How does a tiny generator turn vibrations into electricity?

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I know what it does but what's the mechanism that turns vibration into electricity?
 
The key word you want to search is http://en.wikipedia.org/wiki/Piezoelectricity"
 
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That generator is not piezoelectric. The URL the OP linked to describes how it works very well:

"The sugar cube-sized generator, a smaller version of a design already commercially available, uses cantilever-mounted magnets to induce a current in a copper coil"

Basically, magnets shake around more than the copper coils. Magnetic flux through the coils changes, which induces a current in the coil via Faraday's law of induction.
 
Those readers who are old enough to remember 33 RPM vinyl records may recall the magnetic pickup cartridge:
http://en.wikipedia.org/wiki/Magnetic_cartridge
This pickup came in two configurations; moving magnet and moving coil. Neither had any battery or external power supply. Both work on the Faraday induction law.
Bob S