Those two B747-100 that were used to transport the space shuttle were extensively modified. The entire main cabin was stripped and replaced with strong struts to carry the load from the Space Shuttle directly down to the wing spars. They even added vertical stabilizers to increase stability and escape hatches for the crew. The first flights were carried out by the best test pilots in the world, some of whom had been to the moon. So even though all their models and calculations showed that it would work they were prepared for anything going wrong.
1. Work out how much drag the space shuttle will generate. Given that the drag of the shuttle during its descent is critical you can bet they are good at calculating this.
2. Work out how much extra thrust the 747 needs to maintain desired speed with the extra drag, and to take off with the extra weight.
3. You know how much extra thrust the engines generate, so easy to check if it is enough.
4. Work out the loads on the various bits of the airframe (eg where the wings attach). This is not easy but is the sort of thing you do when designing an aircraft so within the skill set of the team.
5. Go flying!
The carrying capacity of the 747 was known, the weight of the Shuttle was known. The aerodynamic drag of each was also already known, as was the lift of each. Thus, it’s pretty easy to calculate whether it could potentially carry it. After that it’s just engineering to make it work.
As it is, they had to be modified. You have the butt cone on the Shuttle to decrease drag and the extra vertical stabilizers on the 747 because obviously that big thing sitting on top is going to interfere with the airflow over the main vertical stabilizer.
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