The three translational and three rotational equilibrium equations for an orbital vehicle subject to aerodynamic and jet reaction forces are derived using a modified flight-path axis system for the translational equations. The dependent variables of the system are horizontal velocity component, vertical velocity component, and flight-path heading angle. Theresulting equations are shown to have advantages for computer mechanization over alternative axis systems for the translational equations. Complete equations for determining vehicle orientation, instantaneous latitude and longitude, angle of attack, angle of sideslip, areodynamic forces and moments, etc., are presented. Modifications in the translational equations which allow direct solution by an analog computer are also given. Analog computer mechanization of these equations in both real and fast time is described, including a novel technique for division which preserves favorable multiplier scaling. Specific machine results are presented which demonstrate accurate solution of close-satellite trajectories, including re-entry from satellite altitudes to sea level. With no change in circuit or scaling the same computer mechanization yields zero-drag orbits which close within several hundred feet of altitude.

A method of generating sine and cosine functions in analog computers by means of multiplier elements and integrators is discussed. Static accuracy of the method is analyzed and found to be essentially equal to the accuracy of the multipliers employed. The system accepts dθ/dt as the input and generates output voltages of sinθ and cos θ. Amplitude-stabilizing loops are employed to maintain sin2θ+cos2θ = 1. Advantages of the method include representation of unlimited range in angles, dynamic capabilities far beyond that of the multipliers alone, and possibility of employing electronic multipliers. The method has been successfully used to compute Euler angles in analog computer solutions of the three-dimensional flight equations.

The representation of a wide variety of nonlinear functions by means of the interconnection of unstabilized operational amplifiers is discussed. The nonlinear functions described include rectification, saturation functions, coulomb friction, dead space, and starting friction. The use of operational amplifiers alone to produce square waves and triangular waves, as well as gating operations, is also discussed. These latter circuits are combined to give a time division multiplier using only standard operational amplifiers as components. Accuracy capabilities for all of these nonlinear operations are the order of 0.01 to 1 per cent.

A relay device is described which varies resistance in a stepwise continuous manner in order to approximate arbitrary functions of time. The device can be used with electronic differential analyzers to solve linear differential equations with variable coefficients. Sample solutions of Bessel's equation are included as examples. Calibration and measurement techniques which permit computer accuracies, of several hundredths of a percent are discussed.