SEARCH · Search NASA

Results for “harmonic oscillator”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

SourceResearch connectionFrom yearThrough year

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 109 records · Page 6

All sources NASA NTRS DOE OSTI

NASA NTRS1977

A user's guide for V174, a program using a finite difference method to analyze transonic flow over oscillating wings

The design and usage of a pilot program using a finite difference method for calculating the pressure distributions over harmonically oscillating wings in transonic flow are discussed. The procedure used is based on separating the velocity potential into steady and unsteady parts and linearizing the resulting unsteady differential equation for small disturbances. The steady velocity potential which must be obtained from some other program, is required for input. The unsteady differential equation is linear, complex in form with spatially varying coefficients. Because sinusoidal motion is assumed, time is not a variable. The numerical solution is obtained through a finite difference formulation and a line relaxation solution method.

Butler, T. D.↗

NASA NTRS1978

Water-tunnel experiments on an oscillating airfoil at RE equals 21,000

Flow visualization experiments were performed in a water tunnel on a modified NACA 0012 airfoil undergoing large amplitude harmonic oscillations in pitch. Hydrogen bubbles were used to: (1) create a conveniently striated and well preserved set of inviscid flow markers; and (2) to expose the succession of events occurring within the viscous domain during the onset of dynamic stall. Unsteady effects were shown to have an important influence on the progression of flow reversal along the airfoil surface prior to stall. A region of reversed flow underlying a free shear layer was found to momentarily exist over the entire upper surface without any appreciable disturbance of the viscous-inviscid boundary. A flow protuberance was observed to develop near the leading edge, while minor vortices evolve from an expanding instability of the free shear layer over the rear portion of the airfoil. The complete breakdown of this shear layer culminates in the successive formation of two dominant vortices.

Mcalister, K. W.↗

NASA NTRS1973

Calculation of unsteady transonic aerodynamics for oscillating wings with thickness.

An analytical approach is presented to account for some of the nonlinear characteristics of the transonic flow equation for finite thickness wings undergoing harmonic oscillation at sonic flight speed in an inviscid, shock-free fluid. The thickness effect is accounted for in the analysis through use of the steady local Mach number distribution over the wing at its mean position by employing the local linearization concept and a coordinate transformation. Computed results are compared with that of the linearized theory and experiments. Application to a flutter problem is shown.

Ruo, S. Y.↗

NASA NTRS1975

Calculation of unsteady transonic aerodynamics for oscillating wings with thickness

Ruo, S. Y.↗

NASA NTRS1972

Part 1: Classical laser. Part 2: The effect of velocity changing collisions on the output of a gas laser

A classical model for laser action is discussed, in which an active medium consisting of anharmonic oscillators interacts with an electromagnetic field in a resonant cavity. Comparison with the case of a medium consisting of harmonic oscillators shows the significance of nonlinearities for producing self-sustained oscillations in the radiation field. A theoretical model is presented for the pressure dependence of the intensity of a gas laser, in which only velocity-changing collisions with foreign gas atoms are included. A collision model for hard sphere, repulsive interactions was derived. Collision theory was applied to a third-order expansion of the polarization in powers of the cavity electric field (weak signal theory).

Borenstein, M.↗

NASA NTRS1990

Collision-induced gas phase dissociation rates

The Landau-Zener theory of reactive cross sections was applied to diatomic molecules dissociating from a ladder of vibrational states. The result predicts a dissociation rate that is quite well duplicated by an Arrhenius function having a preexponential temperature dependence of about T(sub -1/2), at least for inert collision partners. This relation fits experimental data reasonably well. 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 given by Hammerling, Kivel, and Teare in their analytic approximation for harmonic oscillators, though at very high temperature a correction for the partition function limit is included. The empirical correction for vibration nonequilibrium proposed by Park, which is a convenient algorithm for CFD calculations, is modified to prevent a drastic underestimation of dissociation rates that occurs with this method when vibrational temperature is much smaller than the kinetic temperature of the gas.

Hansen, C. Frederick↗

NASA NTRS1995

Kravchuk functions for the finite oscillator approximation

Kravchuk orthogonal functions - Kravchuk polynomials multiplied by the square root of the weight function - simplify the inversion algorithm for the analysis of discrete, finite signals in harmonic oscillator components. They can be regarded as the best approximation set. As the number of sampling points increases, the Kravchuk expansion becomes the standard oscillator expansion.

Atakishiyev, Natig M.↗

NASA NTRS1972

Classical laser.

In this paper a completely classical model for laser action is discussed. An active medium consisting of classical anharmonic oscillators interacts with a classical electromagnetic field in a resonant cavity. Comparison with the case of a medium consisting of harmonic oscillators shows the significance of nonlinearities for producing self-sustained oscillations in the radiation field. The results for the classical model are found to be similar to those for a semiclassical model of the ammonia-beam maser. The conclusion is that laser action is not intrinsically a quantum-mechanical effect. The classical-laser theory as given in this paper can also be applied to the case of the electron-cyclotron maser.

Borenstein, M.↗

NASA NTRS1995

Numerical Studies of Boundary-Layer Receptivity

Direct numerical simulations (DNS) of the acoustic receptivity process on a semi-infinite flat plate with a modified-super-elliptic (MSE) leading edge are performed. The incompressible Navier-Stokes equations are solved in stream-function/vorticity form in a general curvilinear coordinate system. The steady basic-state solution is found by solving the governing equations using an alternating direction implicit (ADI) procedure which takes advantage of the parallelism present in line-splitting techniques. Time-harmonic oscillations of the farfield velocity are applied as unsteady boundary conditions to the unsteady disturbance equations. An efficient time-harmonic scheme is used to produce the disturbance solutions. Buffer-zone techniques have been applied to eliminate wave reflection from the outflow boundary. The spatial evolution of Tollmien-Schlichting (T-S) waves is analyzed and compared with experiment and theory. The effects of nose-radius, frequency, Reynolds number, angle of attack, and amplitude of the acoustic wave are investigated. This work is being performed in conjunction with the experiments at the Arizona State University Unsteady Wind Tunnel under the direction of Professor William Saric. The simulations are of the same configuration and parameters used in the wind-tunnel experiments.

Reed, Helen L.↗

NASA NTRS1956

On the Kernel function of the integral equation relating lift and downwash distributions of oscillating wings in supersonic flow

This report treats the Kernel function of the integral equation that relates a known or prescribed downwash distribution to an unknown lift distribution for harmonically oscillating wings in supersonic flow. The treatment is essentially an extension to supersonic flow of the treatment given in NACA report 1234 for subsonic flow. For the supersonic case the Kernel function is derived by use of a suitable form of acoustic doublet potential which employs a cutoff or Heaviside unit function. The Kernel functions are reduced to forms that can be accurately evaluated by considering the functions in two parts: a part in which the singularities are isolated and analytically expressed, and a nonsingular part which can be tabulated.

Watkins, Charles E↗

NASA NTRS1955

On the Kernel function of the integral equation relating the lift and downwash distributions of oscillating finite wings in subsonic flow

This report treats the Kernel function of an integral equation that relates a known prescribed downwash distribution to an unknown lift distribution for a harmonically oscillating finite wing in compressible subsonic flow. The Kernel function is reduced to a form that can be accurately evaluated by separating the Kernel function into two parts: a part in which the singularities are isolated and analytically expressed and a nonsingular part which may be tabulated. The form of the Kernel function for the sonic case (Mach number 1) is treated separately. In addition, results for the special cases of Mach number of 0 (incompressible case) and frequency of 0 (steady case) are given. The derivation of the integral equation which involves this Kernel function is reproduced as an appendix. Another appendix gives the reduction of the form of the Kernel function obtained herein for the three-dimensional case to a known result of Possio for two-dimensional flow. A third appendix contains some remarks on the evaluation of the Kernel function, and a fourth appendix presents an alternate form of expression for the Kernel function.

Watkins, Charles E↗

NASA NTRS1983

Energetics and optimum motion of oscillating lifting surfaces

Low-frequency, unsteady, lifting-line theory is used to characterize the energetics and optimum motion of an unswept rigid wing oscillating harmonically in an inviscid, incompressible flow. The energetics calculations account for the leading edge suction force, the power absorbed in the wing oscillations, and the energy loss rate produced by vortex shedding. Optimization is achieved by minimizing the average energy loss rate in relation to a given thrust, and a unique solution is found in the three dimensional case for low, reduced frequencies. The two-dimensional solution is nonunique, a condition which is examined in terms of the normal modes of the energy loss rate matrix. An invisible mode with a hydrodynamic efficiency of 100 pct is obtained in the two-dimensional case, causing the nonuniqueness of the solution by yielding no fixed positive thrust through perfect unsteady feathering.

Ahmadi, A. R.↗

NASA NTRS1975

Cascade with subsonic leading-edge locus

The paper investigates a two-dimensional oscillating cascade with a subsonic leading edge locus in a supersonic flow. The blades are assumed to be of small thickness and camber, and are undergoing small amplitude-harmonic oscillations. The problem is reduced to the solution of a functional integral equation, and an expression is given for the kernel function.

Goldstein, M. E.↗

NASA NTRS1953

Calculations on the forces and moments for an oscillating wing-aileron combination in two-dimensional potential flow at sonic speed

The linearized theory for compressible unsteady flow is used, as suggested in recent contributions to the subject, to obtain the velocity potential and the lift and moment for a thin harmonically oscillating, two-dimensional wing-aileron combination moving at sonic speed. The velocity potential is derived by considering the sonic case as the limit of the linearized supersonic theory. From the velocity potential explicit expressions for the lift and moment are developed for vertical translation and pitching of the wing and rotation of the aileron. The sonic results are compared and found to be consistent with previously obtained subsonic and supersonic results. Several figures are presented showing the variation of lift and moment with reduced frequency and Mach number and the influence of Mach number on some cases of bending-torsion flutter.

Nelson, Herbert C↗

NASA NTRS1995

Geometric phases, evolution loops and generalized oscillator potentials

The geometric phases for dynamical processes where the evolution operator becomes the identity (evolution loops) are studied. The case of time-independent Hamiltonians with equally spaced energy levels is considered; special emphasis is made on the potentials having the same spectrum as the harmonic oscillator potential (the generalized oscillator potentials) and their recently found coherent states.

Fernandez, David J.↗

NASA NTRS1947

Theoretical study of air forces on an oscillating or steady thin wing in a supersonic main stream

A theoretical study, based on the linearized equations of motion for small disturbance, is made of the air forces on wings of general plan forms moving forward at a constant supersonic speed. The boundary problem is set up for both the harmonically oscillating and the steady conditions. Two types of boundary conditions are distinguished, which are designated "purely supersonic" and "mixed supersonic." the method is illustrated by applications to a number of examples for both the steady and the oscillating conditions. The purely supersonic case involves independence of action of the upper and lower surfaces of the airfoil and present analysis is mainly concerned with this case. A discussion is first given of the fundamental or elementary solution corresponding to a moving source. The solutions for the velocity potential are then synthesized by means of integration of the fundamental solution for the moving source. The method is illustrated by applications to a number of examples for both the steady and the oscillating cases and for various plan forms, including swept wings and rectangular and triangular plan forms. The special results of a number of authors are shown to be included in the analysis.

Garrick, I E↗

NASA NTRS2011

A Sub-Hertz, Low-Frequency Vibration Isolation Platform

One of the major technical problems deep-space optical communication (DSOC) systems need to solve is the isolation of the optical terminal from vibrations produced by the spacecraft navigational control system and by the moving parts of onboard instruments. Even under these vibration perturbations, the DSOC transceivers (telescopes) need to be pointed l000 fs of times more accurately than an RF communication system (parabolic antennas). Mechanical resonators have been extensively used to provide vibration isolation for groundbased, airborne, and spaceborne payloads. The effectiveness of these isolation systems is determined mainly by the ability of designing a mechanical oscillator with the lowest possible resonant frequency. The Low-Frequency Vibration Isolation Platform (LFVIP), developed during this effort, aims to reduce the resonant frequency of the mechanical oscillators into the sub-Hertz region in order to maximize the passive isolation afforded by the 40 dB/decade roll-off response of the resonator. The LFVIP also provides tip/tilt functionality for acquisition and tracking of a beacon signal. An active control system is used for platform positioning and for dampening of the mechanical oscillator. The basic idea in the design of the isolation platform is to use a passive isolation strut with an approximately equal to 100-mHz resonance frequency. This will extend the isolation range to lower frequencies. The harmonic oscillator is a second-order lowpass filter for mechanical disturbances. The resonance quality depends on the dissipation mechanisms, which are mainly hysteretic because of the low resonant frequency and the absence of any viscous medium. The LFVIP system is configured using the well-established Stewart Platform, which consists of a top platform connected to a base with six extensible struts (see figure). The struts are attached to the base and to the platform via universal joints, which permit the extension and contraction of the struts. The struts ends are connected in pairs to the base and to the platform, forming an octahedron. The six struts provide the vibration isolation due to the properties of mechanical oscillators that behave as second-order lowpass filters for frequencies above the resonance. At high frequency, the ideal second-order low-pass filter response is spoiled by the distributed mass and the internal modes of membrane and of the platform with its payload.

Ortiz, Gerardo, G.↗

NASA NTRS2017

Thermal Expansion from Stochastic Nonlinear Acoustic Fields

The methods of stochastic mechanics are applied to nonlinear acoustic fields generated by random acoustic radiation sources associated with oscillating lattice sites in nonlinear lattices. The assumption of stochastically independent zero-point and temperature-dependent acoustic fields leads to expressions of the thermodynamic internal and Helmholtz free energies per unit mass in terms of modal energies per unit mass that account for the nonlinearity of the propagation modes. The thermodynamic state functions canonically transform to the familiar results of a system of quantized, simple harmonic oscillators in the linear field limit. The thermal expansion coefficient, derived from the Helmholtz free energy per unit mass, is obtained as a sum of the zero-point modal nonlinearity parameters, weighted by temperature-dependent, modal heat capacities. The relationship is fully anharmonic, in contrast to commonly used quasi-harmonic models, and predicts with excellent agreement the experimentally observed null thermal expansion of vitreous silica at the ‘cross-over’ temperature corresponding to the balance between long wavelength modes that contribute negative nonlinearity parameters to the modal sum and short wavelength modes that contribute positive nonlinearity parameters.

Cantrell, John H., Jr.↗