1991 Maxima -- Car shakes when driving intermittently

I have a 1997 Maxima and I noticed that when I drive, intermittently,
the car starts to shake for a few minutes and then stops. The biggest
shaking occurrs around 50 - 60 mph but the car does shake at lower
speeds. Also, when the car is shaking I can accelerate with no
problem, even though the gas pedal is shaking also, and that if I just
have to tap on the brakes for the car to slow down whereas during
normal driving I would have to press on my brake pedal for the car to
slow down. Its not the alignment for when the car is shaking I let go
of the steering wheel and the car goes straight.

Has anybody experienced a similar issue or knows what the problem
might be.