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Vibration and Waves

Lissajous Curves Animator

Synthesize two orthogonal sinusoidal oscillations and observe beautiful Lissajous figure formation in real-time Canvas animation. Includes oscilloscope waveform view, phase sweep, and frequency ratio map.

Frequency Preset (fx : fy)
Direct Frequency Input
:
Parameter Adjustment
Phase difference δ
XAmplitude Ax
YAmplitude Ay
Animation speed
Trail length
Results
Frequency ratio
1 : 2
Phase difference
90.0°
X-axis node count
2
Y-axis node count
1
Main
X(t) = Ax · sin(fx · t)
Y(t) = Ay · sin(fy · t + δ)
Sweep
Map
Theory & Key Formulas

A Lissajous figure is a plane curve formed by combining two perpendicular sinusoidal oscillations. In parametric form:

$$x(t) = A_x \sin(f_x\, t), \quad y(t) = A_y \sin(f_y\, t + \delta)$$

Here \(f_x\) and \(f_y\) are the angular frequencies on each axis, \(A_x\) and \(A_y\) are amplitudes, and \(\delta\) is the phase difference.

The curve closes when the frequency ratio is rational:

$$\frac{f_y}{f_x} = \frac{p}{q} \quad (p, q \text{ are coprime positive integers})$$

In that case the curve closes with period \(T = 2\pi q / f_x\). The ratio of contacts with the X and Y boundaries is:

$$\frac{n_x}{n_y} = \frac{f_y}{f_x}$$

For the special case \(f_x = f_y\) (a 1:1 ratio), the Lissajous curve satisfies this ellipse equation:

$$\frac{x^2}{A_x^2} - \frac{2xy\cos\delta}{A_x A_y} + \frac{y^2}{A_y^2} = \sin^2\!\delta$$

When \(\delta = 0\) or \(\pi\), it degenerates into a line. When \(\delta = \pi/2\) and \(A_x = A_y\), it becomes a perfect circle.

Theory Note — Mathematics of Lissajous Curves
Learn Through Dialogue: Lissajous Curves
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Lissajous curves look beautiful, but how are they actually used in engineering?
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The most familiar use is oscilloscope XY mode. Feed two signals into the X and Y inputs and a Lissajous figure appears. Its shape immediately tells you the frequency ratio and phase offset between the signals. Engineers used this routinely in analog measurement work.
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With a 1:1 frequency ratio, changing phase changes the shape: a line at δ = 0°, a circle at 90°, and ellipses in between. Why?
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At δ = 0°, x and y are perfectly synchronized, so they increase and decrease together and form a diagonal line. At δ = 90°, y leads x by a quarter cycle, which gives the circle equation x² + y² = A² when amplitudes match. Intermediate phases produce ellipses; algebra gives \(\frac{x^2}{A^2} - \frac{2xy\cos\delta}{A^2} + \frac{y^2}{A^2} = \sin^2\delta\).
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A 1:2 ratio makes a figure eight. The simulator shows X-axis node count = 2 and Y-axis node count = 1. What does that mean?
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They count how often the curve touches the left/right boundaries and the top/bottom boundaries. In a 1:2 figure, the horizontal boundary is touched twice and the vertical boundary once. The relationship is \(n_x : n_y = f_y : f_x\), the inverse of the frequency ratio, so counting nodes can reveal an unknown ratio.
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The phase sweep tab is interesting. It shows the figure morphing as phase changes. Is it listing continuous changes in δ?
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Yes. It displays 20 evenly spaced phases from 0° to 180°. Because δ and δ + 180° are vertically inverted versions of each other, the 0° to 180° half range captures the full set of shapes. This also explains why a slowly shifting reference phase on an oscilloscope can make the figure appear to rotate.
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How is this used in rotating machinery diagnosis?
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Two displacement sensors are mounted near a bearing at 90° to each other, and their outputs are plotted as X and Y. A healthy rotating shaft with some unbalance often traces a clean ellipse. Misalignment can distort it into a figure eight, and cracks can deform it further. This trajectory is called an orbit plot and is still used in predictive maintenance.
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For a complicated ratio such as 3:5, the trajectory looks messy. Is it still theoretically closed?
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Yes. For 3:5 in lowest terms, \(q = 5\), so the curve returns to its starting point after \(2\pi \times 5 = 10\pi\). It looks complicated because it traces many strokes before closing. With an irrational ratio such as \(\sqrt{2}:1\), it never closes and eventually fills the rectangular region densely.
Frequently Asked Questions
It was discovered and systematized around 1857 by French physicist Jules Antoine Lissajous (1822–1880). He built a device that placed two tuning forks at right angles and optically combined their vibrations to observe the resulting figures. Today, similar observations can be easily made using the XY mode of an oscilloscope.
When the frequencies are equal (\(f_x = f_y\)), the phase difference can be found from the ellipse shape using \(\sin\delta = b/a\). Here, \(a\) is the half-width of the maximum X-axis deflection (half the major axis of the ellipse), and \(b\) is the Y value at X = 0 (where the ellipse crosses the Y-axis). This method is called the "ellipse method" and is still used today for probe phase correction checks. In this simulator, you can select a 1:1 frequency ratio and observe the ellipse shape change.
1.5:1 = 3:2 is a rational ratio, so it forms a closed curve. In general, if \(f_y/f_x\) is rational (an integer ratio), the curve is closed. For irrational ratios (like \(\sqrt{2}\), \(\pi\)), the trajectory does not close and, over sufficient time, becomes an ergodic open curve that densely fills a rectangular area. In real electrical circuits, slight frequency differences between two oscillators can cause the pattern to appear as a "slowly rotating Lissajous figure."
Orbit plots are essentially a type of Lissajous figure. If the 1X component (same frequency as rotational speed) dominates, the orbit is elliptical. Misalignment or unbalance can distort the ellipse, and the presence of a 2X component may create a figure-8 shape. Unlike ideal Lissajous curves, real orbits contain multiple frequency components, so interpreting them requires a comprehensive diagnosis combined with FFT analysis.
In transient response analysis of multi-degree-of-freedom vibration systems, displaying the displacement time histories of two nodes in an XY plot yields Lissajous-like trajectories. For example, in a 2-DOF system where the natural frequency ratio is close to 1:2, the XY plot of displacement responses forms a figure-8-like trajectory. This is also used in fluid-structure interaction (FSI) analysis for coupled vibrations in two directions, and in seismic response analysis to visualize horizontal bi-directional responses.
There is no physical meaning; it is a display technique to visualize the passage of time. By making the hue proportional to time \(t\), you can distinguish "older" parts of the trajectory from "newer" ones, intuitively showing the direction of motion. For closed curves, after sufficient time, the start and end points merge with the same color, confirming the curve is closed.

What is Lissajous Curves?

Lissajous Curves is a fundamental topic in engineering and applied physics. This interactive simulator lets you explore the key behaviors and relationships by directly manipulating parameters and observing real-time results.

By combining numerical computation with visual feedback, the simulator bridges the gap between abstract theory and physical intuition — making it an effective learning tool for students and a rapid-verification tool for practicing engineers.

Physical Model & Key Equations

The simulator is based on the governing equations behind Lissajous Curves Animator. Understanding these equations is key to interpreting the results correctly.

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Each parameter in the equations corresponds to a slider in the control panel. Moving a slider changes the equation's solution in real time, helping you build a direct connection between mathematical expressions and physical behavior.

Real-World Applications

Engineering Design: The concepts behind Lissajous Curves Animator are applied across mechanical, structural, electrical, and fluid engineering disciplines. This tool provides a quick way to estimate design parameters and sensitivity before committing to full CAE analysis.

Education & Research: Widely used in engineering curricula to connect theory with numerical computation. Also serves as a first-pass validation tool in research settings.

CAE Workflow Integration: Before running finite element (FEM) or computational fluid dynamics (CFD) simulations, engineers use simplified models like this to establish physical scale, identify dominant parameters, and define realistic boundary conditions.

Common Misconceptions and Points of Caution

Model assumptions: The mathematical model used here relies on simplifying assumptions such as linearity, homogeneity, and isotropy. Always verify that your real system satisfies these assumptions before applying results directly to design decisions.

Units and scale: Many calculation errors arise from unit conversion mistakes or order-of-magnitude errors. Pay close attention to the units shown next to each parameter input.

Validating results: Always sanity-check simulator output against physical intuition or hand calculations. If a result seems unexpected, review your input parameters or verify with an independent method.

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How to Use

  1. Set frequency ratio fxInput and fyInput (e.g., fx=3 Hz, fy=2 Hz) to define the harmonic relationship
  2. Adjust phase difference using phaseSlider (0 to 360 degrees) to rotate and morph the curve pattern
  3. Control amplitude scaling with axSlider and aySlider to stretch the figure horizontally and vertically
  4. Observe real-time Canvas animation updating as parameters change

Worked Example

Set fx=5 Hz, fy=4 Hz with phase=90 degrees and equal amplitudes (ax=100 px, ay=100 px). The 5:4 frequency ratio produces a closed curve with 20 petals (LCM calculation). Increasing phase to 180 degrees inverts the symmetric pattern. For aerospace vibration analysis, a 2:1 frequency ratio with ax=150 mm, ay=100 mm models coupled lateral-vertical motion on a 3-meter cantilever beam experiencing 15 Hz excitation.

Practical Notes