DETERMINATION OF PARAMETERS FOR CART AND MOTOR: SQUARE WAVE INPUT


Square Wave Input:     For omegaltb

Due to static friction inherant in the cart and motor system it is difficult to obtain an accurate Bode plot when pure sinusoidal inputs are used. When the voltage to the motor is near zero the system tends to stick and not move at all. To overcome this problem we will use a square wave input instead of a sine wave input. We now examine the response we expect to get using a square wave input.
A square wave of frequency omega can be expressed as the sum of sine waves as follows:

           p17_1

Since our transfer function for the cart and motor is linear, the output due to a square wave is the sum of the outputs due to each of the sinusoudal terms which make up the square wave.

From our earlier analysis we know the magnitude of the output oscillations due to an input of asinwt is:

           p17_2

For  omegaltb this becomes

           p17_3

For a square wave input expressed as the sum of a series of sinusoidal inputs the magnitude of the output becomes:

           p17_4a arrow p17_4c
 
arrow p17_5c
 
Note:  p17_5d  (Proof)
 
arrow p17_6c

We thus expect the output due to a square wave input to be about 1.233 times the output due to a sine wave input when ( omegaltb)

Expressed in decibels this becomes:

           DB   =   p17_7c
 
  =   p17_8c
 
  =   p17_9c


The output due to a square wave input is about 1.82 DB greater than it would be for a sine wave input when ( omegaltb)
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