Critical Speed

Your question cannot be answered by a formula since the destruction can be due to different perturbations depending on the design of the rotor and without all the dimensions of the shaft and of the discs it is impossible to estimate at which rpm the amplitude could become critical. Your question is one of those which show that the complexity of the phenomenon is not know or even correctly estimated.

Simply measure the RPM and increase speed until failure results. This method is more accurate than calculations.

For even better accuracy, do this two or three times and take the average.

"It is not unusual for a second critical speed (natural frequency) to be sitting above running speed."
The problem is during checking and setting 'trip speed'. For smaller turbines the critical speed is comfortably above trip speed. But for machines with flexible shafts, like you indicated, running between 2nd and 3rd critical speeds, the trip speed comes closer to 3rd critical speed. For the sake of testing, running around critical speed has always been risky.

Hey Steve S. I know this is off-topic, just noticed you live in Friendswood...I live in Alvin.

The set speed of a steam turbine is a design criteria that allows the blades to be pitched at the best lift angle for the passing flow.

Long shafted turbines like steam turbines are subject to rotation rate instabilities that are most apparent during acceleration or deceleration. Running above its set speed is very destructive of its efficiency because the higher speed of the blades causes a change in the apparent wind they see, which reduces the lift they can generate.

While extremely destructive shaft whip instabilities are certain to occur at speeds above the rated speed for the turbine it would be very very difficult to predict exactly where it would blow the casing.

Of more practical concern is an effect know as derating of the turbine wherein the blades erode rapidly due to energetic cavitations occuring on their leading edges or lifting faces. These rapidly destroy the foil characteristics of the blades and permanently reduce the useful power the rotor can produce.

As some of the other commentators have pointed out destructive testing will produce a definitive answer if you really want one, but my hope is that you will reconsider your need for speed and respect the limitations of the machine your working with. They don't operate well for those who do not.

Sincerely,

Mr. Gee

As stated above: natural frequencies and high vibrations may destroy your turbine, these you can measure easily and establish an amplitude you would not like to surpass.

As this is driven by unbalance you may need better balancing.

As stated also above there may be well damped natural frequencies: these you should know and if damping is high (above 0.7) it is difficult to measure amplitude excursions.

So you need phase measurement: phase of the synchronous vibration (taken with accelerometers) with respect to an artificial (dot on shaft) phase null.

Do not try to measure amplitude by direct amplitude measurement as then the geometrical runout is disturbing your measurement and you will not get the correct phase of the added functions of geometry and vibration.

If you have thus measured the first natural frequency you should be extremely careful not to reach twice this speed.

Somewhere near twice the first natural frequency the oil-whip is generating vibrations that cannot be survived! These vibrations have a frequency of half the shaft rotation frequency: the half-frequency whip.

Energy is transferred from rotation into the whirling motion at any frequency and giving the first instability: half-frequency whirl (at lower speed). This is occurring at a speed when the tangential load capacity of the bearing is going to zero. This may be occurring or not and this may be survivable as radial load capacity still exists.

If shaft rotation frequency is near twice the first natural frequency also the radial load capacity vanishes and the amplitude will grow very fast until catastrophic failure.

This half frequency whip is also a forward whirling motion but it is extremely difficult to measure as it is very highly non-damped - despite the good to high damping established by bearing design. The energy transfer is removing any damping and changing into excitation!

We measured a 1000fold increase of vibration amplitude within only 1% shaft speed increasing! (10nm to 10 µm rise of vibration amplitude in an airbearing at nearly reaching this condition at 1500rev/s). And we did drive another airbearing into cold welding the rotor to the stator at reaching this instability and breaking the mounting screws with the ultrafast rundown that followed. This was a sharp and short squeezing sound estimated 0.2 seconds.

RHABE

When large turbines are manufactured or often when they are rebuild, they are put in a large vacuum chamber, (to decrease horsepower needed) and ran through two or three harmonics.

The turbine would then be rated at two or three harmonics, and sometimes may be rated even higher.

When the turbines are ran through the harmonics, you can feel it over the whole shop.

The vacuum chamber is usually located below the ground and shielded to prevent the blades from rendering wide spread destruction.

The destruction of large turbines is not usually from centrifugal force but from excessive harmonics. Or a combination thereof.

I do agree with Nick Name as your ques couldn't be answered in a word. It depends of a scored plausible reasons why did you ask it. These reasons could be three at least:

i Education (degree/graduate) work and so on)

ii Design/developing of new turbine

iii Attempts to assess aftermathes of changed maintaning turbine mode.

case i: You can use significantly simplified mech model having minimized DF(degree of freedom). Crucial parametres you need take in consideration: mass of rotating parts, damping and stiffnes factors as well as imbalance which believed to be ever. I've seen a lot of examples at textbooks concerning not only the turbine critical speed. For instance every vehicle has critical speed being presumed to reach never for safety (even for F1 bolid). You can drive overspeed without any warn be swept off highway. So it's classic issue.

case ii: There are a lot of engineering and scientific schools having used diverse methods of evaluation. At novadays it's thought everywhere should be used CAD/CAE software system for such a task. CAE might be applied even for case i. But even being "armoed" a very powerful CAE (Ansys, Adams and tralala) it's incrediebly hard to interpret properly all obtained data, software generated. Though it's much more easy than handmade calculations. To develop such complex device and evluate its critical speed is task for skilled team.

case iii: If your leadirship staff (or maybe yourself) warns about whatever happens if rotatin speed exceeds nominal one over 10%. You can assure them(yourself) --- nothing daunted happens just in terms of "critical speed" as well-designed turbine as rule has presumed "critical speed" at some times higher than nominal (at least 1,5-2 times). But if your turbine actuates syncronous generator and rotor revolution speed will exceed 10% revoltion of electro-magnetic field --- it will be catastrophic disaster! Your generator will fly out through concrete celing as A380 jumbo. I asure you it'll be. And this case it's does no matter what is a critical speed of turbine. It's another physical aspect.

Excuse me for black humor, but guys at Chernobyl Power station had had just both cases i and case (iii). Default case when safety system with a lot of sensors should be preserve incident had been turned off by devoted explorers.

I wish you every luck. Hope turbine is not located in Nuclear Power Station.