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# Damped versus undamped critical speed calculations

## Damped versus undamped critical speed calculations

(OP)
Hy experts,

I often have the job to calculate lateral critical speeds for pumps. The one who did the job before me only did undamped calculations whereas I do both damped and undamped. Now the problem: most specifications still have the sentence: the distance to rotational speed must be greater than 20%. The (old) undamped calculation fullfills that, the damped calculation does often not. What are you doing in such cases ?

1.) Doing only undamped analysis.
2.) Discussing with the customer about allowable damping ratios

Frank

### RE: Damped versus undamped critical speed calculations

We do lateral vibrations/whirling for ships propulsion installations. For those calculations, the margin of 20% between excitations and natural frequencies is considered a "safe" margin by all major classification societies (some use 10%, but I consider that way to small a margin).
They allow smaller margins, but then a damped vibration calculation is required to prove absence of exceissive resonance vibrations. The problem often is obtaining reliable values for both damping and excitation.

### RE: Damped versus undamped critical speed calculations

Hi,
strange... In fact, due to damping, the frequency shift should be positive, i.e. the damped critical speed should be greater than the undamped. At least, as far as I remember from theory of rotodynamics. And this is also prooved, where I work, by all the analyses of shaftlines we have had to perform until now.
The choice of performing only undamped shaftline analysis should have been conservative, not the opposite...
I'd check and see there is some odd setting in your input...

Regards

### RE: Damped versus undamped critical speed calculations

I think that depends on the installation running under- or overcritical.

### RE: Damped versus undamped critical speed calculations

I am wondering whether undamped critical speed really meant a lot in rotordynamics, but ofcourse in strutural dynamics. Since damping in strutures are typically less, they have minimal effect in structural dynamics.

But in rotordynamics, especially the case where you have cross coupling from hydrodynamic bearings, the damping will significantly influence your critical speeds and you might  even end in instability.

API states " Include 15% separation if operating speed < critical speed Or 7% if operating speed > critical speed"

Regards
Jeyaselvan

### RE: Damped versus undamped critical speed calculations

The damped natural frequency for a single degree of freedom system is sqrt(1-zeta^2)*undamped frequency, where zeta is the critical damping ratio.  So, the damped frequency is lower than the un-damped.

Also, as you approach critical damping the frequency goes to 0.  A critically damped (root) natural frequency has only real part, i.e. 0 frequency - does not oscillate.

Regards,

Bill

### RE: Damped versus undamped critical speed calculations

Hi,
WCFoiles, correct, my fault.
However, in real-life situations, with the real-life rotors I deal with (hydro-electric machinery), undamped real forward-whirl critical speeds are always lower than damped complex forward-whirl critical speeds. These systems can never be simplified down to SDF systems.
In our applications, oilfilm instability is a design fault so it's a phenomenon we never have.
For this kind of machines, we <usually> perform undamped critical speeds calculation. Adding the damping terms of the bearings' oilfilm <generally> shifts the critical speeds not more than 2 - 5%, which is significant "per se" but unrelevant for the verification. Adding the damping terms also means that the oilfilm characteristics must be calculated for a great number of points along the step-up / coast-down ramp in order to be meaningful, and this is a huge amount of calculations in our case (so we try to avoid this...) because oilfilm properties depend upon the shaft running eccentricity, which depends on the transverse loads, which in our case depend upon the shaft running eccentricity etc... so we have a closed-loop which must be solved iteratively for each working point.
Of course, I don't pretend that my particular case can be generalized everywhere.

Regards

### RE: Damped versus undamped critical speed calculations

The forward whirl frequencies being higher than the undamped forward frequencies does not relate to damping; this is caused by a gyroscopic effect.

Regards,

Bill

### RE: Damped versus undamped critical speed calculations

Hi,
sorry, here I disagree. The gyro effect is exactly the same in both cases.

Regards

### RE: Damped versus undamped critical speed calculations

Run the analysis without damping, and you should still see the bifrication of the forward and backward modes.  Add damping and see.

Also, you could turn gyroscopics off and see the effect in the modes.  I assume you have an overhung rotor for which gyroscopics has a large effect.

Regards,

Bill

### RE: Damped versus undamped critical speed calculations

Hi,
WCFoiles, I don't want to be polemic, but what you say is a thing I've already done several times with "MY" shaftlines. I repeat "MY" case may be specific BUT I do believe it can be very close to the one of the O.P.
I give you some data of a typical shaftline of "MINE":
Geo data: 8 [m] vertical shaftline, bearings at 1.5 [m] and 7.5 [m] from DE shaft extremity, equivalent constant diameter 750 [mm] (given as a simplification of the real shaft I have), turbine at DE extremity M = 7000 [kg], Ip = 6000 [kgm^2], Ib = 3000 [kgm^2] (approximations of the real runner's values); polar wheel at 4 [m] from DE extremity, M = 56000 [kg], Ip = 55000 [kgm^2], Ib = 29000 [kgm^2] (approximation of the real polar wheel + fans + etc...), 2 identical isotropic bearings with kii = 1E9 [N/m] and dii = 5E6 [Nm/s] (example values - in reality the two bearings are different and of course the oilfilm properties related to the radial force's direction are not isotropic and incorporate crossed-terms).
Undamped calculation (by switching off the damping):
first natural frequency (generator's mode): 13.89 [Hz], critical speed (forward-whirl): 14.02 [Hz], critical speed (backward whirl): 13.73 [Hz], Shape-and-Directivity-Indexes: +1.0 and -1.0 respectively (not true with real bearings).
Damped complex calculation: first natural frequency: 14.14 [Hz], critical speed (forward): 14.42 [Hz], backward 13.9 [Hz].
"In my home", 14.02 < 14.42, so the undamped calculation is slightly conservative for a machine subjected to an undercritic limit, what do you think?

The O.P. wanted to have indications about if calculating without damping is conservative or not wrt complete damped complex calculation. Well, in "MY" field of application, the response is definitely "YES" (even if only slightly, in most cases), but of course you are free not to believe me.
This is not in contradiction with trying to do the most precise calculations possible, which is a good practice anytime you have enough data and time to do it (performing complex analysis with unprecise or uncomplete data is MUCH more "garbage" in "MY" field rather than doing an undamped calculation.

Regards

### RE: Damped versus undamped critical speed calculations

Hi,
by the way, before you make the remark: the generator's mode is of course limitedly affected by the gyroscopic effect. In this case of isostatic shaftline, the overhung turbine is much more affected but "unfortunately" its mode is the second, which has nothing to do with the specified limit "first critical speed XXX % over the runaway speed". However:
Undamped: Fn = 26.54 [Hz], fw critical = 28.3 [Hz]
Damped: Fn = 26.54 [Hz], fw critical = 28.3 [Hz]

Regards

### RE: Damped versus undamped critical speed calculations

For the system you give I believe that damping interacts slightly with the modeshape to give this effect.  The attached file runs a case that the modeshape remains almost the same (by forcing the rotor to be rigid), and the case with the flexible rotor.

Regards,

Bill

### RE: Damped versus undamped critical speed calculations

Hi,
this exactly reflects what I meant. Hydroset shaftlines are NEVER treated as "rigid". The example I provided was for an isostatic shaftline, but similar considerations are valid for the also very common 3-bearings shaftlines (and also for the horizontal 4-bearings shaftlines).
Under these considerations, as regards the fulfilment of the specifications, the calculation with 0 damping ratio (i.e., with no damping at all) is "slightly" conservative because it gives a lower first forward-whirl critical speed. This is not necessarily true for the second critical speed, as your run also demonstrates, but this goes beyond the spec's needs. Normally the two first crit.speeds, for a well-designed rotor, are not so close as to give "overlapping" problems when damping is added.
Moreover, if we want to be extremely precise we must bear in mind that bearings' oilfilm characteristics do are speed-dependent, and that when the rotation speed increases the stiffness increases while the damping decreases. This is important especially for Kaplan turbines, where the runaway speed can be 2.8 times the nominal speed. I don't think this problematique is interesting for the O.P., however.
Regards

### RE: Damped versus undamped critical speed calculations

It looks like the extra damping at point locations (bearings) in this case is further constraining the shaft at the bearings, meaning greater strain energy ==> greater natural frequency.

The rigid shaft was to prevent modeshape changes.  Different combinations of shaft to bearing stiffness can act similarly.

Regards,

Bill

### RE: Damped versus undamped critical speed calculations

Hi,

"Different combinations of shaft to bearing stiffness can act similarly":

exactly.

The characterization of a shaftline before even to begin doing a calculation can help saving a lot of time and effort. However, for "normally complicated" shaftlines it is a bit hard to decide if the shaftline can fall under the category "rigid rotor on flex supports". In fact, it is very seldom the case, I could even say it is never the case. Exception could be only very "strong", very stiff shafts for isostatic vertical shafts for compact Pelton turbines: they are short, they tend to be slightly over-dimensioned because they have to be standardized.
However, in 99.9% of the cases we have "flex rotor on flex supports" and the behaviour we described is the rule. The margin wrt the runaway speed (+15 / +20 %) is there to "eat up" the calculations uncertainties, so you won't make the shaftline fall into resonance because you have calculated it without damping... For information, I've several times done the comparison and seen that the difference, in absolute value, between undamped and damped calculations remains in a 2 - 5% dispersion band.

Regards

### RE: Damped versus undamped critical speed calculations

Adding damping to the model as in modal damping depresses the frequency.  Adding point occurrences of damping, changes the model.

Regards,

Bill

### RE: Damped versus undamped critical speed calculations

Hi,
I'm strongly surprised this thread is still active after the O.P. hasn't replied for decades...
Yes, adding modal damping lowers the frequencies as it is obvious. However, here's now another problem: how do you decide this modal damping? You could also use Rayleigh proportional damping, but once again: how do you decide alfa and beta?

Regards

### RE: Damped versus undamped critical speed calculations

(OP)
Hi,
in the ANSYS help, there are formulas given. You take 2 frequencies and the modal damping (typicaly 1%) at those frequencies. Then you can calculate the alpha and beta. In between the two frequencies you have a lower modal damping, outside it is higher, so you have to chose the frequencies properly.

### RE: Damped versus undamped critical speed calculations

Hi,
welcome back, VibFrank!
exactly how you say: "typically", "choose" are not good to properly setup a problem, unless you can correlate this kind of things to experimental data.
Just my two-cents, though...

Regards

### RE: Damped versus undamped critical speed calculations

(OP)
You can find references to 1% modal damping in the references of the API Publication 684...but certainly: measurements are the best...

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