What are Lissajous figures ? i know they are something related to simple harmonic motion, but wat are they ?

Question

plz explain clearly.....
thanks a lot for ur help........
i promise to select urs as the best answer, if im convinced.

Answer 1

If you parametrically graph two simple harmonic motions (that is, sinusoidal motions) in the plane, one on the x axis and the other on the y axis, the resulting figure is called a Lissajous figure. They can be quite complicated, if the ratio of frequencies is not simple fraction.

If the ratio of frequencies is rational, a Lissajous figure is actually closed; if the ratio is irrational, the figure never closes on itself, but becomes a space-filling curve.

If you have access to an oscilloscope, you can simultaneously plot two different frequencies on the screen, one horizontal and the other vertical, and produce a real live Lissajous figure that way.

Answer 2

A Lissajous curve is really described as a family of curves where the "x" coordinate and the "y" coordinate are calculated separately. This is often called a set of parametric equations. So, grab some graph paper, label the x axis and the y axis giving yourself plenty of room to represent a set of numbers between -3 and 3 on both the axis (say, 3 inches in each direction positive and negative on each axis). You will also need a calculator.

The family of curves is defined like this:

x = Asin(at + d)
y = Bsin(bt)

where you get to pick A, B, a, b, d. The parameter "t" runs from 0 to 360 degrees (or 0 to 2pi radians, if you prefer).

So, let's assign some numbers and see what happens. Let's define our choices this way (and feel free to change them if you like):

A = 3
a = 5
d = 0
B = 2
b = 3

Now calculate for x and y from t in, say, 5 degree increments:

x = 3sin(5t)
y = 2sin(3t)

if t = 0 then x = 0 and y = 0. So plot the point 0,0 (the origin)
if t = 5 then x = 1.27 and y = .52. So plot the point (1.27,.52)

Now, when you have plotted all the points, connect them in a smooth curve.

This is one of those cases where is you know how to write a simple computer program or you have a programmable calculator you can get the plots "automagically", and changing the parameters lets you "see" what is happening very fast.

HTH

Charles

Answer 3

Here's a good explanation ...

http://en.wikipedia.org/wiki/Lissajous_curve