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A large turbine used in an electric plant can easily have 10+ meters in diameter and rotate at 70+ RPM. Should any asymmetry occur the rotating turbine will produce beat which will sooner or later destroy the plant. Crafting a turbine that large and perfectly symmetrical looks like a tough challenge.
How is this problem solved? Why doesn't the beat from turbine asymmetry cause mechanical damage?
It is done carefully. Lots of other rotating machinery has the same problem, and whole systems exist just to deal with it.
For example, jet engines are usually smaller, but also usually spin much faster. Balancing a jet engine is something that gets lots of attention at manufacturing, and again any time when it is put back together after having been sufficiently disassembled. I have worked on metrology systems designed to measure a few µm non-roundness of circular parts a meter or more in diameter. Large maintenance facilities have such systems, in addition to the manufacturers.
One method of balancing I saw used a stackup of multiple "plates". These plates were ring-like structures that would be part of the spinning system. No plate could be manufactured exactly right. The metrology system would accurately measure the asymmetries of each plate, then calculate what orientation the multiple plates have to be mounted together to make one larger object that was balanced to high precision.
There are also other techniques, like build it best you can, then balance afterwards. Actually in practise multiple techniques are employed, with final assembly balancing always one of them. This works a lot like dynamic balancing of tires. The turbine is spun slowly, and the vibrations measured synchronous to the rotations. From that a computer system calculates how much weight to add or remove where.
The most common scheme I've seen is drilling holes or indentations in a certain area of metal left available for that purpose. The computer will tell you where to drill and what diameter based on the analysis of the vibrations. In another case we has a electronic product that spun as part of its normal operation. In this case we left a few thru-hole solder pads at different angles from the shaft. The vibration analyzer then suggested which pads to fill in with solder, which was done manually by a technician in this case.
This overall process of measure and adjust is repeated until the vibrations are below some specified level. Despite the best measurements and algorithms, any one adjustment doesn't ever seem to null out the vibrations quite as much as it should in theory.
Adjustable counterweights.
The wheels and tires of your car are not manufactured perfectly circular but after the tire gets put on (or you go for a rebalance) the mechanic will tack on weights to put the center of mass closer to the axle. The turbine can be balanced during normal maintenance in a similar way.
Also remember that 70 rpm is roughly 1.2 Hz. And the plant's structure will be engineered to have a natural frequency that is not a harmonic of the typical turbine speed.
Additional counterweights may not be required. The individual blades on the rotor will have slightly different masses because of manufacturing tolerances. In aircraft engines, each blade is accurately weighed and the different masses are arranged around the shaft to improve the balancing.
Also, the rotor can be designed to run faster than its critical whirling speed in normal operation. In that situation, it tends to be self-centering, and the bearings can "float" on a film of oil so the rotor is spinning about its center of mass, not its geometrical center. The large unbalance forces only occur for a short time at startup and shutdown, when the rotor accelerates/decelerates through the whirling speed.
A related issue is that when the machine is shut down, the residual heat naturally rises and the temperature difference between top and bottom can bend the rotor out of balance. To avoid this the rotor may be continuously driven at a slow speed (e.g. 1 RPM) when the machine is nominally "stopped" until the rotor has cooled down, which may take several hours for a large turbine.
