Heavy vibration is not normal, it seems to be very heavy vibration that you are talking about. If it is noise you are talking about, then that could be a bunch of simple things. If it is really a shaking/vibration then you do need to check that out. If the engine runs smooth, then it may be motor mounts or a drive line issue.
A little vibration is normal. Maybe you driving a new one will give you an idea of "normal"..good luck