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From a documentary (at 16:39) on building a Rolls-Royce jet engine:

... no two finished blades are exactly alike, and with twenty in each fan, it will only spin smoothly if the blades are perfectly balanced, so every one is precisely measured and weighed then rung like a bell. Each blade has a different mass and different frequency ...

Is it hard to develop blades of identical mass, or are there other reasons for the different masses and frequencies?

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    $\begingroup$ "despite the precision of the engineering, no two finished blades are exactly alike" $\endgroup$
    – user3528438
    Feb 26, 2020 at 19:35

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Because manufacturing processes are not perfect, and the minute differences between parts as-designed and as-produced are amplified by the high rotational speeds and large diameters of modern jet engines.

Take a following back-of-the-envelope calculation: disregard incoming airflow velocity and assume a hypothetical fan with a 1 meter diameter rotates at such rpm that the blade tip has a tangential velocity of 80% of the speed of sound at 35kft in ISA. The equivalent rpm would be 2817. $$ a_c=\frac{v^2}{r}=\frac{(295\cdot0.8)^2}{1}=55.7\cdot10^3 \; \mathrm{[m/s^2]} $$

That's the acceleration on the blade tip. If this blade were to be just 1 gram overweight, the force exerted on it would be: $$ F_c=m\cdot a_c=0.001 \cdot 55.7\cdot10^3=55.7 \; \mathrm{[N]} $$

that force is more than the weight of a 5kg dumbbell and it alternates its direction almost 3000 times per minute.

Its not that fan blades are hard to manufacture precisely, but that the application in which they are used is unusually demanding and any small imperfection in the blade will make itself known very loudly.

As for why the frequency is different, it is because the vibration modes of any system depend on the spatial mass distribution, like a pendulum which changes its natural frequency if you shorten the distance between the mass and the fulcrum. So the two issues you mention are directly related: a few milligrams of metal missing at the tip will shift the center of mass, and with it the natural frequencies of the blade.

In reality the whole vibration issue is a bit more complicated for continuous bodies, which effectively have infinite natural frequencies, because unlike ideal pendulums, their mass is distributed over infinite points in space, but that is a bit beyond the current point.

The important part is not to have any natural frequency coincide with the rotational frequency of the fan, because that would quickly lead to rapid unplanned disassembly: the vibration amplitude would be amplified with each revolution until the blade failed.

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    $\begingroup$ Very well written and a nice illustrative example! I'd love to add the keywords "resonance" or "eigenfrequency" to your last point, as you can find a lot of examples when searching for these. $\endgroup$
    – flawr
    Feb 26, 2020 at 16:20
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    $\begingroup$ I deliberately avoided speaking about eigenvalues because the question was quite simple and I wanted to keep the language accessible, but yeah, maybe it can be revised. I'll have a look when I get home. $\endgroup$
    – AEhere supports Monica
    Feb 26, 2020 at 16:23
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    $\begingroup$ I understand your point, and I agree, you certainly don't have to use them in your explanation - I thought however, that it would help people find other occurences of the same phenomenon, maybe just as some pointers. $\endgroup$
    – flawr
    Feb 26, 2020 at 16:36
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    $\begingroup$ "...rapid unplanned disassembly". That made me laugh. As for using mathematical terms like eigenvalues, perhaps you could edit your post to leave the bulk of it in layman's terms but add something like ("If you want to learn more on the topic, you can search on terms like 'resonance frequnency' and 'eigenvalue'.") $\endgroup$
    – Duncan C
    Feb 27, 2020 at 0:12
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    $\begingroup$ @Vlad You need to not think about the turbine like a spinning disk, and instead think of the centre of the turbine as an Olympic hammer thrower, and the blade as the hammer. Centrifugal force (actually centripetal force, but that's a separate conversation) acts down the blade. The force down the blade does not change, but the blade itself rotates. If the blades aren't perfectly balanced, the rotor shaft will pull in the direction of the heaviest blade(s), and that force rotates with the blades. $\endgroup$
    – Graham
    Feb 27, 2020 at 0:36
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There are advantages in the blades not being "perfectly identical". If all the vibration frequencies were exactly the same, when the blades were assembled to make the complete fan the vibrations would be coupled together and the blades could resonate with a large amplitude, causing increased noise and potentially reducing the life of the blades because of the increased oscillating stresses.

The blades are arranged in a pattern which minimizes not only the amount of unbalance, but also any "regular patterns" in the different vibration frequencies. For example if there are 20 blades in the fan, you don't want every fourth blade or every fifth blade to have close frequencies since these form regular repeating patterns.

Computer software is used to create the "best" arrangement of blades to meet all the criteria. For a fan with 20 blades this might not seem such a huge problem to solve, but some stages in the turbine may have more than 100 blades around a single disk.

The measurement of the blade frequencies when new is also important for safety checks later.If any internal cracks start to develop frequencies will change, and this is a useful way to check for damage as a supplement to X-raying the blades, etc. In some engines the vibration frequencies are "measured" continuously from the vibrations of the engine while it is running, and a change in the frequency spectrum gives an early warning of problems developing.

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    $\begingroup$ So if a blade has to be replaced, does the entire disc have to be re-balanced, possibly including changing other blades? $\endgroup$
    – Andrew Morton
    Feb 26, 2020 at 19:26
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    $\begingroup$ @AndrewMorton yes - if any blade is replaced the unit will need rebalancing. How could it not ? $\endgroup$
    – Criggie
    Feb 27, 2020 at 0:27
  • $\begingroup$ From the number of nasty fan (and to a lesser extent, compressor and turbine) failures in recent history, it sounds to me like like engine designers may have gone a little too far in reaching for performance. That, or airlines are just getting sloppy in their maintenance procedures... $\endgroup$
    – Phil Perry
    Feb 27, 2020 at 1:18
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    $\begingroup$ ETOPS ratings suggest engines as a whole are becoming more reliable every decade. $\endgroup$
    – MSalters
    Feb 27, 2020 at 12:52
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    $\begingroup$ @PhilPerry or there are lots more airplanes and lots more engines, so a rare event can happen more often, and catch your attention. That and selection bias - you probably don’t remember all the engine failures from 10 years ago. $\endgroup$
    – Tim
    Feb 27, 2020 at 21:53

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