Honda Lawnmower Flywheel Key Sheared?
Have a look underneath.  There are several failure modes which could cause strong shaking:


A cutting blade is broken, missing, or bent
A rock, scrap metal, lead weight, or other miscellany is attached to the blade
The crankshaft is bent
The crankshaft bearing or housing is cracked


A close look should be able to rule out and identify the problem, the last being the most difficult.


  How do I tell if the crankshaft is bent?


Using your hand, turn the blade at least one half rotation and watch to see if its path is level.  Be sure the engine cannot start by disconnecting the spark plug and having the throttle at its full stop position.