2007 Chevy Trailblazer Steering problems

I have a 2007 Chevy Trailblazer. It's been happening more often lately. When I drive the car all seems okay until I get on the highway. After about 5 minutes or so of driving at about 60 -65 mph the whole sterring wheel starts shaking very bad and the whole car seems to vibrate. If you try to brake during this occurence the sterring and car shake very violently. After about 10 minutes or so of this the car goes back to normal and it drives smooth as usual. I'm worried because I travel in this car with my wife and 2 little kids. Your help would be greatly appreciated. I took the car back twice to the dealership and they could not duplicate the problem. Thank you in advance.