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A brief outline of the method of transient analysis which employs half-period transient-gain (HPTG) characterizing functions to linearize the nonlinearity in control systems is first given. Two theorems concerning these functions are then presented. The first theorem shows that the HPTG of a nonlinearity, which is formed by adding N other characteristics, is the sum of the HPTG functions of the N individual characteristics. The second theorem shows that the imaginary part of the HPTG is proportional to the area of the loop in the nonlinearity. With these theorems, a general method for calculating the HPTG of any single- or double-valued nonlinearity is developed which involves only arithmetic computations. The method is illustrated by application to a magnetic hysteresis characteristic with variable loop width and saturation. With the HPTG assumed calculated, a method of evaluating the transient response is developed which avoids the step-by-step calculations necessary with an earlier approach and also leads to direct ways of finding rise time, overshoot, time to first overshoot, and settling time of a step-function response. This work is illustrated by application to a position control system in which nonlinear friction is the important nonlinearity. It is shown that the method accurately predicts heavily damped transients and the initial sticking present.
Date of Publication: October 1967