Draw Lissajous figures by composing two sinusoidal oscillations. Adjust frequency ratio, amplitude, and phase difference in real time while simultaneously viewing side projections of each component.
Parameters
Preset ratios
XFrequency f_x
Hz
YFrequency f_y
Hz
Amplitude A_x
Amplitude A_y
Phase difference δ
π
Automatically sweep δ from 0 to 2π
Drawing points
Playback Controls
FigureComparison
Results
1:2
Frequency ratio f_x:f_y
0.50π
Phase difference δ
Closed curve
Curve type
1.00 s
period T
Visualization
Drag horizontally on the canvas to adjust phase difference δ
CAE Applications
Structural resonance frequency identification (phase difference δ=90° between excitation/response indicates resonance) / Shaft runout and imbalance detection in rotating machinery / Frequency ratio and phase measurement via oscilloscope XY mode / 2-DOF system vibration mode visualization.
Theory & Key Formulas
Combine two sinusoidal oscillations in orthogonal directions:
What exactly is a Lissajous figure? I see the simulator draws a looping shape from two waves.
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Basically, it's the path traced by a point undergoing two independent perpendicular vibrations. In this simulator, you control the sine wave for the X-direction and the Y-direction. The resulting pattern you see is the combination of both motions over time. Try setting both frequencies (f_x and f_y) to 1 – you'll get a simple ellipse or line.
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Wait, really? So the shape changes based on the frequency numbers? What does a 2:1 ratio look like?
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Exactly! The frequency ratio is the key. A 2:1 ratio, like f_x=2 and f_y=1, often creates a figure-8 or a more complex "bow-tie" shape. The pattern is closed and repeats because the vibrations eventually resynchronize. Go ahead and use the sliders to set f_x to 2 and f_y to 1, then slowly adjust the "Phase difference δ" slider. You'll see the shape rotate and morph beautifully.
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That's cool! But when would this ever be useful outside of making pretty patterns?
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Great question. It's a fundamental tool for measurement. For instance, in an electronics lab, you feed two signals into an oscilloscope in "XY mode" to compare their frequencies and phase. If you know one signal's frequency, the Lissajous pattern tells you the other. In our CAE world, it helps identify structural resonance—when the phase difference δ hits 90°, it often means you've found a natural frequency. Try setting f_x and f_y to the same value and sweep δ from 0 to 180° to see the transition from a line to an ellipse to a line again.
Physical Model & Key Equations
The motion of the point is defined by two independent harmonic oscillations, one along the x-axis and one along the y-axis. The position at any time \( t \) is given by these parametric equations.
Where:
\( A_x, A_y \): Amplitudes (controlled by sliders) – they stretch the figure along each axis.
\( \omega_x = 2\pi f_x, \omega_y = 2\pi f_y \): Angular frequencies, set by the frequency sliders.
\( \delta \): Phase difference (the "Phase difference δ" slider) – it shifts one wave relative to the other, changing the figure's rotation and shape.
\( t \): Time. The "Drawing points" (Number of Points) controls how many time samples are used to draw the curve smoothly.
The resulting figure is closed and periodic only if the frequency ratio \( f_x / f_y \) is a rational number (a ratio of integers). The specific integers determine the pattern's complexity.
$$\frac{f_x}{f_y}= \frac{m}{n}\quad (m, n \text{ are integers})$$
The pattern will have \( m \) lobes in the vertical direction and \( n \) lobes in the horizontal direction. For example, a 3:2 ratio (\(f_x=3, f_y=2\)) typically creates a pattern with 3 vertical loops. Try it in the simulator and count them!
Frequently Asked Questions
For the figure to close (become periodic), the frequency ratio between the X and Y directions must be a rational number (integer ratio), such as 2:3 or 3:4. If the ratio is irrational (e.g., 1:√2), the trajectory will never close and will produce a pattern that fills the screen. Try adjusting the frequency ratio slider to a rational number.
The phase difference δ represents the offset in the start timing of the X-direction oscillation. At δ=0°, the figure is a straight line; at 90°, it becomes a perfect circle (if the amplitudes are equal); for other values, it becomes an ellipse or a tilted figure. Moving the slider while checking the waveform offset of each component in the side projection will deepen your understanding.
In addition to the Lissajous figure on the main screen, the side projection displays the simple harmonic motion (sine wave) of the X and Y directions along the time axis. This allows you to intuitively compare how the amplitude, frequency, and phase difference of each component affect the composite figure. A particular advantage is the ability to visualize the effect of phase difference as a waveform offset.
The amplitudes Ax and Ay determine the horizontal and vertical spread of the figure. A larger Ax results in a horizontally elongated ellipse, while a larger Ay results in a vertically elongated ellipse. When the amplitudes are equal, the figure fits within a square range, and circles or straight lines are more likely to appear depending on the frequency ratio and phase difference. Fine adjustments can also be made via numerical input.
Real-World Applications
Oscilloscope Frequency & Phase Measurement: This is the classic lab use. By connecting two electronic signals to an oscilloscope in X-Y mode, engineers can quickly determine an unknown frequency by comparing it to a known reference. The resulting Lissajous figure reveals both the frequency ratio and the phase difference between the signals.
Structural Dynamics & Resonance Identification: In CAE and experimental vibration testing, the phase difference \( \delta \) between an applied force (input) and the structure's response (output) is critical. At a system's natural resonance frequency, this phase shift approaches 90°. Monitoring the Lissajous-like relationship between input and output helps engineers identify and avoid dangerous resonant conditions in buildings, car parts, or aircraft wings.
Rotating Machinery Diagnostics: Imbalance or misalignment in shafts, turbines, and motors creates vibrations. By measuring vibrations in two perpendicular directions on a bearing housing, analysts can plot the orbit of the shaft centerline—which is a Lissajous figure. The shape of this orbit helps diagnose the specific type of fault, such as unbalance, bent shafts, or oil whirl.
Audio & Signal Processing: Lissajous figures are used in some audio software and hardware visualizers to represent stereo signals or the relationship between different modulation effects. They provide an intuitive, geometric representation of the correlation between two channels.
Common Misunderstandings and Points to Note
First, the amplitudes A_x and A_y are often misunderstood as only determining the "size" of the figure, but in reality, the "aspect ratio" significantly affects shape recognition. For example, if you try to draw a circle with a frequency ratio of 1:2, even with a phase difference of 90°, you will not get a circle if the amplitudes are equal. To obtain a correct circle, the amplitude ratio must also be adjusted according to the frequency ratio. Specifically, when f_y=2*f_x, try setting A_y to about half of A_x. You will gain a deeper understanding by verifying this with the simulator.
Next, if you feel that "the figure doesn't change even when you vary the phase difference δ". This tends to happen when the frequency ratio is not 1:1. The effect of phase difference is most pronounced when the frequencies of the two oscillations are equal or very close. Even if you change δ from 0° to 180° with settings f_x=1, f_y=2, the figure will only rotate or translate symmetrically and will not become a straight line or a circle. For learning about phase difference, start with the setting f_x=f_y=1.
A practical pitfall is that the presence of "noise" is often forgotten. This simulator uses ideal sine waves, but real-world measurement signals always contain noise. Therefore, the Lissajous figure you see on an oscilloscope is blurry and not a sharp line like in textbooks. Judging whether a figure is closed requires experience; note that even with an irrational frequency ratio, the figure may appear closed in the short term.