How is health monitoring of large aircraft engines and components done?
Methods of recording data and what's monitored differ between manufacturers but speaking as a Rolls Royce employee, the engine I'm responsible for, the AE2100D3 on the Hercules, is done via downloads. The flight computer records many different channels during flights and ground runs which are downloaded daily, then trended and analyzed. RR HQ then flag any anomalys and send out data for customers to act on, i.e. vibration problems, reduced performance etc. Big engine such as Trents are monitored in real time whilst they fly which is really impressive.


As regards to what exactly is measured, I could list lots of stuff but primarily it's broken down into vibration (all rotating components vibrate, and have allowable limits for this) and performance (referencing pressures and temperatures through the engine to give a performance margin or figure) Anything that either exceeds or drops below set limits raises a maintenance alert.
Engine control units/FADECs constantly monitor all sensors, as do the crews, again anything that needs attention will trigger some sort of warning or information system
Generally through trending and graphs, or in terms of component lifing, etc there will be a count.