Abstract

Lissajous Figures for Flight Test Data Analysis In Lissajous Ž gures, when a point executes two motions simultaneously at right angles to one another in a plane, it traces out a two-dimensional trajectory that lies within a rectangle. If there is no relation between the two motions, the point does not return to its original position, and its trajectory Ž lls up the rectangle by its repeated passages. However, recognizable stationary patterns emerge whenever the two motions are either of the same frequency or are commensurate. Two motions are commensurate with each other if the ratio of their frequencies is a rational number and the initial phases are a simple fraction of 2p. Commensurateness does not require that either motion be absolutely periodic (constant frequency). These stationary patterns are Lissajous Ž gures named after J. A. Lissajous, who made an extensive study of such motions. Figure 1 shows commensurate motions at two different combining frequencies and Ž ve different initial phases. The motions combined at right angles to each other are both simple harmonics, and their frequencies ratio is a rational number. Different initial phase differences are readily apparent by inspection. Further, the trajectory is tangential to the boundary lines of the rectangle enclosing the Lissajous Ž gure. Inspection of the ratio of the number of tangencies made by the trajectories with the vertical and horizontal adjacent sides of the rectangle gives the ratio of the combining frequencies. For example, at v1/v2 = 4 and f = 90, inspection reveals four tangencies on either vertical side and one tangency on either horizontal side of the rectangle, which means the frequencies are at a ratio of 4. The only exception to this rule is when the initial phase is either 0 or 180 deg. Historically, Lissajous Ž gures have been used extensively in electrical and mechanical engineering. A signal generator generates one signal of known frequency, and an oscilloscope compares the known signal with an unknown by combining them at right angles to each other. If there is a small difference

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