The floor and walls of the inner room can easily be isolated from the concrete garage floor by putting down a 25mm (1") layer of polystyrene sheet. Despite being polystyrene, this material is very strong in compression. It will provide very good thermal insulation, but more importantly, it will for prevent vibration from being transmitted from the walls down into the floor and therefore into the main structure of the building..

Polystyrene sheeting laid on the floor

Erect the inner stud wall

The stud frame for the inner wall was put up next. It would have been better to avoid connecting the top of the uprights to the rafter joists. This causes sound to be transmitted from the inner walls into the exterior frame of the building. This turned out not to be a massive problem, but the room would have performed better had the situation been designed around.