Hyundai Repair: Mis-firing engine on XG350, sludge problem, purge solenoid

Question
2004 XG350

Car vibrates badly above 55 to 70 MPH.  Have to "feater" the accelerator to keep speed above 60. Vibratiion can get so bad it seems like the frot wheels are going to come off.  Have had two dealers diagnose it.  One changed out the purge solenoid and the other could not find anything. Told me there was an engine sludge problem, even though I have changed the oil on regular basis. Suggested an engine cleaner from GumOut, but it has not helped. Vibration is now happening at lower speeds (50-55). When straing out on trip, the first 100 miles or so are trouble free. Then, the proble re-appers and gets progressively worse.

Answer
A dealer replaced a purge solenoid for a bad vibration at highway speeds??  As you might guess, I'm not very surprised it isn't fixed.  In regard to the second dealer, do you mean they were unable to duplicate the condition?  You should also expect that addressing the sludge issue will have no effect.  Few dealers are willing to test drive a vehicle 100 miles to duplicate the condition.  If you really want this addressed, you'll probably need to drive the car long enough to get the condition to happen and then have them ride with you so they can experience it.  If it's a misfire, the first things to typically check are the secondary ignition components (plugs, wires, coil).  And these should still be covered by your warranty if you're under 60k.  

Based on your description of the severity of the vibration, I'm not sold on the misfire idea.  I suspect that the vibration may be coming from the transmission or the brakes (perhaps staying applied).