I didn't preclude resonance, but I precluded flexible rotor resonance. The two plane "exact" solution is based on a rigid rotor, isn't it?

The two plane solution does not exclude flexible rotors. One needs is linear repeatable response.

You're right Bill. I was remembering that there is a distinction that the rigid rotor can be balanced in 2 planes to remove vibration in all planes, while flex rotor may have vibration remaining in other planes even after balancing in 2 planes. But since we have only two measurement planes in the original post, my comments are not relevant.

Back to the original question: I came up with a fictional set of data that does roughly what David described (attached).

The math explanation is roughly what I described above. Maybe the numbers explain it better than the words.

I analysed three cases:
Case 1 – 2-plane balance with “exact” (accurate values)
Case 2 – 2-plane balance with inaccurate values representing measurement error
Case 3 – Static couple balance using same inaccurate data as case 2.

Case 1 represents the "true"/exact solution since it is assumed to use the exactly correct input data.

In case 2 a measurement error is introduced (<=1.5%). This completely confuses the 2-plane case 2 solution which comes up with a solution very far different than the exact 2-plane solution of case 1.

The static couple solution of case 3 does a much much better job at recreating the exact solution from case 1, even though it had the same inaccurate input data as case 2.

I admit I had to work a little bit to come up with data that would act that way. It would seem to represent a machine that has a high couple unbalance and a high couple unbalance sensitivity (and low static unbalance and low static unbalance sensitivity). The little amount of measurement error (compared to the large couple vibratios) creates havoc on the very small numbers associated with static balance. Maybe it is similar to what Bill said... trying to model a system with too detailed a model when fewer parameters describe it accurately.

With a 2-plane program, you can verify the results for yourself. Verify that the data in case 1 and and case 2 gives the widely different balance corrections shown, even though the vibration numbers are not far different. You should also be able to verify the static/couple solution, although it's a little more difficult than it has to be because the input data consists of trial runs adding to one bearing at a time, rather than adding couple and static trial weights (but those can be simulated/calculated mathematically).

Even with a numerical example (representing a symmetric rotor) I still am not really sure what makes static/couple better especially in the case of overhung rotor.

Any comments?

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StaticCoupleBeats2Plane.doc (36 Kb, 31 downloads)

Here are the input and output numbers excerpted from the above attachment so you can see at a glance that a small amount of measurement error between case 1 and 2 causes a big error in correction weight for case 2 but doesn’t cause much error for case 3:

#=== Case1 = 2-plane balance with "Exact" (accurate) Values ===
Original
O1 := 10.100
O2 := -9.9000
Add 1<0 trial at bearing 1
T1A := 10.650
T2A := -10.350
Remove above and add 1<0 trial at bearing 2.
T1B := 9.6500
T2B := -9.3500
2-plane solution of this “exact” data
CW1 := -11.000
CW2 := 9.000

#=Case 2=2-Plane Bal. of same machine with Inaccurate Data ======
Original
O1 := 10.200
O2 := -10.000
Add 1<0 trial at bearing 1
T1A := 10.700
T2A := -10.500
Remove above and add 1<0 trial at bearing 2.
T1B := 9.8000
T2B := -9.4000
Two plane balance solution using the inaccurate data
CW1 := -21.200
CW2 := -1.000

#==== Case 3 - Static/Couple Solution with SAME Inaccurate Data===
Same inaccurate input data as case 2. calcs in attachment give:
CW1 := -11.100
CW2 := 9.1000

Note - phase angles are ommitted because there are only two angles in the problem 180 degrees out. Positive numbers should be interpretted as phase angle 0 and negative numbers should be interpretted as phase angle 180

I just noticed that this scenario had only 5% change in magnitude due to trial weight (but that didn't bother the static/couple solution!). I do have a few other scenario's with similar behavior that had larger changes (33%). I can post one of those if anyone is interested.

Also, the reason that I considered case 1 to be more accurate than case 2 was that the case 1 data was generated assuming no cross-coupling from static to couple. It seems that would be the case for a perfectly symmetrical (end-to-end) machine. I'm sure there are cases where this is not so and I'm sure there are cases where the static/couple may give worse solution than the 2-plane. I'm not sure how to tell ahead of time which approach we expect to perform better.

This message has been edited. Last edited by: electricpete,

In the above attachment, I used the effect vectors E1A, E1B, E2A, E2B as input to calculation of the static/couple unbalance. It worked with the equations written because the trial weights A and B both had magnitude/angle exactly 1<0 (A at bearing 1 and B at bearing 2). A more general solution should replace
E1A -> IC11
E1B -> IC12
E2A -> IC21
E2B -> IC22
where IC11,IC21,IC12,IC22 are the same as in the post dated 02 September 2006 09:32 AM

In case it is not obvious, here is an explanation of how the static/couple influence coefficients were calculated from run A and run B where:
Run A = 1<0 at bearing 1. Resulting change in vibration at bearing 1 is IC11 and change at bearing 2 is IC21.
Run B = 1<0 at bearing 2. Resulting change in vibration at bearing 1 is IC12 and change at bearing 2 is IC22.
(IC stands for influence coefficient)

As before we have defined.
IC11 = Effect on vib at bearing 1 of unbalance 1<0 at bearing 1
IC12 = Effect on vib at bearing 1 of unbalance 1<0 at bearing 2
IC21 = Effect on vib at bearing 2 of unbalance 1<0 at bearing 1
IC22 = Effect on vib at bearing 2 of unbalance 1<0 at bearing 2

Similarly define
ICS1 = Effect on static vib of unbalance 1<0 at bearing 1
ICS2 = Effect on static vib of unbalance 1<0 at bearing 2
ICC1 = Effect on couple vib of unbalance 1<0 at bearing 1
ICC2 = Effect on couple vib 2 of unbalance 1<0 at bearing 2

From the above, we can see:
ICS1 = 0.5*(IC11 + IC21)
ICS2 = 0.5*(IC12 + IC22)
ICC1 = 0.5*(IC11 - IC21)
ICC2 = 0.5*(IC12 - IC22)

Now define the following quantities:
ICSS = Effect on static vib of static unbalance 1<0
ICCC = Effect on couple vib of couple unbalance 1<0

We can compute these last two quantities by superposition of the previous four quantities:

the effect on static vib of static unbalance 1<0 is sume of static unbalance from 1<0 at bearing 1 plus static unbalance from 1<0 at bearing 2:
ICSS = ICS1 + ICS2 = 0.5 * (IC11+IC21)+0.5*(IC12+IC22)

the effect on couple vib of couple unbalance 1<0 is sum of couple unbalance from 1<0 at bearing 1 plus couple unbalance from 1<180 at bearing 2 = sum of couple unbalance from 1<0 at bearing 1 MINUS couple unbalance from 1<180 at bearing 2:
ICCC=ICC1+ICC2= 0.5*(IC11 - IC21)-0.5*(IC12 - IC22)

These last two equations are the equations used in the above attachment to calculate the static/couple I.C's and correction weights from the 2-plane trial weight run influence vectors IC11,IC12,IC21,IC22

This message has been edited. Last edited by: electricpete,

There seems to be some errors in my numerical values on the static/couple solution. For example E1A = T1A-O1 should be 10.7 - 10.2 should be 0.5 but is listed as 0.6.

I need to go back and recheck what happened in my program there. I'll post a correction to the numbers in the static/couple calculation. I don't think it will affect the results since I calculated those same results a number of different ways.

A comment regarding 2 plane balancing.

I guess we have to conclude that relationship between calculated CW vs. errors incurred during the test procedure is highly non-linear. Under "errors" I mean mostly inaccurate installation of the trial weights, as oppose to vibration (amplitude and phase ) measurement errors. Apparently, higher sensitivities will affect the error more.

If so, the question 'Why the above relationship is highly non-linear?' still remains.
I am not sure but apparently it has to do with vectorial nature of this calculation.

David

I think the comparison of cases 1 and 2 above shows that a small error in measurement (or trial weight position) can in some circumstances result in large change in calculated correction weight, even though the system remains linear. I agree that this behavior is associated with the linear/vectorial nature of the calculation.

(Adding non-linearities to the system would probably make it act even weirder.)

Yes, I meant that for a 2 plane balancing even of a linear mechanical system, the CW vs. measurement error is non-linear, i.g. quadratic or possibly exponential. If true, that imposes high accuracy requirements while doing 2 plane balancing (unlike 1 plane balancing).

So that theory confirm my gut feeling that make me use 1 plane balancing whenever I can and especially in cases with more than normally unlinear machines and where weight size and position is less than accurate. Thanks. Olov

I don't think we want to go too far with the conclusions, based on a few simple examples. Use as many planes as it takes and look at what the particular situation tells you.

Keeping it as simple as possible without over simplifying the problem is a good way to balance. Many problems take two planes. The static-couple approach is overused and can result in poor balances - Lindsey (in his paper from the late 60's on this) listed some problem areas for the method that he had observed.


J. R. Lindsey used static and couple balance weights combined with a sensitivity factor and a high spot number. The modal component of the vibration is determined by graphical means using vibration data from each end of a machine in one plane. The first mode (one loop) vibration is taken to be the vector average of the two vibrations, and the second mode component is taken to be the vector difference of the two vibration components (Three modes can also be accommodated.). Sensitivity factors relate to the magnitude of either the first or second mode components of vibration calculated as previously stated.


The high spot number relates to the phase angle. Sensitivities and high spot numbers have been developed over time for a variety of machines. The strength of this technique derives from using this historical data, and the method is most often used as a one shot balance method. General Electric Company's field engineers use this technique, as do many utilities to balance their turbine generators. From 1960 until the presentation of the paper [88], Lindsey said that this technique had been used to balance more than one hundred rotors. The desire with this method is to arrive at an adequate balance | usually not the best achievable balance | in an e±cient manner; the technique provided good results with turbine generator units consisting of multiple rotors with long spans. A weakness of the method involves its lack of concern with the cross effect of a static weight on the couple vibration and the effect of a couple weight on the static component of the vibration. Lindsey stated, "When extensive coupling exists among unbalances in several rotors, the methods errs signicantly."

[88] J. R. Lindsey, Signicant developments in methods for balancing high-speed rotors, in Paper No. 69-Vibr-53, Philadelphia, Pa., 1969, American Society of Mechanical Engineers Vibration Conference.

This message has been edited. Last edited by: William_C._Foiles,

Good comments from David, Oli and Bill. I can't argue with the voice of experience. I have mentioned before I don't have much experience balancing (other than observing a few), so I'm just throwing out some ideas.

I would like to learn a little more about the sensitivity discussion Bill is talking about because it sounds like it touches on the same issues we're talking about (when is a certain balance technique likely to fail or be adversely affected by errors). For the time being I'm going follow up on some items from my previous posts.

Let's call the previous numerical example (3 cases) "example 1". I have attached a new corrected example 1 which has the following changes:
1 - use the terminology IC11 vs E1A as mentioned in my post above dated 06 September 2006 05:01 AM above.
2 - Corrected math errors in the blue numbers in the static/couple calculation section as discussed in my post 06 September 2006 10:17 AM. The resulting calculated correction weight stayed the same. Nothing changed in cases 1 or 2. Everything listed in the text of my summary post dated 05 September 2006 10:29 PM remains the same.

Example1Corrected.doc (34 Kb, 11 downloads)

Example 2 – this is very similar to example 1 with the same three cases studied. The only difference is that the numerical values have been changed to give trial weight effect vectors of approximately 25% of the initial vibration. The error between the case 1 and case 2 input data are 3% or less. Still the case 2 fails miserably and the case 3 does fantastic at recreating the exact solution defined in case 1 for this particular set of numerical data (same disclaimers as before… the values are somewhat contrived to produce this effect… may or may not represent any real machines). Summary as follows (details attached):

=== Case 1: 2-plane balance with "Exact" (accurate) Values ===
# Original vibration:
O1 := 2.0200
O2 := -1.9800
#Trial run A after adding trial weight 1<0 to bearing 1
T1A := 2.5450
T2A := -2.4550
> #Remove weight from bearing 1
> # Trial run B after adding trial weight 1<0 to bearing 2
T1B := 1.5450
T2B := -1.4550
> # Two plane balance solution of above data (exact):
CW1 := -2.400
CW2 := 1.600

===Case 2: 2-Plane Balance of same machine with Inaccurate Data===
> #Original vibration at bearing1, bearing 2:
O1 := 2.0800
O2 := -2.0400
> #Trial run A after adding trial weight 1<0 to bearing 1,
T1A := 2.5050
T2A := -2.5150
> #Remove weight from bearing 1
> # Trial run B after adding trial weight 1<0 to bearing 2
T1B := 1.6050
T2B := -1.4150
> # Two plane balance solution using the inaccurate data
CW1 := -8.275
CW2 := -3.025

== Case 3: Static/Couple Sol'n with SAME Inaccurate Data=
#Same data as above for original, trial run A, trial run B
Calculations shown in attachment
CW1 := -2.4600
CW2 := 1.6600

Example2.doc (33 Kb, 13 downloads)

While a balance program can't tell when a solution is correct, a balance program could easily be programmed to identify when the solution is sensitive to small variations in the input.

Given the input data, a balance program could very easily be programmed to calculate 4 types of "sensitivity factors" along with each solution:
1 - % change in correction weight magnitude per % change in input (weight or vibration) magnitude
2 - % change in correction weight magnitude per degree change in input (weight or vibration) angle
3 - degree change in correction weight angle per % change in input (weight or vibration) magnitude
4 - % degree change in correction weight magnitude per degree change in input (weight or vibration) angle
The program could also calculate sensitivity factors in a slightly different form associated with errors in CW magnitude/angle (expected vibrattion error as a functio of percent CW magnitude error or degree error).

There could be preprogrammed limits and a warning could be displayed when limit is exceeded.

Or if an analyst has difficulty he could be able to view these parameters.

High sensitivity to vibration might be a cue to look more closely at whether magnitude is stable/repeatable. (high sensitivity and unstable or unrepeatable measurements would be a bad combination).

High sensitivity to weight angle may or may not be a problem depending on how angle is determined.

If high sensitivity is seen, the analyst might try calculating another solution (static/couple) and view the sensitivity of that solution.

Low sensitivity would not be a guarantee of a good solution but high sensitivity would be a warning flag for a solution method, especially if there is some variability noticed in the data.

Has this been done before? Would it be valuable?

This message has been edited. Last edited by: electricpete,

Pete,

What values were used for the ICs?

What is the math expression for calculating CW? Its analysis may explain the effect of measurement/installation error for a 2 plane balancing.

It appears that examples provided deal with predominantly couple unbalance. Is there where the problem occurs?

quote:

Given the input data, a balance program could very easily be programmed to calculate 4 types of "sensitivity factors" along with each solution:
1 - % change in correction weight magnitude per % change in input (weight or vibration) magnitude
2 - % change in correction weight magnitude per degree change in input (weight or vibration) angle
3 - degree change in correction weight angle per % change in input (weight or vibration) magnitude
4 - % degree change in correction weight magnitude per degree change in input (weight or vibration) angle

Some of this works best with an exact point balance, and a number of people are using least squares (number of measurement points not equal to the number of balance planes).

The condition number (ratio of minimum and maximum singular values) of the influence coefficient matrix gives the sensitivity. Also, the singular value decomposition gives the sensitive and insensitive directions for balance weights and response.

Knowing the sensitive directions can help attack a balance problem. I have used IC’s from similar machines to derive combination shots to have certain desired effects and keep the problem simple (pseudo single plane balance).

One can use the process in reverse to derive weird balance problems. Try the static couple method with this one.

Initial vibration
R0=
[10 @ 0 deg end 1
10 @ 0 degree end 2] pure static

Trial weight end 1, 10 @ 180 degrees
Response R1=[
15.811 @ 251.6 end 1
15.811 @ 18.4 end 2]

Remove weight and add 10 @ 0 deg to end 2
Result with weight on end 2 R2=[
15.811 @ 18.4 end 1
15.811 @ 251.6 end 2]

Will a static weight pair balance this well?

I'm in the process of moving over the next week, so I don't have as much time as I'd like to participate in this interesting discussion.

David - the IC's for the 1,2 coordinate system are calculated and displayed in the attachments for case 3 (applies to cases 2 and 3). Those IC's wouldn't look much different for case 1 in the 1,2 coordinate system. The IC's can also be expressed in the static/couple coordinate system. I described those earlier. I set up this true (case 1) problem with high couple IC and very low static IC and 0 cross-effect IC. This might be typical of symmetric machine (no cross effect) operating very near a resonance which creates out-of-phase motion at bearing 1 and 2 (causes couple influence coefficient to be much higher than static). The measurement error associated with case 2/3 creates an apparent non-zero cross-effect between static and couple IC. Therefore when necessary balance weight is added to correct the large initial couple, it creates a (false) apparent static unbalance which must be removed by static weights. But since the static influence coefficient is very small, it requires a lot of static weight to remove this apparent static error (which isn't even real) which throws case 2 way off. The static/couple solution is immune to this particular error because it ignores that false non-zero static/couple cross effect and therefore no false static unbalance is created by the couple correction (but there are of course other scenario's were normal 2-plane would outperform static/couple - I think this is Bill's example below). Couple doesn't have a unique role in the math - I could come up with a parallel example switching the roles of couple and static with similar results.

Bill - I'll be interested to try your problem when I get a chance. In the mean time, can you explain any more in general terms about what is singular value decomposition why it can help us identify sensitive directions?

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