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Time Vector Analysis
The time vector analysis method provides an excellent insight into the modes of airplane motion. The method came about as an incidental result of debugging one of the world’s first electronic analog computers, built to represent generalized airplane longitudinal dynamics. This computer’s inventor was Dr. Robert K. Mueller; his device, now in the MIT Museum, was built to support his 1936 MIT ScD thesis.
The fundamental concept of time vector analysis is that for any oscillatory transient generated by a linear system having a certain undamped natural frequency and damping ratio:
1. the amplitude of the transient derivative is the transient amplitude multiplied by the undamped natural frequency, and
2. the phase of the transient derivative is the phase of the transient advanced by 90 degrees plus the angle whose sine is the damping ratio.
With this concept, one can construct time vector polygons representing each term in any system equation corresponding to a particular modal solution of the characteristic equation. The time vector polygons show which terms are dominant and how the amplitude and phase relations among the variables arise (Figure 18.9). In Mueller’s thesis example, the time vector polygons give insight into the wind axis equations of longitudinal motion and suggest correction of the phugoid mode instability with pitch attitude feedback. At the urging of his then-supervisor at the Glenn L. Martin Company, James S. McDonnell, he presented a paper on the topic at a meeting of the Institute of Aeronautical Sciences (Mueller, 1937).
In Germany, Dr. Karl-H. Doetsch used the time vector method to study lightly damped airplane-autopilot combinations. Working at the Royal Aircraft Establishment (RAE) after World War II, K-H. Doetsch and W. J. G. Pinsker applied time vector analysis methods to the Dutch roll problems of jet airplanes.
There was an early application of the time vector analysis method by Leonard Sternfield of the NACA Langley Laboratory to the Dutch roll oscillation. Around 1951 he built two bridge-table-size mechanical analogs of the roll and yaw time vector polygons to predict the Dutch roll characteristics of new airplanes. Around the same time E. E. Larrabee made what he thought was the first use of time vector analysis to extract stability derivatives from flight time history measurements, although Doetsch had done much the same in England.
