1998 Jeep Cherokee vibration on 60 miles speed

My guess is that it does NOT shake or vibrate much before 60 mph. I'll also guess that at some point after 60 mph the vibration decreases a bit. Reason: Assuming that your tires are properly inflated and balanced- It could be your brakes.

Sometimes the rotors go out of balance either during hard use or because they are rather cheap to begin with and weren't balanced all that well at the factory. If you need a brake job anyway, get it done at a reputable repair shop and insist on better than average parts.