Research: Lissajous Figures
Wittelsbachergymnasium München, Germany 2020 – 2021
Seminar project with the objective of visualizing and analyzing Lissajous figures—complex curves formed by the superposition of two perpendicular harmonic oscillations—through both digital simulation and physical experimentation. Designed and assembled construction that creates Lissajous figures and analyzed the Curves using an Oscilloscope.
To view the Physics Seminar Paper on Lissajous figures, Click here.
Methodology
1 / Digital Simulation with GeoGebra
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Created interactive simulations visualizing how changes in frequency ratio, amplitude, and phase shift affect curve shape.
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Demonstrated how rational vs. irrational frequency ratios impact whether curves close or fill space.
2 / Analog Visualization with Oscilloscope
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Used two signal generators to produce orthogonal sine waves.
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Created real-time Lissajous figures by applying signals to the oscilloscope’s X and Y channels.
3 / Mechanical Fabrication
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Designed and built a wooden frame for a dual-string pendulum. The pendulum dispenses sand or paint to trace Lissajous curves as it oscillates.
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Calculated ideal pendulum lengths using physics-based derivations, and adjusted string lengths to generate target frequency ratios (e.g., 3:4, 2:3, 4:5).
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Verified results through experimental data and mathematical analysis, analyzing frequency ratios through time measurements and geometric curve analysis.
The Double-Pendulum
In practice, Lissajous figures can be visualized using harmonographs—mechanical setups that combine two orthogonal harmonic oscillations into a single curve. One common implementation uses a double-pendulum string setup, where the pendulum can swing simultaneously in two perpendicular planes.
This setup consists of a main string suspended at both ends, with a second string tied at the midpoint to hold the pendulum mass. As a result, the effective string lengths in the x and y directions differ, which directly affects the frequency ratio of the resulting oscillations.
Two exceptions allow for isolated motion in only one plane: (1) when the full pendulum is displaced perfectly perpendicular to the suspension line, or (2) when the upper string is held fixed and only the lower string is allowed to oscillate. These cases enable the measurement of individual oscillation periods in the x- and y-directions. From these, the frequency ratio between the two oscillations can be determined.
See the report for the derivation of the equation to determine the pendulum lengths, the total length of the pendulum 𝑙1, and the length of the second string tied at the midpoint to hold the pendulum mass 𝑙2, to create the figures for frequency ratios 3/4, 4/5 and 2/3.
Assembly
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Two hooks mounted on the central beam
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Suspension strings joined at the center and tied to a lightweight carabiner
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The pendulum body made of a cut plastic bottle
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A hole drilled in the center of the bottle cap
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Three equally spaced holes added 1 cm below the cut edge of the bottle, through which three strings were threaded
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The total length of the pendulum, denoted as 𝑙1, is 121 cm.
Results
Evaluation
The observed Lissajous figures were analyzed using two methods to verify their frequency ratios. First, the ratios were calculated from the measured oscillation periods in the x- and y-directions, as seen below.
Figure 1:
T2/T1 = 0,75 ± 0,04 = 3/4
Figure 2:
T2/T1 = 0,65 ± 0,03 = 2/3
Figure 3:
T2/T1 = 0,79 ± 0,03 = 4/5
Second, the ratios were inferred visually from the shape of the figures. This can be verified by observing how many oscillations the pendulum completes in the x- and y-directions before returning to the starting point P(A). In both cases, the measured values closely matched the theoretically predicted frequency ratios, confirming the results.