balancing of rigid shafts

[chichuck]

I am developing a spreadsheet for analysis/design of foundations for vibrating and rotating equipment. This post is about input for that design, specifically the eccentricity

of the rotor(s) in a turbine generator. It also should be applicable to other types of rotors as well. I am concerning myself right now with rigid rotors, not flexible rotors.

The textbooks I have available do not give much information about the eccentricity of the rotor as input. The made-up examples merely state the eccentricity resulting from a

static balance, and demonstrate how to use that value. That value is for a flexible rotor, and is not my concern here.

I have a real-world example from an upcoming project. I have in hand a proposal from a vendor which gives incomplete data, so at this point I will have to make some

assumptions to proceed.

The data I have is as follows:

W rotor turbine = Later (I will assume W rotor turbine = 50 k, because I need a value)
W rotor generator = Later (I will assume W rotor generator = 30 k, because I need a value)
c.g. location along rotor length = not provided (I will ask for this, for now I will assume that cg is mid-length of rotor for each rotor)
operating speed = 3600 rpm
normal operating speed range = not given
maximum overspeed = not given

Rotors are balanced to ISO 1940 Grade G2.5.

I have calculated the eccentricity of the rotor mass in two different ways. First using Eq. 6 from the ISO standard. I go through a calculation that includes a parameter U
per, and from there, I calculate the eccentricity of the rotor mass in mm or inches. A second method is to start with the operating speed of the rotation in mm/sec (for a
Grade G2.5 balance, this is 2.5 mm/sec, per the ISO standard), and divide by the operating frequency in rad/sec, and get a value of the eccentricity in mm / rad, which is the
same as mm. Both calculations give the same result. Please see the attached spreadsheet on the tab iso1940. Look in rows 18-41.

ISO 1940 applies to rigid rotors only.

It isncludes a definition of rigid rotors: that the unbalance does not change within the speed range, and that the position of all the mass elements of the rotor relative to
each other remains sufficiently constant within the speed range. I have read essentially the same definition of a rigid rotor in Haris' Shock & Vibration Handbook 5th Edition
in Chapter 39 Part 1. I have also read other definitnions of rigid rotors in other articles, such as: rigid rotor is one that operates below the critical speed; or that a
rigid rotor is one that operates at 70-75% of critical speed.

It seems to me that if for a rigid rotor, if the eccentricity does not vary significantly with rotor speed, then it really does not have a critical speed. So neither of the
latter definitions make sense.

I also have found that the eccentricity of a rotor balanced to, say, G2.5 varies greatly from zero to about 1000 rpm, above that the eccentricity is almost constant. Refer to
the graphs of the eccentricity in my spreadsheet, one on log-log plot, and the other plot of the same data on semi-log plot. I also noted that the size (weight) of the rotor
has no effect on this eccentricity. All rotors balanced to G2.5 have the exact same eccentricty and any speed, regardless of the weight.

Based on the definitino of the rigid rotor given in ISO 1940, the plot of the eccentricity does not have one, two or three peaks corresponding to critical speeds, because it
does not "flex" like a flexible rotor does. In other words, the radial centrifugal force due to the rotating eccentric mass does not cause any lateral deflection of the shaft,
so the resonance does not happen.

I have no idea if hte balancing process done to the limits in ISO 1940 is a static balance or a dynamic balance, but I don't think this is an important distinction for what I
am trying to measure and input into my foundation design spreadsheet.

I don't know if the unbalance I've calculated is to be applied to one or to more than one plane. I don't know if it should be applied to one or more tolerance planes or to
balance planes. I don't know if either if these is significant for my goal or not.


So, now to my questions:

1) Which of the several definitions for a rigid rotor is correct?

2) It seems to me that the weight of the rotor ought to affect the eccentricity of a balanced rotor, but my calculations do not indicate that. Can someone exlain this to me?

3) Does a rigid rotor have critical frequencies?

4) Are either of my calculations for eccentricity of the rotor mass (weight) at the operating speed correct?

5) Please see my attached spreadsheet, on tabs Rotating - critical speed (e dyn in cells I40-J40 and I50-J50) and rotating limited input (e dyn in cells AT121-AU121 and
AT129-AU129) for examples of what I will do with the rotor eccentricity once I have it. (Please note, in the first tab, that is a calculation for a flexible rotor, I have to
modify that slightly for a rigid rotor.) Is my application of the results from the ISO 1940 balancing an appropriate one, or is the nature of the unbalance something
different entirely from what I am trying to use it for?

I hope that someone can shed some light on this subject for me. My thanks in advance for any insight that may be offered.

Regards,

chichuck

 

[GregLocock]

Just on the rigid rotor definition thing - those definitions make sense from a practical view, that is, the unbalance won't vary much up til X% of critical, so the rotor is /effectively/ rigid at low rpm, even if at high rpm it goes into whirl.

Effectively a rigid rotor/shaft balancing op ignores the bending of the shaft.



Cheers

Greg Locock


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[chichuck]

Greg Lacock,

Thank you for your quick response. Your short answer explains a lot to me, and also gives rise to many new questions about this balanced rigid rotor. I’d like to summarize what I think is the condition of the balanced turbine rotor.

First, the turbine rotor has several stages, each with a different diameter, mass and eccentricity at various positions along the shaft length. All of the texts and articles I’ve read about this subject idealize the rotor as a thin shaft with one disk on it. More precisely, it is a thin shaft with several disks on it. Each disk has a mass, and each one has its own “eccentricity”. By that I mean that for each disk, the true center of mass is not on the cl of the shaft, due to manufacturing conditions, variable material properties, etc. That eccentricity has a different magnitude and a different radial direction for each disk. While it may be theoretically correct to simplify and idealize a multi-stage rotor as a single disk, for me that masks some of the intricacies of the actual equipment.

With enough information I could calculate the location of the cg of all the disks, along the shaft length. (this is simply a statics calculation, of the resultant of the weights) I don’t have enough information from this vendor to do that, and I would simply ask the vendor for the location of the rotor cg and I wouldn’t calculate it. Also with enough information, I could calculate the location along the shaft length of the resultant of all the eccentricity vectors for all the disks. The resultant of the weight X eccentricity vectors is obviously not necessarily located at the same point along the shaft as the cg, because both the eccentricity and the direction of that eccentricity varies for each component. Can someone confirm this for me? I would also simply ask the vendor for the location of this resultant eccentricity along the shaft length, I would not try to calculate it.

The resultant eccentricity vector, again, at its location different from the cg location, causes a centrifugal force when the shaft is rotated. That centrifugal force has both a horizontal and a vertical component. (the horizontal component is 90 degrees out of phase with the vertical component.) ISO 1940 indicates in Section 4.2 how to represent the unbalance. They use a resultant unbalance and a resultant couple. I understand that there are many different ways to express the unbalance, but I don’t understand why there is no resultant horizontal component, only a couple. Can someone explain this to me?

To get a complete representation of the unbalance, I need to get from the vendor the vertical unbalance (actually I have that from the 2.5 mm/sec limit and the operating speed), its location along the length of the shaft, and the horizontal couple, which is defined by an eccentricity in g*mm and a length distance.

The balancing process itself I don’t understand. It does seem that the goal in balancing is to make this eccentricity zero or near to zero. To do that requires a way to determine the location along the shaft of the resultant eccentricity, its angular direction, and its magnitude. Then, weight is added to the opposite side of the shaft, or taken away from the same side of the shaft to make the eccentricity near zero, or more accurately, within the tolerance from the ISO standard. This might be done by adding or subtracting weights on only one disk plane, (I know, that is static balancing) or two or more disk planes (Dynamic balancing). For what I am doing, this method is not important, so I won’t discuss it here. Perhaps a new thread to learn about the science/art of balancing a rotor would be appropriate.

Now. If the vendor dynamically balances this rotor to ISO 1940 Grade 2.5, then the result will be that the rotor, with its resultant eccentricity of the total mass, will be rotating at 2.5 mm/sec when it is running at the operating speed (3600 rpm). The balance needs to refer to a specific speed. That seems to be the proper way to express the balance, not merely with the speed of the eccentric mass. With that additional information, I can then calculate the eccentricity of the rotor mass, by simply dividing the speed by the angular velocity. (2.5 mm/sec / 376.9911 rad/sec = .006631 mm/rad = .006631 mm). So, for my case, after balancing, the total mass of the rotor has no more than .006631 mm eccentricity. Then, when this rotor starts rotating from 0 rpm on up to operating speed of 3600 rpm, this eccentricity remains constant, but the velocity of the eccentric mass varies with the operating speed at any time. Unlike a flexible rotor, this rigid rotor does not “whirl” appreciably, that is to say, the centrifugal force from the rotating eccentric mass does not cause any appreciable lateral deflection of the shaft. At least not up to the top of the operating speed range, and not up to a practical overspeed limit. The rotor does have a critical speed, one where resonance would occur were it not for damping, but I take it as a given that the rotor first critical speed is at least, say, 30% higher than the operating or overspeed, so that this resonance will not be encountered.

To sum up, I now think that I don’t have a complete report of the balancing of this rotor. Here’s what I (think I) know:

At 3600 rpm, the eccentric mass rotates at 2.5 mm/sec ,then the eccentricity is 0.006631 mm.
The ISO standard for this condition also specifies that U per = 150373.15 gram-mm, which also leads to an eccentricity of 0.006631 mm. This result occurs for any weight of rotor.

What I need to know:

The location of that eccentricity along the shaft length.
The location of the cg of the rotor, along the shaft length.
The orthogonal couple of the remaining unbalance, a number in g-mm plus a length (or moment arm).

What is the correctd way to ask for this? Should I refer to section 4.2 of the ISO standard, and ask for a residual unbalance, its location and a horizontal couple?

Once I have this information, I know what to do with it in order to have the appropriate input for my foundation design. I can then compute and check the dynamic loads on the anchor bolts. I can then use appropriate soil and concrete properties and calculate the horizontal and vertical vibration of any point on the foundation. I could then determine if the vibrations at any point are above some allowable value, for example the value where the vibration sensor is set to trip the machine. That is another, separate subject that I won’t go into in this thread.


I apoligize for the very long post. This is a subject that I don’t understand well, so I am having a hard time expressing my thoughts and questions succinctly.



Regards,

chichuck

 

[GregLocock]

Have you read a book on 2 plane balancing of rigid rotors, and worked through an example? because, frankly answering posts like that needs pages of explanation.

One way of expressing an imbalance is as a force and couple. Both rotate at shaft speed, and create both horziontal and vertical forces at the journals, as sine waves 90 degrees out of phase.



Cheers

Greg Locock


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[chichuck]

Greg,

No, I haven’t read a book on the subject. I have looked in several books on vibration and on machine design. Also at a number of articles on rotating shafts, vibration of shafts, machine maintenance, etc. None of them deal with the subject at all completely. Just bits and pieces that I have to try and put together. Can you recommend a text that treats this subject, especially the basics, comprehensively? And maybe one or two that deal with some of the more advanced concepts and issues related to balancing of shafts? And I’ve certainly not found a good complete example of balancing a rotating shaft to work through. Can you point me to that as well?
And I do understand your point about needing to make pages and pages of explanation to reply to my posts. I suspect that a complete answer might be found in a university course. But I’m afraid that a university course would likely be one on fundamentals of vibration, with all the treatment I’ve already read, about one DOF systems, and the block mass-spring-dashpot simplified models that are used to analyze those systems, with the corresponding differential equations. What I am seeking is an understanding of practical situations, with multi-stage rotors, and non-rigid bearings, and real-world balancing processes that do not achieve the theoretical “perfect balance conditions”. And also how to apply the results of those balancing processes to other real-world situations. (eg. Foundation design). I suspect that the subjects I’m trying to learn about would be a footnote or an afterthought in a university course.
But of course, I might be wrong about that, and I’ll consider it further.

Thanks again for your replies.

Regards,

chichudk

 

[GregLocock]

It's fundamental to dynamics, but I suppose a good one is Mechanical Vibration and Shock measurements by Broch, published by bruel and kjaer.

Cheers

Greg Locock


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[dhengr]

chichuck:
Another good ref. on the general subject is “Shock & Vibration Handbook,” Edited by Harris and Crede, Pub. by McGraw Hill.

 

[Tmoose]

How are these rigid shaft supported in service, and what support features will be available for the balance shop?

ISO 1940 includes some guidelines for repeatablity expectations when checking unbalance, and also some regarding conditions that may be required to achieve a particular "balance" (residual imbalance) for various grades. It is all about being able to re-create center of rotation in the balance machine as in the actual installation. Some balance shops are unaware of this, and will provide a balance report claiming G 1.0 when the fixturing and design of the part is such that potential centering variations don't justify G 6.3

 

[electricpete]

Take a look here:

http://www.concrete.org/GENERAL/ACI 351.3R(04)-Foundations-for-Dynamic-Equipment.pdf

=====================================
(2B)+(2B)' ?

 

[chichuck]

Greg & dhengr, thank you for the references, I will check both of them out.

Electricpete, thanks also for that referral. I’ve got a copy of that, and looked at it carefully when considering concrete and geotechnical aspects of the foundation design. But I had skipped over the parts about the input loads because I thought it was cursory treatment, and empirical and approximate in nature. I looked at it again, and it might be helpful when I don’t have useful input from the vendor.

Tmose, based on the general arrangement for the turbine and generator that I have in hand, both rotors are supported on bearings/pedestals that are at or near the shaft ends. (I mean that the rotating masses are between the bearings, and its not an overhanging or cantilever type.) I don’t know anything about the type of bearings or about the design of the pedestals. I also know nothing about what support features are available to the balancing shop.

I’ve looked at ISO 1940, and I actually didn’t give any real thought to those guidelines for repeatability. I didn’t realize their importance. For the turbine generator I’m looking at, the vendor claims they will balance the rotors to G2.5. I simply assumed that they will be achieving that, since it will become part of the contract when it is signed. How they do it, and whether or not they follow all the provisions of ISO 1940, I wasn’t concerned with. As the designer of the foundation, I don’t know how to question that intelligently. The mechanical engineers in our office, the people who wrote the original equipment specs, and who will ultimately make the selection of the vendor and equipment, probably should be the ones to evaluate that. Maybe I have blinders on, but I thought that our specs had the proper QA/QC provisions in it so that we will be able to verify that we get what we specified.


Thank you to all who have responded.



chichuck