The variational method 2 - Perturbation theory
Variational method to approximate solutions of perturbation equations, perturbation analyses, and use of perturbation theory to infer variational principle
Epstein, S. T.↗
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An analysis of the expected eccentricity perturbations for the second Radio Astronomy Explorer (RAE B)
Analytical prediction of expected eccentricity perturbations for the RAE 2 lunar orbit shows that the eccentricity will grow linearly in time. Parametric inclination studies and analysis of perturbation equations establish a critical retrograde inclination of 116.565 at which the positive perturbation slope vanishes for a circular orbit about 1100 m above the lunar surface with an eccentricity constraint of less than 0.005 during a period of about one year.
Murphy, J. P.↗
The responses of rising or falling spherical wind sensors to atmospheric wind perturbations.
Analysis of the responses of rising or falling spherical wind sensors to atmospheric wind perturbations on the wind profile in the vertical, using Fourier transform techniques. The linearized equations of motion of a sensor that is subject to drag and gravitational body forces are developed by perturbing a sensor about an equilibrium uniform motion with wind fluctuations which have vertical variations. The wind environment and sensor velocities are decomposed with stochastic Fourier-Stieltjes integrals, and the linearized equations of motion are used to derive the response functions and phase angles of the sensor motions. The results of the analysis are used to analyze the response properties of the Jimsphere balloon wind sensor. It is shown that, in general, the transfer functions associated with the horizontal sensor motions are smaller than the transfer functions associated with the vertical sensor motions in the omega times T product range from zero to infinity, omega being the wind perturbation frequency and T a time constant of the system. Thus, the sensor is more responsive to vertical than to horizontal air motions.
Fichtl, G. H.↗
On optimizing the treatment of exchange perturbations.
Most theories of exchange perturbations would give the exact energy and wave function if carried out to an infinite order. However, the different methods give different values for the second-order energy, and different values for E(1), the expectation value of the Hamiltonian corresponding to the zeroth- plus first-order wave function. In the presented paper, it is shown that the zeroth- plus first-order wave function obtained by optimizing the basic equation which is used in most exchange perturbation treatments is the exact wave function for the perturbation system and E(1) is the exact energy.
Hirschfelder, J. O.↗
Attitude perturbations of a spinning Jupiter Orbiter spacecraft.
A Pioneer-class spinning spacecraft, serving as an orbiter of the planet Jupiter, depends on close-tolerance pointing of the spin axis toward earth for communications, and on a uniform spacecraft spin rate for control and correlation of scientific measurements. However, these attitude characteristics are subject to perturbations due to the gravity gradient and the magnetic field associated with Jupiter. The mechanisms of these perturbing effects are identified, and their magnitudes are estimated in parametric analyses. The conditions which can lead to perturbations which are significant to the mission and the means of alleviating these conditions are indicated.
Dixon, W. J.↗
The response of the upper atmosphere to perturbations from diffusive equilibrium
It is generally assumed that, in the atmosphere above 120 km, the deviations from diffusive equilibrium are small, though a minor constituent may show some deviation. The response of such a constituent to perturbations from diffusive equilibrium was analyzed qualitatively. It is shown that the magnitude of these deviations is mainly determined by the characteristic diffusion time. For a time-dependent perturbation, the ratio of the characteristic time of the perturbation to the characteristics diffusion time is an important parameter.
Blum, P. W.↗
A determination of the earth tidal amplitude and phase from the orbital perturbations of the Beacon Explorer C spacecraft
An analysis of the orbital inclination of the Beacon Explorer C spacecraft over a period of nearly five months in 1970 revealed a very clear and distinctive perturbation caused by the earth and ocean tides. The perturbation has a full amplitude of about 1.8 seconds of arc and a period of about 85 days. This amplitude is approximately 15% smaller than would be expected from the solid-earth tide alone and is shifted slightly in phase. The perturbation can be represented almost exactly by a Love number of k sub 2 = 0.245 with a phase lag of 3.2 degrees. The data used were laser range measurements obtained by a NASA Goddard Space Flight Center tracking system in Greenbelt, Maryland.
Smith, D. E.↗
Applications of the Kustaanheimo-Stieffel transformation of the perturbed two-body problem
The Newtonian differential equations of motion for the two-body problem can be transformed into four linear harmonic-oscillator equations by simultaneously applying the regularization step dt/ds = r and the Kustaanheimo-Stieffel (KS) transformation. The regularization step changes the independent variable from time to a new variable s, and the KS transformation transforms the position and velocity vectors from Cartesian space into a four-dimensional space. A derivation of a uniform, regular solution for the perturbed two-body problem in the four-dimensional space is presented. The variation-of-parameters technique is used to develop expressions for the derivatives of ten elements (which are constants in the unperturbed motion) for the general case that includes both perturbations which can arise from a potential and perturbations which cannot be derived from a potential. This ten-element solution has mixed secular terms that degrade the long-term accuracy during numerical integration. Therefore, to eliminate these terms, the solution is modified by introducing two additional elements.
Bond, V. R.↗
Lunar perturbations of artificial satellites of the earth
The disturbing function for the lunar perturbations of an artificial satellite are derived, using ecliptic elements for the moon and equatorial elements for the satellite. Secular, long-period, and short-period perturbations are then computed, with the expressions kept in closed form in both inclination and eccentricity of the satellite. Alternative expressions for short-period perturbations of high satellites are also given, assuming small values of the eccentricity. The moon's position is specified by the inclination, node, argument of perigee, true (or mean) longitude, and its radius vector from the center of the earth. The results can then be applied to numerical integration by using coordinates of the moon from ephemeris tapes or to analytical representation by using results from lunar theory, with the moon's motion represented by a precessing and rotating elliptical orbit.
Giacaglia, G. E. O.↗
Perturbation treatment of mixing in Josephson junctions
The mixing properties of a current-biased resistively-shunted Josephson junction irradiated at two frequencies are calculated by a perturbation technique, treating both signals as perturbations. The size of the mixing steps is obtained from second-order calculations. It is shown that subharmonic mixing steps are absent in the case of a single frequency. The amplitude of the voltage oscillations at the difference and sum frequencies is shown to be nonzero at all voltages. The microwave resistance is calculated for one frequency to third order in the perturbation.
Levinsen, M. T.↗
Pressure perturbation in EHD contacts due to an ellipsoidal asperity
The pressure fluctuations around an ellipsoidal asperity tip at the inlet region of an elastohydrodynamic contact is determined by solving a perturbed Reynolds equation assuming that the asperity shape is unaffected by the perturbed pressure. Results are presented as the amplitude of the perturbed pressure as a function of ellipticity ratio, maximum Hertzian pressure, nominal EHD film thickness, asperity size, asperity height, and pressure viscosity coefficient. For the case of simple sliding between a smooth surface and a stationary asperity, a comparison is also made between the results for large ellipticity ratios and the pressure fluctuations obtained by using a previous line contact EHD analysis for a two-dimensional asperity ridge. The agreement is found to be close for moderate Hertzian pressure and relatively thick EHD film thickness.
Chow, L. S. H.↗
On the equations governing the electromagnetic perturbations of the Kerr black hole
Teukolsky's (1973) radial wave equation for electromagnetic perturbations about a rotating black hole is transformed into a one-dimensional wave equation with a real and short-range potential. A general technique for transforming one wave equation into another is described, and it is shown how an equation with a complex potential can be transformed into one with a purely real potential. These general techniques are then used to transform Teukolsky's equation into a wave equation whose potential is purely real and of short range. This equation is found to provide a simple formulation of the concepts of superradiant scattering and the conservation of energy underlying the present electromagnetic perturbations. It is noted that the techniques employed are general enough to be applicable when the spin weight is plus or minus 2, making a 'real-potential' formulation of the problem of gravitational perturbations about a Kerr black hole most likely.
Detweiler, S.↗
A perturbation solution for interlaminar stresses in bidirectional laminates
The paper considers a thin elastic bidirectional symmetric composite laminate subjected to a uniform axial strain. All interlaminar stresses are determined by a zeroth-order perturbation analysis of the governing equations. The method of solution considers the laminate to be composed of an interior region (removed from the free edge) and a boundary layer region. The interior region is analyzed by dropping terms related to powers of the thickness-to-width ratio in the dimensionless elasticity equations. The boundary layer region is studied by introducing a stretching transformation to the governing equations. A uniform solution is then formed by satisfying Prandtl's matching principle of perturbation theory. Results for a four-layer bidirectional graphite-epoxy laminate are compared with existing finite-difference results. It is shown that the perturbation analysis provides more insight into the stress distribution near the free edge.
Hsu, P. W.↗
Perturbation solutions for blade-to-blade surfaces of a transonic compressor
The paper describes a perturbation method for turbomachinery calculations, particularly where it is necessary to carry out a number of calculations for closely-related flows such as are needed in a parametric study. The method is applied for solving a model problem involving blade-to-blade surfaces of a transonic compressor. Basically, the method makes use of a previously calculated base solution to determine first-order changes in the flow field due to variations in one or more of a variety of geometrical or flow field parameters. The fundamental assumption associated with the perturbation solution is that the magnitude of the deviations from the base solution lies within the range of a linear perturbation analysis. Comparisons are made for results obtained, by varying the thickness ratio of an unstaggered nonlifting cascade composed of biconvex profiles in a flow with an oncoming freestream Mach number of 0.60.
Stahara, S. S.↗
Application of perturbation methods to improve analytical model correlation with test data
A linear perturbation method developed by White (1972) to improve the correlation between analytical models and experimental data is applied to modeling of the Viking Propulsion Subsystem (VPS), flown successfully as part of the Viking Spacecraft launch of 1975. Analytical models and modal test data developed for the VPS are employed in the application; a set of mathematically generated data and a corresponding perturbed analytical model, representative of actual flight hardware, are also subjected to the White method. It is found that the linear perturbation method does not yield a single unique improved model.
Garba, J. A.↗
Interactions of simultaneous perturbations of stratospheric photochemistry
The combined effects of two specific ozone perturbation mechanisms are examined. The need to look back to predictions of the near past as well as the future as a means of testing model results is emphasized. The two perturbations examined are that due to chlorofluoromethane (CFM) release and that due to the air pollutants CO and nitrogen oxide. A steady-state model is used to investigate the characteristics of the stratospheric-tropospheric photochemical system for differing levels of CO and nitrogen oxide fluxes at the ground and for simultaneously changing the stratospheric from CFM release. Two basic scenarios chosen to illustrate the competing effects of the two types of perturbation use the time-dependence for chlorine radical buildup calculated for constant continued release of CFM's.
Butler, D. M.↗
A study of the application of singular perturbation theory
A hierarchical real time algorithm for optimal three dimensional control of aircraft is described. Systematic methods are developed for real time computation of nonlinear feedback controls by means of singular perturbation theory. The results are applied to a six state, three control variable, point mass model of an F-4 aircraft. Nonlinear feedback laws are presented for computing the optimal control of throttle, bank angle, and angle of attack. Real Time capability is assessed on a TI 9900 microcomputer. The breakdown of the singular perturbation approximation near the terminal point is examined Continuation methods are examined to obtain exact optimal trajectories starting from the singular perturbation solutions.
Mehra, R. K.↗
Singular perturbations and the sounding rocket problem
In this paper, Goddard's problem of maximizing the final altitude of a sounding rocket (a singular problem of optimal control) is analyzed using singular perturbation methods. The problem is first cast in singular perturbation form and then solved to zero order by adding boundary-layer corrections to the reduced solution. For a quadratic drag law, a closed-form solution is obtained, although consideration of a numerical example indicates that this solution is not useful for practical sounding rockets. However, use of state variable transformations allows a very accurate numerical approximation to be constructed. It is concluded that application of singular perturbation methods to the well-known sounding rocket problem indicates that these methods may have utility in dealing with singular problems of optimal control.
Ardema, M. D.↗