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Lissajous Figure from Wikimedia Commons: Image by Peter D Reid
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Image generated by a Pendulum with two natural Frequencies.
The figures commonly called Lissajous Figures were investigated by Jules Antoine Lissajous in the 1850s, but had earlier been generated and studied by Nathaniel Bowditch of Salem, Massachusetts in 1815, who built a special pendulum to generate them. Later, a Scottish Professor named Hugh Blackborn would create a similar device, which would develop into the "Harmonograph". The figures it generated would become a 19th century craze.
The basic Lissajous figure is made up of two sinusoidal oscillations along directions at right angles to each other, and when the periods of the oscillations along both directions are in whole-number ratio to each other (such as 1:2, or 2:3. The above figure has a ratio of 3:4). The simplest way to achieve this is by using a pendulum whose strings form a "Y" shape. The Lissajous figures are the paths taken by the pendulum bob during its swings, if viewed from directly above or below.
Solution
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System:
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Each of the two major directions of oscillation can be independently treated as a case of .
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Interactions:
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Each direction is an independent case of with Gravity and the Tension in the String acting as the .
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Model:
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Approach:
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Diagrammatic Representation
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First, consider the Y support for the pendulum:
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The pendulum effectively has length L2 when swinging in the horizontal plane in and out of the page, but length L1 along the horizontal direction in the plane of the page.
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Mathematical Representation
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We ignore the distribution of tensions in the upper cables, and simply view the pendulum as a simple pendulum along either the plane of the drawing or perpendicular to it. In the plane perpendicular to the drawing (where the mass oscillates toward and away from the reader) the pendulum length is L2 and the angular frequency of oscillation is given by the formula for the Simple Pendulum (see Simple Harmonic Motion.
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|!Lissajous Figure.PNG!|
|Lissajous Figure
from Wikimedia Commons: Image by Peter D Reid|
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{excerpt:hidden=true}Image generated by a Pendulum with two natural Frequencies.{excerpt}
The figures commonly called *Lissajous Figures* were investif=gated by Jules Antoine Lissajous in the 1850s, but had earlier been generated and studied by Nathaniel Bowditch of Salem, Massachusetts in 1815, who built a special pendulum to generate them. Later, a Scottish Professor named Hugh Blackborn would create a similar device, which would devlop into the "Harmonograph". The figures it generated would become a 19th century craze.
The basic Lissajous figure is made up of two sinusoidal oscillations along directions at right angles to each other, and when the periods of the osillations along both directions are in whole-number ratio to each other (such as 1:2, or 2:3. The abiove figure has a ratio of 3:4). The simplest way to achieve this is by using a pendulum whose support has a "Y" shape.
h4. Solution
{toggle-cloak:id=sys} *System:* {cloak:id=sys}Each of the two major directions of oscillation can be independently treated as [simple oscillator].{cloak}
{toggle-cloak:id=int} *Interactions:* {cloak:id=int}Each direction is an independent case of Simple Harmonic Motion with Gravity and the Tension in the String acting as the Restoring Force.{cloak}
{toggle-cloak:id=mod} *Model:* {cloak:id=mod} Simple Harmonic Oscillator.{cloak}
{toggle-cloak:id=app} *Approach:*
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{toggle-cloak:id=diag} {color:red} *Diagrammatic Representation* {color}
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First, consider the *Y* support for the pendulum:
!Lissajous Figure 1.PNG!
The pendulum effectively has length *L{~}2{~}* when swinging in the horizontal plane in and out of the page, but length *L{~}1{~}* along the horizontal direction _in_ the plane of the page.
!Lissajous Pendulum 2!
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{toggle-cloak:id=math} {color:red} *Mathematical Representation* {color}
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The force is supplied by a belt around the smaller wheel of radius *r* (in a 19th century factory, it would probably be a circular leather belt attached to the water wheels). This means that the direction the force is applied along is always tangential to the circumference of the wheel, and hence *Torque = r X F = rF*
{latex}\begin{large}\[ \vecomega_{\tau2} = \vecsqrt{r} X \vecfrac{F} = rF = Ig}{L_{\rm total} \alpha2}}} \]\end{large}{latex}
The Moment of Inertia of combined bodies about the same axis is simply the sum of the individual Moments of Inertia:
{latex}\begin{large}\[ I_{\rm total} = I_{\rm small} + I_{\rm large} \]\end{large}{latex}
The Moment of Inertia of a solid disc of radius *r* and mass *m* about an axis through the center and perpendicular to the plane of the disc is given by:
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Along the plane lying in the page, where the mass moves left and right, the pendulum length is the shorter L1 and the angular frequency is
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\begin{large}\[ I = \frac\omega_{1}{2}mr^2= \] sqrt{\endfrac{large}{latex}
So the Moment of Inertia of the complete flywheel is:
{latex}\begin{large}\[ Ig}{L_{\rm total} = \frac{1}{2}(m r^2 + M R^2 ) 1}}} \]\end{large}
the ratio of frequencies is thus:
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The expression for the angular velocity and the angle as a function of time (for constant angular acceleration) is given in the *Laws of Change* section on the [Rotational Motion] page:
{latex}\begin{large}\[ \omega_frac{\rm f} = \omega_{\rm i} + \alpha (t_{\rm f} - t_{\rm i}) \] \end{large}{latex}
and
{latex}\begin{large}\[ \theta_{\rm f} = \theta_{\rm i} + \omega_{\rm i} ( t_{\rm f} - t_{\rm i} ) + \frac{1}{2} \alpha ( t_{\rm f} - t_{\rm i} )^2 \]\end{large}{latex}
We assume that at the start, *t{~}i{~} = 0* , we have both position and angular velocity equal to zero. The above expressions then simplify to:
{latex}\begin{large}\[ \omega_{\rm f} = \alpha t_{\rm f}\]\end{large}{latex}
and
{latex}\begin{large}\[ \theta_{\rm f} = \frac{1}{2} \alpha {t_{\rm f}}^2 \]\end{large}{latex}
where
{latex}\begin{large}\[ \alpha = \frac{rF}{I_{\rm total}} = \frac{2rF}{mr^2 + MR^2 }\]\end{large}{latex}
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2}}{\omega_{1}}= \frac{\sqrt{\frac{g}{L_{2}}}}{\sqrt{\frac{g}{L_{1}}}} = \sqrt{\frac{L_{1}}{L_{2}}} \]\end{large}
in order to have a ratio of 1:2, one thus needs pendulum lengths of ratio 1:4. In order to get a ratio of 3:4 (as in the figure at the top of the page), the lengths must be in the ration 9:16.