just a guess but try ckecking the homokinetic joints. That's what
happened to me a few years ago. Do you feel the shaking in the
steering wheel?
or hopefully maybe the wheels need balancing.
try checking the pressure in the tyres, and see how it goes. I
wouldn't reccomend high speeds with this shaking anyhow.

A couple-few of things could be happening, some of which may be
instantly curable, some of which could be instantly nasty. In no
particular order, they are:

Tread separation: check your tires for lumps, bubbles, or strange
markings on the visible surface. Early tread separation may not be
discernable at low speed. Typically, this happens on one tire, and is
most overt if that is a front tire. It can be caused by defective
tires, hitting too many potholes, jumping too many curbs, and so
forth. If the tire blows, especially at-speed, results could be
nasty.

Bad ball-joint, steering damper or other suspension bushing. This
could kill you if a joint fails at any speed as you will instantly
lose control of the car (the affected wheel pretty much falls off, or
steering may jamb).

You lost a wheel-weight or two. Happens quite often if one is driving
in bad weather, or runs against a curb, and so forth. In the process
of checking your suspension and the tires, consider having the wheels
rebalanced. Balance should be done at rotation in any case.

You picked up a heavy nail or screw in your tire, and it is holding
the seal. But at speed, it will cause vibration.... and eventually
come out with subsequent flattening of the tire... also at speed.

You picked up a chunk of ice or crud frozen into your rim. You would
be greatly surprise how often this happens. Check.

One/both of your CV joints is on the verge of seizing. At slow speeds,
you may not notice. This may not cause the tire to flatten or the
wheel to fall off, but it may cause loss of power.

Your shocks/struts are shot. In this case, the slightest tire
imbalance can set up vibrations on that wheel as the shock/strut is no
longer damping anything. This can cause loss of control in a critical
situation. If there are more than say.... 80,000 miles on this beast,
suspect them.

GET IT CHECKED ASAP.

Only two scenario aboves are not potentially instantly deadly. And
even the least-dangerous of the rest is VERY expensive if failure
actually occurs.

Peter Wieck
Wyncote, PA

It makes perfect sense that you feel the vibration strongly at a
certain speed and then it even decreases as you go faster throught
that point.  I recommend that you stay well below that point though!
It sounds as though you have a rotating imbalance -- like a missing
wheel weight or road goop stuck inside the wheel rim.  Your car is a
dynamic system.  If you were to take spring with a weight on the end
of it, and you began to push the weight at a certain number of
repetitions per second, you can see that if you time it right, the
oscillation get larger. If you push too fast or to slowly, you will
decrease the oscillations of the spring.  The particular frequency is
called the natural harmonic frequency.  When an imbalanced car tire
rotates it exerts a force on the suspension system spring.  As your
speed changes, the frequency of the rotating imbalance changes.  When
the oscillating tire force matches the natural harmonic frequency of
your suspension system, you feel it most.

Sorry I wrote so much.  I just wanted to strut my stuff.  It took a
lot of work to get a degree in mechanical engineering... But the
answer is probably to just get your tires checked!

Good luck

There are other good suggestions, but when you say shaking I presume you
checked that it's not something in the engine that is shaking?

For instance is this an rpm-related phenomenon, or is it in fact related to
your speed on the road? If you rev the engine while in neutral without
moving the car, do you observe the shaking?