Coupe GT--Vibration Damper? Motor Mount?

[Jonathan Monetti <jmonetti@us.net>]

I searched the archives, but could not find any info about Vibration
Dampers and problem diagnosis thereof.  Car is 86 Coupe GT, 2.2 83.5k on
odometer, motor otherwise in pretty nice shape. 

When I shut my motor down, I get a heavy shudder--not run on or dieseling,
either, but a shudder like I left a pair of sneakers in the spin cycle
and turned it off--definitely the feeling of a rotating mass out
of balance.  It only lasts about 1.5 seconds.  
           
Stupid question, do vibration dampers go bad?  If so, what would be
the symptoms? Is there much to this component?  The GT is my first 5
cylinder car.  Hell, it's my first Audi.  

Damper, motor mount, poltergiests?   Any input helpful, mechanic still
in the hospital, days getting longer, sunroof aching to leave car,
windows itching to be down, foot getting heavier, Jonathan wanting to
drive nuisance free miles.  

Jonathan Monetti
86 Coupe GT, 83k