My first thought was that it was an engine, which seemed logical for what we had just experienced, but when I scanned the instruments, both N1s and N2s, temperatures and oil pressures were parallel and showing normal for takeoff power setting, with no disparity between either engine. Then I thought of a blown tire but quickly dismissed that idea, as we were well airborne when the muffled explosion happened, and that would not have caused yaw and roll. As we climbed out, I got on the radio, declared an emergency and gave the pertinent information to ATC. The airplane was still shaking violently but was still controllable, and the sound had changed to a loud clacking noise similar to a large baseball card in the spokes of a bicycle tire coming from the back. I was confused: In the simulator, an engine failure is usually accompanied by warning lights, but we had nothing out of the ordinary (other than the noise and the shaking). What was going on? I knew we hadn't imagined this - it was real, but why couldn't we see it? Then, as if we didn't have enough going on, the VIP came up between the seats and asked if we knew that there was a problem with the right engine. We thanked him and told him we were aware and to please return to his seat and strap in. As we tried to concentrate on the instruments through all the shaking, the engine temperature for the right engine started to climb, taking aim at the upper temperature limits. "Ah, now I see it." Just then, we saw a flight of four F-4s in the overhead break and firmly asked ATC for the F-4s to make room for us as we started a turn to crosswind - and, man, did those F-4s scatter everywhere. (When we later talked to the F-4 pilots in operations, they said that just before they initiated their break they had heard us on the radio and they were ready to make room for us.) After briefly discussing the engine problem and then confirming the left engine was still operating satisfactorily, we shut the right engine down and ran the appropriate checklists. Even with the right engine shut down, the airplane was still shaking violently and making that noise. We finally made it to pattern altitude, entered downwind and confirmed that we had done all we could do for now. After "adjusting our gross weight" and double-checking everything as we turned base and then final, we asked ATC if it could see anything unusual, and it reported no smoke or fire.