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Chapter 8: Problem 5

(Plotting Lissajous figures) Using a computer, plot the curve whose parametric equations are \(x(t)=\sin t, y(t)=\sin \omega t\), for the following rational and irrational values of the parameter \(\omega\) : (a) \(\omega=3\) (b) \(\omega=\\}\) (c) \(\omega=\frac{5}{3}\) (d) \(\omega=\sqrt{2}\) (e) \(\omega=\pi\) (f) \(\omega=\frac{1}{2}(1+\sqrt{5})\). The resulting curves are called Lissajous figures. In the old days they were displayed on oscilloscopes by using two ac signals of different frequencies as inputs.

Short Answer

Expert verified
Plot Lissajous figures for given values of \(\omega\), using the parametric equations \(x(t) = \sin t\) and \(y(t) = \sin \omega t\). For example, when \(\omega = 3\), plot the curve with coordinates \(\sin t\) and \(\sin (3t)\). Use a graphing tool to visualize the curves, and observe how their shapes depend on the frequency ratios. Lissajous figures are used in oscilloscopes to display the relationship between two AC signals with different frequencies.

Step by step solution

01

1. Set up the coordinates and the parameter t

Define the parameter t within a certain range, for example \(t \in [0, 2\pi]\). Then, specify the functions \(x(t) = \sin t\) and \(y(t) = \sin \omega t\) for each selected value of \(\omega\).
02

2. Ploting the Lissajous figure for \(\omega = 3\)

Replace \(\omega\) with 3 in the parametric equations to obtain \(x(t) = \sin t\), \(y(t) = \sin (3t)\). Plot the resulting curve using a graphing tool.
03

3. Ploting the Lissajous figure for \(\omega = \frac{5}{3}\)

Replace \(\omega\) with \(\frac{5}{3}\) in the parametric equations to obtain \(x(t) = \sin t\), \(y(t) = \sin (\frac{5}{3}t)\). Plot the resulting curve using a graphing tool.
04

4. Ploting the Lissajous figure for \(\omega = \sqrt{2}\)

Replace \(\omega\) with \(\sqrt{2}\) in the parametric equations to obtain \(x(t) = \sin t\), \(y(t) = \sin (\sqrt{2}t)\). Plot the resulting curve using a graphing tool.
05

5. Ploting the Lissajous figure for \(\omega = \pi\)

Replace \(\omega\) with \(\pi\) in the parametric equations to obtain \(x(t) = \sin t\), \(y(t) = \sin (\pi t)\). Plot the resulting curve using a graphing tool.
06

6. Ploting the Lissajous figure for \(\omega = \frac{1}{2}(1+\sqrt{5})\)

Replace \(\omega\) with \(\frac{1}{2}(1+\sqrt{5})\) in the parametric equations to obtain \(x(t) = \sin t\), \(y(t) = \sin (\frac{1}{2}(1+\sqrt{5})t)\). Plot the resulting curve using a graphing tool.
07

7. Lissajous figures and oscilloscopes

Lissajous figures are used in oscilloscopes to visually display the relationship between two AC signals with different frequencies. By plotting these figures for various values of \(\omega\), we can observe how the shape of the curve depends on the ratio of frequencies.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parametric Equations
Parametric equations are a way of representing mathematical functions where each axis is expressed as a function of one or more parameters, rather than a single variable. In the context of Lissajous Figures, parametric equations express the coordinates of the curve in terms of a parameter, usually denoted as \( t \). The equations for Lissajous figures are \( x(t) = \sin t \) and \( y(t) = \sin(\omega t) \). Here, \( t \) is the parameter that denotes time or a certain angle progression, and \( \omega \) is a variable influencing the curve's frequency.

These parametric equations allow us to create complex and often beautiful curves by changing the value of \( \omega \). These changes affect the trajectory traced out by the function in the \( xy \)-plane, leading to the unique figures recognized as Lissajous figures. Understanding parametric equations helps in visualizing the path taken by these figures over time by tracing out both the x and y components simultaneously. With Lissajous figures, different values of \( \omega \) yield varied patterns, capturing the essence of the relationship between x and y as they vary with \( t \).
Oscilloscopes
An oscilloscope is an electronic test instrument that graphically displays varying signal voltages. Historically, Lissajous figures were used with oscilloscopes to analyze the relationship between two periodic signals, usually for determining their frequency and phase differences.

By inputting two sinusoidal signals of different frequencies into the oscilloscope, each having its own channel, the display would intricately plot their interaction in real-time. The x-component of the signal would drive the horizontal axis, and the y-component would drive the vertical one. The resulting Lissajous figure represented the continuous interaction between the two signals.

This technique provided a practical visual method for measuring frequency ratios and assessing phase differences between signals without needing complicated equipment or calculations beyond the simple plotting on the oscilloscope screen. The shapes helped technicians and scientists easily interpret what was happening with the signal properties, making it a valuable tool in both educational and professional settings.
Frequency Ratio
The frequency ratio in Lissajous figures is a crucial element in determining the resulting pattern of the figure. This ratio is defined by the frequencies of the two sinusoidal inputs. For the parametric equations \( x(t) = \sin t \) and \( y(t) = \sin(\omega t) \), the frequency ratio is \( 1:\omega \).

If \( \omega \) is a rational number (such as \( \omega = 3 \)), the resulting figure will close and repeat after a complete cycle. This closure indicates a stable, repeated pattern like a simple loop or figure-eight shape. However, when the frequency ratio becomes irrational (values such as \( \omega = \sqrt{2} \)), the Lissajous figure does not repeat perfectly, resulting in what appears as a more chaotic or densely filled pattern.

By adjusting this frequency ratio, different Lissajous figures represent the harmonious or discordant interaction of the two signals. In the context of real-world applications, understanding frequency ratios aids in tasks such as tuning musical instruments or studying waveforms in physics and engineering.

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Most popular questions from this chapter

(Interacting bar magnets) Consider the system $$ \begin{aligned} &\dot{\theta}_{1}=K \sin \left(\theta_{1}-\theta_{2}\right)-\sin \theta_{1} \\\ &\dot{\theta}_{2}=K \sin \left(\theta_{2}-\theta_{1}\right)-\sin \theta_{2} \end{aligned} $$ where \(K \geq 0\). For a rough physical interpretation, suppose that two bar magnets are confined to a plane, but are free to rotate about a common pin joint, as shown in Figure \(1 .\) Let \(\theta_{1}, \theta_{2}\) denote the angular orientations of the north poles of the magnets. Then the term \(K \sin \left(\theta_{2}-\theta_{1}\right)\) represents a repulsive force that tries to keep the two north poles \(180^{\circ}\) apart. This repulsion is opposed by the \(\sin \theta\) terms, which model external magnets that pull the north poles of both bar magnets to the east. If the inertia of the magnets is negligible compared to viscous damping, then the equations above are a decent approximation to the true dynamics. a) Find and classify all the fixed points of the system. b) Show that a bifurcation occurs at \(K=\frac{1}{2} .\) What type of bifurcation is it? (Hint: Recall that \(\sin (a-b)=\cos b \sin a-\sin b \cos a .)\) c) Show that the system is a "gradient" system, in the sense that \(\dot{\theta}_{i}=-\partial V / \partial \theta_{1}\) for some potential function \(V\left(\theta_{1}, \theta_{2}\right)\), to be determined. d) Use part (c) to prove that the system has no periodic orbits. e) Sketch the phase portrait for \(0\frac{1}{2}\).

For each of the following systems, a Hopf bifurcation occurs at the origin when \(\mu=0\). Use the analytical criterion of Exercise \(8.2 .12\) to decide if the bifurcation is sub- or supercritical. Confirm your conclusions on the computer. In Example 8.2.I, we argued that the system \(\dot{x}=\mu x-y+x y^{2}\), \(\dot{y}=x+\mu y+y^{3}\) undergoes a subcritical Hopf bifurcation at \(\mu=0\). Use the analytical criterion to confirm that the bifurcation is subcritical.

(Numerical exploration) Fix the parameters \(k=1, b=4, F=2\). a) Using numerical integration, plot the phase portrait for the averaged system with \(a\) increasing from negative to positive values. b) Show that for \(a=2.8\), there are two stable fixed points. c) Go back to the original forced Duffing equation. Numerically integrate it and plot \(x(t)\) as \(a\) increases slowly from \(a=-1\) to \(a=5\), and then decreases slowly back to \(a=-1\). You should see a dramatic hysteresis effect with the limit cycle oscillation suddenly jumping up in amplitude at one value of \(a\), and then back down at another.

Consider the system \(\dot{x}+x=F(t)\), where \(F(t)\) is a smooth, \(T\)-periodic function. Is it true that the system necessarily has a stable \(T\)-periodic solution \(x(t)\) ? If so, prove it; if not, find an \(F\) that provides a counterexample.

(Explaining Lissajous figures) Lissajous figures are one way to visualize the knots and quasiperiodicity discussed in the text. To sce this, consider a pair of uncoupled harmonic oscillators described by the four-dimensional system \(\ddot{x}+x=0, \ddot{y}+\omega^{2} y=0\) a) Show that if \(x=A(t) \sin \theta(t), y=B(t) \sin \phi(t)\), then \(\dot{A}=\dot{B}=0\) (so \(A, B\) are constants) and \(\dot{\theta}=1, \dot{\phi}=\omega\). b) Fxplain why (a) implies that trajectories are typically confined to two- dimensional tori in a four-dimensional phase space. c) How are the Lissajous figures related to the trajectories of this system?
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