Saw it on a 747 freighter nose door, see it again on an F16's M61A1 gun assembly. How do you design with a flexible shaft? In what situation do you use it to replace a bunch of rigid shafts and gears? Does it provide the strict angular synchronization of normal rigid driveshafts? Or is the synchronization only approximate, and can only be used in situations like either the 747 nose door or M61A1 gun-to-drum drive where there is only a requirement for a rough 1-1 translation of motion and power and no real demand to perfectly synchronize? Can you guarantee that if there is a constant speed revolution in one side you would get a constant speed revolution in another?
Flexshafts are used on a lot of flap systems to interconnect an array of screwjacks. They are more or less aerospace versions of weedeater shafts, inside a hydraulic line style casing (typically stainless braid over a teflon liner). They are used because the envelope and clearance requirements are much easier to meet than with solid shafts with angle drives and gearboxes. You can route them through very tight spaces and around gentle corners without having to maintain clearances to adjacent parts; you just have to keep the flexshaft housings from chafing on adjacent parts, like hydraulic lines.
Flexshafts aren't meant to take high torque like solid drive shafts and are normally used in high RPM low torque applications. Or better to say that if you want to use a flexshaft, you design your drive system to run at high speed/low torque. A drive line might be designed to run at 500-1000 rpm with solid shafts and a moderate gear reduction at the actuators, whereas a flexshaft will run at 2 or 3000 rpm at much less torque, but with a much higher gear reduction in the actuator to get the same output force result.
Flap system flexshafts as used on CRJ200s and Embraer 145s will only have to sustain a little less than 100 inch pounds of torque at operating load, while producing several thousand pounds of force at the end of each screw jack thanks to the gear reductions in the actuators. You get some torsional displacement, or wind-up twist, from one end of a drive line to the other, maybe 1/3rd of a turn, but with the high speed and large number of revolutions (say 800 to 1000 turns) to cover an operating range, a third of a turn of wind-up doesn't have a major effect on position sensing, so synchronization isn't a problem.
It'd be the same deal on a 747 nose door. The flexshaft connecting the door actuators runs at a high enough speed so that synchronization down to a fraction of a turn is not really necessary because each turn is a very small part of the total motion. Cascade thrust reversers on the CF-34 engine are also run by screwjacks interconnected by flexshafts; same thing there.
The big downside to flexshafts if used in exposed areas is sensitivity to contamination through imperfectly sealed connections. Also the carbon steel wire windings tend to suffer from fatigue where they are crimped at the terminals. There is a lot of back and forth flex as they spin. They usually start to break down as individual strands fracture, and the bundle swells out and jams in the casing or sometimes lets go completely.
Flexible shafts do are just shafts and do not contain any gearing or otherwise change of angular rate. They may, depending on their construction, be subject to a load related skew (torsion) - like any 'rigid' shaft as well. Nothing in this universe is absolute inflexible.