I was taught a quick way to align flexible couplings by an old millwright. Here is the method, and I've used it many times since:
1. Using a straightedge, assuming the coupling halves are the same outer diameter, line them up with the straightedge accross the coupling hubs at 12:00 & 3:00 & 9:00. You will need to decide which end of things is the "fixed end" and which end can be moved into alignment. SOmetimes, something like a pump which is coupled to piping is the fixed end, and the motor gets moved into alignment.
2. I use a piece of brass and a hammer to bump the mounting feet to get things into straightedge alignment. You should get the side-to-side ( 3:00 & 9:00) OK, and be able to measure how much you need to shim under the lower side's mounting feet to get things into rough alignment.
3. Once you have rough shimmed, check the faces of the coupling halves for parallel. I use a wedge or taper gauge with a light film of Prussian blue. Where the blue rubs off, that is where the coupling halves contact the taper gauge. I then see how the coupling faces are for parallel and move things accordingly.
4. When you have completed rough alignment with straightedge and taper gauge, it is time to put the coupling together. After you have done this, putting the center element (like the rubber "biscuit" in a "Lovejoy" type coupling) in place, you use a dial indicator. I braze a nut or a bushing tapped to take a post for a dial indicator to a common hose clamp. Different coupling hub sizes, use the appropriate hose clamp. The indicator post sticks out radially from the hose clamp, perpendicular to the shafting. clamp a dial indicator to the post so it spans the coupling and contacts the opposing coupling hub. The indicator support forms an "inverted L" so that one leg of the L is parallel to the shafting and bridges the center element of the coupling. Clamp a dial indicator to this bracket so the contact point of the indicator touches the opposite coupling hub. Position the indicator so it is at about half travel and clamp it. Bring the indicator to 12:00 and zero it.
5. Revolve the coupled shafting 90 degrees and read the indicator. Note the reading and move another 90 degrees to 6:00. You may need a mirror to read the indicator in this position. I use a mechanic's inspection mirror for this. Roll the shafting another 90 degrees and note the reading, then return to 12:00 and be sure the indicator returns to zero. If it doesn't, chances are your setup has some looseness or slop, so find it and fix it. I use just a hose clamp with a brazed on tapped lug, and whatever posts and clamps are in my dial indicator's kit.
As you revolve the shafting, the coupling halves will "work" or move in relation to each other if there is misalignment. You make a sketch, and write the readings. Sum the opposing readings ( such as +0.002"/ -0.003" at 3:00 & 9:00). This gives you a total of 0.005", and you can need to move HALF that amount for alignment. Looking at the sign ( + or -) on each reading, you see that you need to move the + side "in". You can do this with a piece of brass or hardwood and a hammer and very light tapping. Once you have this correction, reset your indicator and take another set of readings. I try to bring in the 3:00-9:00 readings first, then go after the shimming for 12:00/6:00.
6. Shims can be cut from sheet metal, or shim stock, but use as few as possible. If you use numerous thin shims to get the first alignment, it is good practice to reshim with thicker shims to avoid a "leaf spring effect". Any time you change shims, pull the bolts down hard before checking alignment with the dial indicator.
7. Once you have alignment made, you can really pull the bolts down or torque according to bolt size/grade, assuming a steel mounting and cast iron pillow blocks. Some millwrights drill and ream the feet of mounted machinery for dowel pins. The dowel pins insure the alignment will not be lost.
8. Another trick to both facilitate alignment and hold alignment in service is to make "jacking dogs". These are steel lugs, tapped for jacking screws. The steel lugs may be bolted to the bedplate or welded to the bedplate. The jack screws are often made from all thread rod, and push against the feet of the pillow blocks or motor or gearbox, pump, etc. The jack screws allow minute adjustments to be made to "dial in" an alignment, and are check-nutted off to hold the alignment in service.
The single dial indicator used accross a flexible coupling is an old trick. It is essentially used as a strain gauge. It is a very accurate way to align flex couplings. A flex coupling has some forgiveness for misalignment, but it is not a license to put machinery with shafts misaligned into service. A misalignment, while tolerated by the flex coupling, puts a sideload into the bearings and seals of pumps or motors. Shorter seal life and shorter coupling life result. SOmetimes, vibration results from this misalignment. It is a good idea to start off with the best alignment job you can do. Another thing to check for is the rigidity of the actual mounting. Some engines, motors, pumps, etc are mounted on fabricated bedplates that are too "twisty", or are pulled down on a floor that is not flat. I always start any alignment job by checking what the components are mounted on.
Incidentally, in February of 1987, I sustained a broken wrist which was set and casted. We had a number of high pressure IMO rotary screw oil pumps driven by electric motors on a job I was on. There were no millwrights, only linemen. The pumps were coupled with Lovejoy couplings. No alignment had been done or checked. The pumps were put into service, handling insulating oil in a series of "pipe type cables" that crossed under a river. In short order, the pump seals were leaking and pump shaft bearings were howling. The pumps were also eating Lovejoy coupling elements. The contractor's response was to put in a heavier duty Lovejoy coupling "spider" (different composition material). This did not fix the problem, and things continued to get worse. I was an engineer on the job, and being a mechanical engineer, got asked to take a look. It was pretty obvious that the frames these pumps and motors were mounted on were lightly built and then bolted to a concrete floor and pulled down hard. If there ever was an alignment done at the factory, it was lost when the units were mounted on the concrete floor. I said the motors needed to be aligned to the pumps, since the pumps had steel piping connecting them to all sorts of system components. This got blank looks. I said we needed millwrights, and was told this was a lineman's job, and no millwrights would be on site. I said I'd bring my own tools and align the motors. I filled a 5 gallon pail with my 6" machinist rule, combination square, wedge gauge, Prussian Blue, scriber, micrometer, dial indicator, hoseclamp/lug, snips, shim stock, hammer, brass bar, small pinch bar, and mirror. I had the pumps removed and taken to a service shop off site. There, we had new mechanical seals and bearings installed. When the pumps came back, I had the linemen remount them and reconnect the piping. I then did the alignments with one arm in a cast. Linemen took turns being my other arm, and I took everyone thru the basic steps in alignment. It became a joke that I could align pumps with one arm in a cast. The pumps ran smoothly and quietly for years afterwards, no problems. The concept of using a single indicator to span a coupling is a bit strange when you first hear of it, and doing an alignment with the coupling assembled rather than the more traditional method of indicating one coupling to the other is a bit hard to grasp. But, when you see it happen, you realize it is a strain gauge. The new laser shaft alignment systems use a similar approach, relying on spanning the coupled shafts with a laser beam and target and rotating the coupled shafts. The laser beam does the same thing the single dial indicator does.
As a safety precaution, do not work on any alignment unless you have protected yourself from accidental starting of the motor or engine, and have bled off any pressure trapped within pumps.