The choice of trial functions in the perturbation-variation method.
Nonlinear variational trial function for atomic hydrogen dynamic polarizability, discussing first order perturbed wave function and perturbation- variation method
HYDROGEN ATOM↗
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First-order secular perturbations of an artificial earth satellite due to the sun and moon.
Artificial earth satellite orbital elements first order secular perturbations due to sun and moon mass derived, using Lagrangian perturbation equations
Dearman, C. C., Jr.↗
Measured thermal-neutron-flux perturbations in cylinders in a test reactor.
Thermal neutron flux perturbations in cylinders in test reactors, using regression analysis to obtain polynomials for flux perturbation, depression, self shielding factors, etc
Lynch, J. H.↗
A simple linear approximation for perturbed motion about moderately elliptic orbits
Moderately elliptic reference orbit perturbed motion, applying linearized perturbation equations for circular orbit
Koble, H. M.↗
Classical adiabatic perturbation theory
Classical Poincare-von Zeipel canonical perturbation theory extension to adiabatically perturbed systems with slow dependence on time or dynamic variables
Stern, D. P.↗
On optimizing the treatment of exchange perturbations
A method using the zeroth plus first order wave functions, obtained by optimizing the basic equation used in exchange perturbation treatments, is utilized in an attempt to determine the exact energy and wave function in the exchange process. Attempts to determine the first order perturbation solution by optimizing the sum of the first and second order energies were unsuccessful.
Hirschfelder, J. O.↗
On the analytic lunar and solar perturbations of a near earth satellite
The disturbing function of the moon (sun) is expanded as a sum of products of two harmonic functions, one depending on the position of the satellite and the other on the position of the moon (sun). The harmonic functions depending on the position of the perturbing body are developed into trigonometric series with the ecliptic elements l, l', F, D, and Gamma of the lunar theory which are nearly linear with respect to time. Perturbation of elements are in the form of trigonometric series with the ecliptic lunar elements and the equatorial elements omega and Omega of the satellite so that analytic integration is simple and the results accurate over a long period of time.
Estes, R. H.↗
Lunar fossil magnetism and perturbations of the solar wind.
Perturbations of the solar wind downstream of the moon and lying outside of the rarefaction wave that defines the diamagnetic cavity are used to define possible source regions comprised of intrinsically magnetized areas of the moon. A map of the moon is constructed showing that a model in which the sources are exposed to the grazing solar wind during the lunation yields a selenographically invariant set of regions strongly favoring the lunar highlands over the maria. An alternative model with the source due to electromagnetic induction is explored. The ages of the field sources should be consistent with those based on the basalt ages and possibly far older if the sources are connected with the formation of the highland rocks themselves. The perturbations are tentatively identified as weak shock waves, and a Mach angle in accord with nominal values for the solar wind is found.
Sonett, C. P.↗
Coordinate perturbation and multiple scale in gasdynamics.
A simple way of applying the coordinate perturbation technique is developed which bypasses the usual transformations of the equations to perturbed coordinates, and which provides the appropriate coordinate stretching by simple algebraic manipulations on the nonuniformly valid expansion obtained by a straightforward expansion from the original equations. Interesting results are obtained by applying the procedure to two gasdynamical problems. In the first, the flowfield around a supersonic two-dimensional wing is determined up to third order, including a uniformly valid representation of the front shock shape. The second concerns the oscillations in a closed tube following an arbitrary initial disturbance, both when the two ends are closed, and when one of the two ends contains an oscillating piston.
Crocco, L.↗
Perturbed nonlinear differential equations
The existence of a solution defined for all t and possessing a type of boundedness property is established for the perturbed nonlinear system y = f(t,y) + F(t,y). The unperturbed system x = f(t,x) has a dichotomy in which some solutions exist and are well behaved as t increases to infinity, and some solution exists and are well behaved as t decreases to minus infinity. A similar study is made for a perturbed nonlinear differential equation defined on a half line, R+, and the existence of a family of solutions with special boundedness properties is established. The ideas are applied to integral manifolds.
Proctor, T. G.↗
Tidal perturbations on the orbits of GEOS-1 and GEOS-2
Analysis of the luni-solar tidal perturbations of the inclination of GEOS-1 (1965-89A) and GEOS-2 (1968-002A) yielded the values k2 = 0.22 (sigma = 0.02) and 0.31 (sigma = 0.01) respectively for the second degree Love number. For GEOS-1 a new, purely numerical method involving osculating elements was employed. For GEOS-2 it was necessary to analyze the variations of the mean elements because of the very long period (450d) of the dominant solar tidal perturbation. An additional analysis of the variation of the mean elements of GEOS-1 confirmed the value of k2 obtained from the osculating elements.
Douglas, B. C.↗
A model for estimating the cost impact of schedule perturbations on aerospace research and development programs
The problem of determining the cost impact attributable to perturbations in an aerospace R and D program schedule is discussed in terms of the diminishing availability of funds. The methodology from which a model is presented for updating R and D cost estimates as a function of perturbations in program time is presented.
Bishop, D. F.↗
System design optimization for a Mars-roving vehicle and perturbed-optimal solutions in nonlinear programming
Work in two somewhat distinct areas is presented. First, the optimal system design problem for a Mars-roving vehicle is attacked by creating static system models and a system evaluation function and optimizing via nonlinear programming techniques. The second area concerns the problem of perturbed-optimal solutions. Given an initial perturbation in an element of the solution to a nonlinear programming problem, a linear method is determined to approximate the optimal readjustments of the other elements of the solution. Then, the sensitivity of the Mars rover designs is described by application of this method.
Pavarini, C.↗
Perturbation solutions for the influence of forward flight on helicopter rotor flapping stability
The stability of the flapping motion of a helicopter rotor blade in forward flight is investigated, using a perturbation technique which gives analytic expressions for the eigenvalues, including the influence of the periodic aerodynamic forces in forward flight. The perturbation solutions are based on small advance ratio (the ratio of the helicopter forward speed to the rotor tip speed). The rotor configurations considered are a single, independent blade; a teetering rotor; a gimballed rotor with three, four, and five or more blades; and a rotor with N independent blades. The constant coefficient approximation with the equations and degrees of freedom in the nonrotating frame represents the flap dynamic reasonably well for the lower frequency modes, although it cannot, of course, be completely correct. The transfer function of the rotor flap response to sinusoidal pitch input is examined, as an alternative to the eigenvalues as a representation of the dynamic characteristics of the flap motion.
Johnson, W.↗
Control optimization of a lifting body entry problem by an improved and a modified method of perturbation function
A study of the solution problem of a complex entry optimization was studied. The problem was transformed into a two-point boundary value problem by using classical calculus of variation methods. Two perturbation methods were devised. These methods attempted to desensitize the contingency of the solution of this type of problem on the required initial co-state estimates. Also numerical results are presented for the optimal solution resulting from a number of different initial co-states estimates. The perturbation methods were compared. It is found that they are an improvement over existing methods.
Garcia, F., Jr.↗
Aircraft Range Optimization Using Singular Perturbations
An approximate analytic solution is developed for the problem of maximizing the range of an aircraft for a fixed end state. The problem is formulated as a singular perturbation and solved by matched inner and outer asymptotic expansions and the minimum principle of Pontryagin. Cruise in the stratosphere, and on transition to and from cruise at constant Mach number are discussed. The state vector includes altitude, flight path angle, and mass. Specific fuel consumption becomes a linear function of power approximating that of the cruise values. Cruise represents the outer solution; altitude and flight path angle are constants, and only mass changes. Transitions between cruise and the specified initial and final conditions correspond to the inner solutions. The mass is constant and altitude and velocity vary. A solution is developed which is valid for cruise but which is not for the initial and final conditions. Transforming of the independent variable near the initial and final conditions result in solutions which are valid for the two inner solutions but not for cruise. The inner solutions can not be obtained without simplifying the state equations. The singular perturbation approach overcomes this difficulty. A quadratic approximation of the state equations is made. The resulting problem is solved analytically, and the two inner solutions are matched to the outer solution.
Oconnor, Joseph Taffe↗
A numerical-perturbation method for the nonlinear analysis of structural vibrations
A numerical-perturbation method is proposed for the determination of the nonlinear forced response of structural elements. Purely analytical techniques are capable of determining the response of structural elements having simple geometries and simple variations in thickness and properties, but they are not applicable to elements with complicated structure and boundaries. Numerical techniques are effective in determining the linear response of complicated structures, but they are not optimal for determining the nonlinear response of even simple elements when modal interactions take place due to the complicated nature of the response. Therefore, the optimum is a combined numerical and perturbation technique. The present technique is applied to beams with varying cross sections.
Nayfeh, A. H.↗
Perturbation theory for a resistivity shunted Josephson element
We present a systematic perturbation theory, extendable in principle to all orders of magnitude, for the solution of the equations of motion of an ideal Josephson element shunted by a resistance and driven by a dc current source and a small time-dependent source. We present second-order results for the case in which the time dependence is that of a single sinusoid, and these results are compared with other numerical and analytical calculations. Near, but not on, the first constant voltage step where the perturbation theory appears divergent, the current-voltage characteristic is calculated by means of a nonperturbative adiabatic procedure. The impedance and responsivity agree with earlier results.
Thompson, E. D.↗