What Mac said, followed by more motion after the packing fails and exhaust blows thru the corrugations, then it stops moving as oxidation builds up, then it fails completely from corrosion and fatigue. Unless it's a sealed bellows type, which just oxidizes and fatigues.
What do you mean by the word "move"? Are you referring to elongation, angular displacement or deformation of the shape of the flex pipe? and under what condition(from temperature changes, pull of gravity or stretching from an external force) heat is the movement generated? and how is the pipe supported during the "move". As others have said on this forum the more information the better the quality of responses.
You can measure static and dynamic movement with one or more accelerometers or LVDT sensors. High temperature can be an issue when selecting and mounting sensors. The ease of doing this depends on whether the situation is an engine dynamometer, vehicle dynamometer, or vehicle on tests track or on/off road. What is the situation? What do you need static and/or dynamic movement?
The way I measured engine motion, for zero cost, was to ink up some flat sheets of steel, screw them to the body of the car, and make pointy sticks from coathanger and screw them to the engine. Go for a drive over whatever test circuit is of interest, and look at the scratches on the plates. If I was feeling posh I'd use LVDTs.
I dare say you could make a stab at estimating it if you know the internal construction of the engine mounts etc.
Cheers
Greg Locock
New here? Try reading these, they might help FAQ731-376