equations are written for a rigid typical section undergoing pitching (torsion) and plunging (bending) motion. The solution methods include harmonic oscillation method, influence coefficient method, pulse response method, and time integration method. For harmonic oscillation method, example inputs and outputs are provided for pitching motion and plunging motion. For the rest of the methods, input and output for pitching motion only are given.

Reddy, T. S. R.↗
NASA NTRS1993

Geometric model from microscopic theory for nuclear absorption

A parameter-free geometric model for nuclear absorption is derived herein from microscopic theory. The expression for the absorption cross section in the eikonal approximation, taken in integral form, is separated into a geometric contribution that is described by an energy-dependent effective radius and two surface terms that cancel in an asymptotic series expansion. For collisions of light nuclei, an expression for the effective radius is derived from harmonic oscillator nuclear density functions. A direct extension to heavy nuclei with Woods-Saxon densities is made by identifying the equivalent half-density radius for the harmonic oscillator functions. Coulomb corrections are incorporated, and a simplified geometric form of the Bradt-Peters type is obtained. Results spanning the energy range from 1 MeV/nucleon to 1 GeV/nucleon are presented. Good agreement with experimental results is obtained.

John, Sarah↗
NASA NTRS1993

Geometric model for nuclear absorption from microscopic theory

A parameter-free geometric model for nuclear absorption is derived from microscopic theory. The expression for the absorption cross section in the eikonal approximation taken in integral form is separated into a geometric contribution, described by an energy-dependent effective radius, and two surface terms which are shown to cancel in an asymptotic series expansion. For collisions of light nuclei, an expression for the effective radius is derived using harmonic-oscillator nuclear density functions. A direct extension to heavy nuclei with Woods-Saxon densities is made by identifying the equivalent half density radius for the harmonic-oscillator functions. Coulomb corrections are incorporated and a simplified geometric form of the Bradt-Peters type obtained. Results spanning the energy range of 1 MeV/nucleon to 1 GeV/nucleon are presented. Good agreement with experimental results are obtained.

John, S.↗
NASA NTRS1976

Diatomic molecule variations

The rotational energy is separated from vibrational energy in the two-particle, steady-state wave equation and to first order the solutions are harmonic oscillator functions. The classical phase integral gives a partition function valid only at high temperature, but the quantum summation is easily performed to give analytic expressions for all the thermodynamic properties of the harmonic oscillator at all temperatures. Anharmonic effects are treated by small perturbation solutions to the wave equation and the relation between energy levels and a series expansion of the perturbation potential is derived. Next, the quantum solutions for an oscillator with a Morse-function potential are derived in terms of Laguerre polynomials.

Source record↗
NASA NTRS1984

Friction and particle-hole pairs

The effect induced by dissipation on quantum phenomena has recently been considered, taking into account as a starting point a phenomenological Hamiltonian in which the environment is simulated by an appropriately chosen set of harmonic oscillators. It is found that this approach should be adequate to describe the low-energy behavior of a wide class of environments. The present investigation is concerned with an analysis of the case in which the environment is a gas (or liquid) of fermions, and the relevant low-energy excitations are particle-hole pairs. A study is conducted regarding the extent to which the quantum results obtained for harmonic oscillators are also valid in the considered situation. Linear-response theory is used to derive an effective action which describes the motion of an external particle coupled to a normal Fermi fluid.

Guinea, F.↗
NASA NTRS1978

Numerical integration of nearly-Hamiltonian systems

The reported investigation is concerned with the solution of systems of differential equations which are derived from a Hamiltonian function in the extended phase space. The problem selected involves a one-dimensional perturbed harmonic oscillator. The van der Pol equation considered has an exact asymptotic value for its amplitude. Comparisons are made between a numerical solution and a known analytical solution. In addition to the van der Pol problem, known solutions regarding the restricted problem of three bodies are used as examples for perturbed Keplerian motion. The extended phase space Hamiltonian discussed by Stiefel and Scheifele (1971) is considered. A description is presented of two canonical formulations of the perturbed harmonic oscillator.

Bond, V. R.↗
NASA NTRS1974

Effects of curvilinear motion in large-amplitude bending of C3

The geometry of the bending of a linear triatomic molecule is analyzed, and an expression for the average rotational constant is derived. A harmonic oscillator model of C3 is fitted to the observed rotational constant within 0.6%. The bond distance between atoms at zero bending is 1.287 A according to this model; this is noticeably larger than the average internuclear distance of 1.277 A for the vibrational ground state. The first order perturbation solutions for the vibrational energy levels, taking into account the effect of a quartic perturbation potential, closely match observed levels. For a square well potential model of C3, the effect of curvilinear motion in bending is similar to that found for the harmonic oscillator model, though the decreases in energy are about twice as large. In both models, the average energy decrease is relatively constant at approximately 10% over a wide range of vibrational quantum number.

Hansen, C. F.↗