Engine Belts And Pulleys problems of the 2010 GMC Yukon

We have owned the car since sept. 2013. On more than 15 occasions it feels like the transmission does not engage the next gear. When it misses the next gear while driving the rpms shoot way up, nose of the car dips, we decelerate or coast, and occupants get jerked and thrusted forward triggering the seatbelt to harness you in place. This has happened on most of the occasions. At times when I am pulling out of a parking spot in reverse the transmission does not engage. Normally when vehicles are put into gear you can feel the car it, not on these instances. I would shift the car in gear but not feel it actually get into gear, press the pedal and watch/feel the rpms increase, and the vehicle does nothing - stays in the same place. Accelerating on a green-light is another situation where the vehicle has missed the next gear. Similar to the driving description above, only this is from a dead stop. The car will move forward then try to shift into 2nd gear and miss. Again the nose dives, rpm's shoot way up, we are jolted into our seatbelts. I guess that it's the transmission but I am not a car guy at all, supposedly the 2 dealerships (even longer story) that I took the vehicle to rebuilt it (twice) under warranty. The vehicle has been in the shop at least 4 times for long periods, weeks each time, due to whatever the issue is. Each time they say they have fixed the problem and that they are sorry that they didn't fix it the previous time. It had been a months since it happened last, until 2 weeks ago while on my way to work doing about 10mph exiting a round-a-bout. I only heard about this site today (3/19/15) on the news so I am reporting as soon as I heard. I did get video of some of the problems and showed it the dealers (unimpressed), only problem is they could not get the car to miss gears, jerk, jolt, or nose dive while in their possession to experience it happen.