(picture from http://www.daerospace.com/MechanicalSystems/DwellLinkFig1.png)The system has a relationship similar to that of the crank and shaft mechanism pictured above and a mathematical representation similar to piston equations, describing rotary motion converted into reciprocal.
- L = length of rod (distance between piston join and crank join)
- r = crank radius
- theta = angular velocity
- x = reciprocal displacement
to get the velocity with respect to angular position, we differentiate using chain rule:
to get the velocity with respect to time:
Plotting the relationship between angular position and change in reciprocating velocity, we find that when the source is far away, the relationship represents one that is sinusoidal in nature and as it gets closer, it converges on a tangential relationship.
Graph above is the reciprocating velocity relationship, head size 20cm and source distance 30cm.
Graph above is the reciprocating velocity relationship, head size 20cm and source distance 1 metre.Rotating Binaural Dummy Head
Using the change in velocity from the above graph and putting it into the equation for doppler effect as a variable, the relationship for change in frequency can be found.
In the graph above, the line that is positive from 0 to (pi) and then negative between (pi) and 2(pi) is the left ear of the binaural microphones. The above equation in the graph describes this relationship. The other line on the graph corresponds to the lower equation and is the same as the upper equation, with a half-phase difference.
The change in frequency can be seen on the y-axis, only varying 6 Hz, changing from 503 - 497Hz and having a rest frequency of 500Hz.
Above image shows the effect of the rotating binaural dummy head at 500Hz.
Rotating at 84.5rpm
Rotating at 84.5rpm
I admire the valuable information you offer in your articles. I will bookmark your site!!
ReplyDeleteWhat is localization