Lissajous curve

Lissajous figures are waveform graph that result from the superposition of two harmonic, at right angles to each other standing vibrations. They are named after the French physicist Jules Antoine Lissajous ( 1822-1880 ). Later they played for example in the training to a deeper understanding of alternating currents by means of the oscilloscope a role.

They are often used for purely aesthetic purposes. A particularly fascinating sight provides the curve with minor deviations between the vibrations because the slowly rotating figure creates a 3D effect.

Mathematical Description

Mathematically, it is parametric graphs of functions of the form

Such a function is precisely then periodically when the frequency ratio

Is rational, that can be converted into an integral fraction. In this case, one obtains a closed figure. For example, give the frequency ratios

The same curve. Otherwise, the graph is not periodic and is placed close to the rectangle.

Note: The picture above shows the view similar to an oscilloscope. There, a lack of coordination of the two oscillations not a filled rectangle, as due to the limited persistence of the picture tube only part of the curve is always shown.

The amplitudes Ax and Ay scale the figures only horizontally or vertically. The appearance of the graph depends mainly on the frequency v and the phase φ from. V has the value 1, yields Δφ = φ1 - φ2 the phase shift between the oscillations. If v is a rational number not equal to 1, specifying Δφ is usually done for the minimum phase difference. Furthermore, it is required for which the vibration information is provided.

The portions pictures for frequency ratio 1: n and n: 1 and figures for frequency ratio n1: n2 show Lissajous figures for various frequency ratios and phase difference, the next section Lissajous figures in the oscilloscope and then explains methods for metrological determination of the figures.

Pictures for frequency ratio 1: n and n: 1 ( amplitude ratio 1:1)

The phase difference Δφ always refers in the following figures to the greater frequency. Is the frequency on the horizontal axis higher, is not completely formed Reconciled frequency of appearance of a rotation around the vertical axis and in the opposite case to the horizontal axis.

Pictures for frequency ratio n1: n2 ( amplitude ratio 1:1)

For ratios, which carry neither the numerator nor the denominator is 1, Δφ does not reach the maximum value of 2 · π, the repetition of the curve pattern begins already in first. This effect arises because a number of oscillations of the first signal n1 and n2 of the second oscillation signal are required for the formation of FIG. Accordingly, it is now more zero crossings for the determination of the maximum phase difference into account.

Lissajous figures in the oscilloscope

When working with the oscilloscope to obtain Lissajous figures when one invests in disconnected sweep both the input for the y and for the x-deflection harmonic AC voltage.

The shape of the figures, allows accurate conclusions about the frequency and phase position of the two voltages. With the same frequencies ( For example: v = 1:1) can be read on the elliptical figure, the phase difference. For two nearly equal frequencies (or a frequency ratio that is very close to a simple rational ratios ) shows the screen of the oscilloscope while a closed, but time-varying character. So you can measure with high sensitivity small frequency differences.

Therefore Lissajous figures were, for example, in the workshop of television and tube technicians a common sight. On the other hand, they act in their diversity especially ( but not only) on the technical laymen extremely fascinating, especially in light of the animated form. Therefore also monitors were often decorated with sets with Lissajous figures when an area should look very modern or futuristic, as in science fiction movies and series in cinema and television.

Mechanical generation of Lissajous figures

Lissajous figures can be, for example, with a mechanical, designed as Harmonograph writers produce (see figure).

A frequently used method is simpler drawing by a fortified under an oscillating ground pin in a sand surface over which the mass oscillates. If you want to combine two oscillations with different frequencies, one can for example use a sand-filled plates and these also hang pendulous ( hanging chords at the bigger picture, so that in the middle on the same hook hanging mass movement game has ). It is also possible to draw the track with swinging from the mass leaking ink or ausrinnendem sand, where the vibrations are less braked.

Film model in DIY: It prints a sine curve with one or more periods - landscape - along, rather flat, perhaps a rectangle B: H = pi: 1 inscribed - on a transparent sheet and arches of it as a cylinder that by appropriate overlap of the film - and possibly also the line - an infinite, uniformly rotating sine curve is obtained. Is used only a third or half the height of cut strips from the DIN Landscape, so can the cylinder already with two cylindrical edge near- staples or paper clips or Klebstreifenstücken good and wiederzerlegbar staple together. With skill can alternatively paint in this way, a cylindrical glass. By rotating the cylinder you get when looking through the side (on a white surface ), the various views of the respective Lissajous figure. Good pose can thus be integer frequency multiples such as 2:1, 3:1, etc. By 2 windings of the film strip is also the relation 3:2 can be realized, if not better from the outset a braid is printed from two shifted by 180 ° sine curves and them exactly 3 bellies are used when the first is overlapped with the fourth node.