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Experimental results are presented that verify the theoretical conclusion that an inverted pendulum may be stabilized with second-order stationary, random parametric excitation having a discrete-power spectral density. Experiments also verified that the theoretical condition on frequency spacing is critical.

A new method for the solution of the problem of heat conduction in a melting slab, where the molten material is immediately removed upon formation, is presented. No restrictions are placed on the boundary conditions which may be imposed on the slab and the material properties are allowed to be temperature dependent. The problem of determining the temperature distribution in the slab and the amount of material melted is reduced to finding the solution of an ordinary differential equation on the amount of material melted. This reduction from a partial differential equation problem is accomplished by determining a Taylor's series expansion in space for the temperature distribution. The equation so obtained for the determination of the amount of material melted is of a form readily solved numerically. Comparisons with known results for a slab insulated on one face and subjected to a constant heat input on the other face are given.

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Terminal guidance of a soft-landing lunar spacecraft can be achieved by perturbing a nominal gravity-turn trajectory This perturbation is imposed by offsetting the thrust vector for some period of time from the nominal attitude, anti-parallel to the instantaneous velocity vector The offset is removed when the predicted and required landing points coincide The vehicle then proceeds along a gravity-turn descent trajectory until landing Closed-form solutions obtained determine the thrust acceleration required and the predicted landingpoint coordinates For trajectories following a shallow descent, as from a parking orbit, the moon is represented by spherical geometry For trajectories that initially make large angles with the local horizon, or have small velocities, the moon is considered flat During descent an on-board computer compares the predicted and desired landing-point coordinates A pin-point guidance channel drives the down-range and cross-range components of the landing point error to zero A soft-landing control channel constrains the thrust level so as to terminate the trajectory at the lunar surface with zero velocity Separation of the two control functions has the advantage of assuring a safe landing in the event of component failui es in the pin-point guidance subsystem Simulation studies have shown this technique to be quite effectiveSubscripts 0 f Nomenclature thrust acceleration vehicle energy per unit mass acceleration at lunar surface due to gravitational attraction, 5 322 ft/sec 2 vehicle angular momentum altitude above lunar surface radial distance from center of moon lunar radius 5 702 X 10 6 ft path length traversed time vehicle velocity circular velocity free-fall velocity down-range distance traversed at any time during descent, flat moon height above lunar surface, flat moon flight path angle central angle traversed by vehicle at any time during descent lunar gravitational parameter reciprocal of normalized thrust acceleration = initial value = value at termination of landing maneuver Problem Statement P REVIOUS analyses of the lunar problem have been principally concerned with either the basic trajectory considerations for various types of approach 1 " 3 or implicit guidance systems with varying amounts of landing-point prediction capability 4~8 The attendant complexity of some of these systems, which require high-speed repetitive computation, has led the authors to consider the possibility of developing an explicit guidance technique It is the purpose of this paper to present a method of accomplishing soft landings at predetermined points on the lunar surface The guidance system used is self-contained and may be applied to either a hyperbolic approach or a descent from a parking orbit In addition to the analytical determination of the descent trajectories, a possible system implementation of the pinpoint landing scheme is described Requirements for onboard sensing and navigation equipment are established, and procedures for automatic and manual guidance operations are discussed Results show...