Page History

{table:cellspacing=0|cellpadding=8|border=1|frame=void|rules=cols}
{tr:valign=top}
{td:width=310|bgcolor=#F2F2F2}
{live-template:Left Column}
{td}
{td}

|!Lissajous Figure.PNG!|
|Lissajous Figure
from Wikimedia Commons: Image by Peter D Reid|



{composition-setup}{composition-setup}

{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:*  

{cloak:id=app}

{toggle-cloak:id=diag} {color:red} *Diagrammatic Representation* {color}

{cloak:id=diag}

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.PNG!
{cloak:diag}

{toggle-cloak:id=math} {color:red} *Mathematical Representation* {color}

{cloak:id=math}

The force is supplied by a belt aroundWe ignore the smallerdistribution wheel of radius *r*tensions (in athe 19thupper century factorycables, itand wouldsimply probably be a circular leather belt attached to view the waterpendulum wheels).as Thisa meanssimple that the direction the force is applied pendulum along is always tangential to either the circumferenceplane of the wheel,drawing andor henceperpendicular *Torqueto = r X F = rF*

{latex}\begin{large}\[ \vec{\tau} = \vec{r} X \vec{F} = rF = I_{\rm total} \alpha \]\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:it. In the plane perpendicular to the drawing (where the mass oscillates toward and away from the reader) the pendulum length is *L{~}2{~}* and the angular frequency of oscillation is given by the formmula for the Simple Pendulum.


{latex}\begin{large}\[ I = \fracomega_{1}{2}m r^2= \] sqrt{\endfrac{large}{latex}

So the Moment of Inertia of the complete flywheel is:

{latex}\begin{large}\[ Ig}{L_{\rm total} = \frac{1}{2}(m r^2 + M R^2 ) 2}}} \]\end{large}{latex}

TheAlong expression for the angularplane velocitylying andin 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}\[ \omega_{\rm f} = \omega_{\rm i} + \alpha (t_{\rm f} - t_{\rm i}) \] \end{large}{latex}

and

page, where the mass moves left and right, the pendulum length is the shorter *L{~}1{~}* and the angular frequency is

{latex}\begin{large}\[ \thetaomega_{\rm f1} = \theta_sqrt{\rm i} + \omega_{\rm i} ( t_{\rm f} - tfrac{g}{L_{\rm i} ) + \frac{1}{2} \alpha ( t_{\rm f} - t_{\rm i} )^2  \]\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 toratio of frequencies is thus:

{latex}\begin{large}\[ \omega_frac{\rm f} = \alpha tomega_{\rm f}\]\end{large}{latex}
and
{latex}\begin{large}\[ \theta_{\rm f} 2}}{\omega_{1}}= \frac{1}{2} \alpha {t_{\rm f}}^2 \]\end{large}{latex}
where
{latex}\begin{large}\[ \alpha = \frac{rF}{I_{\rm total}} = \frac{2rF}{mr^2 + MR^2 }sqrt{\frac{g}{L_{2}}}}{sqrt{\frac{g}{L_{1}}}} \]\end{large}{latex}


{cloak:math}
{cloak:app}



{td}
{tr}
{table}
{live-template:RELATE license}