In Ron’s post “unbalance or ?”, Steve C and William F mentioned an approach for balancing using sum-of-squares.

At the time we were talking about single-plane balance of an overhung rotor. The vibration showed up on both radial directions and in axial direction at more than one bearing. As William said, we could in theory do several individual balance solutions each using a different vib measurement location. If they all agree, there is no need for anything more. If they differ, there might be need for a "compromise" balance which places the correction weight in a position to minimize the sum of squares of the predicted residual unbalances at the various points.

For kicks, I tried it out in an excel spreadsheet (attached) if anyone is interested.

========= CALCULATION DESCRIPTION: ==============

Assumptions: All vibration is due to (static) unbalance and the machine responds linearly.

The initial vibration measurements before placing trial weight are Oi, were i=1,2,3 etc correspond to your various measurements 4H, 4V, 4A etc

Add trial weight: Utrial

Get another set of vibration measuremetns (Oi+Ti)

Calculate the trial weight vibration effect estimates Ti = (Oi+Ti) – Oi (vector subtraction)

Calculate Influence coefficients ki = Ti/Utrial (vector division)

For a given correction weight, the predicted residual unbalance vibration at each point would be

Ri = Oi + ki * CW (vector arithmetic).

Sum-of-Residual-Squares = R1^2 + R2^2 + R3^2 +etc

Vary CW and look for a solution that minimizes the sum-Sum-of-Residual-Squares

Rotation convention - My assumption in translating results between vector arithmetic and positioning of the weights is that the angle increases as we go in a direction opposite of rotor rotation.

======== SPREADSHEET NOTES ==========
The spreadsheet requires two "add-ins" to work.
1 - Analysis tool-pack. This provides the vector functions: complex, improduct, imdiv, imsum,imsub,imabs,imargument etc.
2 - Solver add-in. Solver is a very powerful tool. It allows you to build a function from many inputs to one output and then it will automatically determine the value of the inputs necessary to achieve a given value of the output (in this case the input was the CW magnitude and angle and the output was the sum-of-residual-squares and we searched for the value of inputs to minimize the outputs). To launch solver, select tools/solver. A dialogue box appears. The selections I used in the dialogue box are shown in the tab entitled "DialogueBoxGraphic" in the attached spreadsheet.

Add-ins are a part of excel which are optionally not installed to save disk space. If you see garbage in the cells, then you don't have the Analysis tool-pack installed. If you don't have the choice "solver" on the tools menu, then you don't have the solver add-in installed.

To install an add-in, select tools/add-ins and check the appropriate boxes, and press OK. At work the add-ins were immediately installed. At home on my Office2000 computer, I put the office CD in, but I'm not sure if it was required. At home on my Office97 computer, "solver" did not appear to be an available add-in although I didn't try running the Office97 installation disk. After you install the solver add-in, I think you need to exit and reenter excel before the Tools/solver choice shows up.

To use the spreadsheet after you have installed the add-ins, enter the data into the green boxes. This is the original vibration at all desired points, the trial weight position, and the vibration after installation of the trial weight. Then run solver with choices as shown in the DialogueBoxGraphic tab. Select "keep solver solution" to plug the solution into the trial weight position, and you can view the programs prediction of what the residual vibration would be at each point.

The spreadsheet is currently set up for 6 points. If you had less, you could just 0's into the O and (O+T) column vibrations for the points you don't need.

Note that there are two formats for displaying the vectors. One is magnitude and angle as we are familiar with. The other is the "complex" for that excel works in. I included all three for each variable. The inputs and outputs use the magnitude and angle format. The program calculates using the complex format.

When you launch the program it will warn you that there is a macro. That is the "getformula" macro that I installed in the spreadsheet. It allows you to see the formula's used in the calculations in (column I). It is harmless and I promise it won't hurt your computer.

========== TRIAL CASE ======================
I ran a trial case. If anyone else has a similar program, I'd be interested to know if their results agreed.

***Input data:***
Original vibration O was as follows (magnitude and angle)
4H 0.29 29
4V 0.403 34
4A 0.34 54
3H 0.194 60
3V 0.26 53
3A 0.343 218

Trial weight was assumed 1 Unit at angle 45

Vibration O+T was assumed as follows (I created these somewhat randomly, assuming that all points increased slightly and rotated slightly toward higher angle).
0.4 45
0.5 75
0.45 90
0.3 75
0.4 90
0.5 250

I initially plugged in random Correction Weight of 1 Unit at angle 0 (actually that initial guess doesn't affect anything). I ran the spreadsheet as described above to find the CW that would minimize sum of squares of residual vibrations.

***Output data:***
The results was CW -1.22 units at angle -33 degrees which is predicted to give the following residuals in an ideal world: (highest vib 0.16)

0.16 61.49
0.11 -47.35
0.04 -10.08
0.12 110.91
0.04 -142.47
0.03 -62.95

=============== TWO PLANE BALANCE ===============

I don't see any reason why the same thing couldn't be done for two plane balance at planes A and B with more than two monitored vibration points.

Measure original vibration Oi for i=1,2,3 etc
With trial weight Uta measure Oi+Tai for i=1,2,3. etc
Then repeat with weight Utb to find Oi+Tbi for i=1,2,3. etc
Calculate Tai = (Oi+Tai)-Oi and Tbi = (Oi+Tbi)-Oi
Calculate Kai = Tai/Uta and Kbi = Tbi/Utb
If we add correction weights Ucwa and Ucwb, we predict residuals are
Residual Ri = Oi + Kai Ucwa + Kbi Ucwb
Use excel solver as above to find values of Ucwa and Ucwb that would minimize the sum of the squares of the residuals.

==============FINAL QUESTIONS OR COMMENTS ======

OK, now after all this work, I wonder when is this procedure really beneficial. As Willliam said, when the individual balance solutions based on individual vibration measurements would not reduce all the points satisfactorily. It seems to me that occurs either when there is something other than unbalance or a large measurement error. Maybe compromise balance is not the perfect solution in those cases (would rather correct the other factors or correct the non-unbalance force). But maybe sometimes it's the best we could do in that situation. I would be interested to hear what you guys think... when would this type of approach be beneficial?

By the way, it sounds like Ron had a somewhat easier method that works for him on overhung rotors... a 2-plane balance run with one sensor in the vertical on the nde bearing and the other plane being axial on the de. It kind of makes sense to me that one axial and one radial would be nice. I wonder what are the advantages of selecting those two particular choices.

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

SinglePlaneBalanceMultipleVibPoints.xls (84 Kb, 22 downloads)

I ran some test cases on a balance program. Try these out if you like some test examples.

This is from test one. The program printed out the IC’s; I can’t find the trial run, but you could synthesize it if you need.

The underlying principle was based on a few balances. Look at what least squares does to this. Of course I don’t use least squares often – actually, I don’t balance much anymore.

Origianal run
Channel 1X Amp 1X Phs
ch1 4 mil 0 Deg
ch2 4 mil 90 Deg
ch3 4 mil 180 Deg
ch4 4 mil 270 Deg



Influence Vectors
Weight Plane Channel Speed Load Magnitude Phase
bp1 ch1 3600 rpm 0.000000 kg/hr 0.009769 mil/g 234
bp1 ch2 3600 rpm 0.000000 kg/hr 0.009769 mil/g 324
bp1 ch3 3600 rpm 0.000000 kg/hr 0.010549 mil/g 235
bp1 ch4 3600 rpm 0.000000 kg/hr 0.009769 mil/g 144

------------------------

A second example that can give many balance programs heck. This comes from a linear computer model, no tricks.

Again the program printed out the ic’s, but what ever I did I didn’t get the trial run. This one is based on another idea to give balance programs trouble.

Influence Coefficients as from Balance Program
Station 2 Bearing 1 0.0104 um/g 153 Deg
Station 2 Station 3 0.01823 um/g 168 Deg
Station 2 Bearing 2 0.02818 um/g 176 Deg
Station 3 Bearing 1 0.01655 um/g 166 Deg
Station 3 Station 3 0.01975 um/g 169 Deg
Station 3 Bearing 2 0.02258 um/g 172 Deg
Station 5 Bearing 1 0.02818 um/g 176 Deg
Station 5 Station 3 0.02191 um/g 171 Deg
Station 5 Bearing 2 0.0104 um/g 153 Deg
Station 2, 3, and 5 aree the locations where the balance weight was added.

Small Amplitudes of Vibration to start – Case a
Bearing 1 5.015 um 0 Deg
Station 3 7.97 um 180 Deg
Bearing 2 3.364 um 0 Deg

What do you get for balance weights, and does it surprise you?

Balance Program Solution for the above vibration
Station 2 28781.62 g 210 Deg
Station 3 42331.11 g 31 Deg
Station 5 14573.27 g 211 Deg
Note: The units above and in other solutions in this section should have units of kg-μm not g.

Same ic’s as 2 a case,
Larger Amplitudes of Vibration to start – Case b – Same IC’s
Bearing 1 54.24 um 168 Deg
Station 3 59.26 um 169 Deg
Bearing 2 59.56 um 170 Deg



Balance Program Solution for the above vibration
Station 2 913.3738 g 218 Deg
Station 3 1595.761 g 149 Deg
Station 5 917.7816 g 197 Deg

Predicted vibration with balance shot
Channel 1X Amp 1X Phs
Bearing 1 0.993098 um 259 Deg
Station 3 0.732208 um 262 Deg
Bearing 2 0.84387 um 254 Deg
Note: This is a reasonable solution. Add the noise vibration in case a (or a portion of it) to this and an un-reasonable solution results.

Pete - I've attached a spreadsheet with some real data.

This is from a vertical motor driving a two-stage fan (one wheel above the motor and one wheel below the motor). We have tri-axial accelerometers mounted at each motor bearing. The naming convention is bearing location (Upper or Lower) and accelerometer axis (1, 2, or 3). Axis 3 is the axial measurement, axis 1 and 2 are radial measurements.

The first table shows the initial data, the 1st trial weight data and the resulting correction weight calculations.

The second and third tables show the predicted responses of all 6 measurements using each of the calculated correction weights. I've highlighted the largest amplitude in each group.

The fourth table shows the predicted responses of all 6 measurements using an averaged correction weight and using the least square method. The largest amplitudes are highlighted. As you can see, the least square method minimizes the highest vibration amplitude.

In reality, we performed a two-plane balance on this fan resulting in a maximum vibration amplitude of 0.09 ips.

Another note - The Excel solver will be tough to utilize for a two-plane balance because you need to vary the correction weights in both planes simultaneously.

Balancing_Example.xls (18 Kb, 26 downloads) Balancing Data

Thanks William and Steve.

I will have to think about your examples for awhile William. At present my spreadsheet is not equipped to solve problems in that format. Interesting to see such high balance weight recommendations.

Steve - I tried your example and came up with same results. Since you had exactly 6 points (same as my example), I just cut and paste/transpose the O and O+T into my spreadsheet and added the TW. The result was -18.6 angle -86. To convert to positive magnitude I had to add 180 degrees which gives 18.6 angle 83.9.

Your example provided a good demonstrationt that no amount of calculation will solve a problem using single-plane solution if there exists a 2-plane unbalance. (The two plane worked better than the 1-plane-minimized-sum-of residuals-squared).

Regarding minimizing sum of residual squareds for 2-plane - I don't see why excel can't do it. As far as I know there is no limit on the number of inputs for a solver solutions. My spreadsheet used 2 inputs (mganitude and angle of trial weight0. 2-plane version would require four inputs (2 magnitudes and 2 angles).

My older computer with Office 97 has a stripped down version of solver called "goal seek". It only allows one input.

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

I created a 2-plane spreadsheet very similar to the above single-plane spreadsheet.

Since I don't have machine data to work with, I generated the input data using a linear model (see tab Create2PlaneInputData). I assumed complex influence coefficients (different for each point) and a plane A and plane B original unbalance. Each point original vibration was computed as the vector sum of a plane A unbalance Times a plane A influence coefficient, a plane B unbalance times a plane B influence coefficient, and a random noise component between 0 and 0.08 (also random angle). The results of trial run were generated using the same model. The solver program/worksheet didn't "know" anything about the linear model other than the original vibration, trial run weights and vibrations for trials at planes A and B. (The solver worksheet doesn't know the stiffness coefficients used, that would be cheating).

The input data to the solver worksheet was as follows:
Original vibration:
0.16 23.34
0.42 46.75
0.24 124.07
0.31 179.61
0.22 163.73
0.36 164.44

Result of trial adding 1 angle 0 at plane A:
0.26 22.83
0.60 36.69
0.37 109.40
0.30 176.40
0.23 155.12
0.36 163.16

Result of trial adding 1 angle 0 at plane B:
0.10 8.88
0.39 48.46
0.27 136.29
0.41 147.35
0.27 140.79
0.41 137.02

The solver solution was
Plane A CW -2.15 at 37.63 degrees = 2.15 at 218 degrees
Plane B CW -1.55 at 88.31 degrees = 1.55 at 268 degrees

The resulting predicted residual vibrations were:
0.053 0.022 0.026 0.034 0.017 0.042

These residuals were a little lower than the random noise added in the linear model. I guess to prove the value of the method I should try again with a little more noise, and compare the results to a solution using only 2 sets of bearings. Not today.

TwoPlaneBalanceMultipleVibPoints.xls (124 Kb, 22 downloads)

I created another file which is the same as the last, but adds at the bottom of the main sheet a standard 2-plane balance solution for each possible pair of the six measurement points (1/2, 1/3, 1/ 4, 1/5, 1/6, 2/3, 2/4, 2/5 2/6, 3/ 4 etc)

I added a higher “noise” error to the 6 input points (added initial vibration error between 0.05 and 0.25 at varying angles for each point).

The solutions are tabulated and highest residual vibration listed in a new column on the right. The minimize-sum-of-squares method gives a highest residual of 0.28. The individual pair solutions give highest residuals ranging from 0.42 to 1.82. Also if you look at the corerction weights, the sum of squares method comes closest to equal and opposite the original unbalance (shown in Create2PlaneInputData tab as 2.15 angle 35 at plane A and 1.43 angle 86 at plane B).

This spreadsheet appears to demonstrate superior performance of the sum of squares method WITHIN THE ASSUMPTIONS of this model.

In my mind the effect of the “noise” added to the initial vib data in this model would be equivalent to the presence of a forcing function like misalignment in addition to the unbalance. The initial vibration from that non-unbalance function distorts the calculated influence coefficients somewhat.

The minimize-sum-of-squares process inherently tends to ignore this noise somewhat, since trying to correct noise on one point may result in higher vib on another point and doesn’t help the program meet it’s objective of simultaneously minimize all points. The minimize-sum-of-squares process tends to focus on corrections which benefit all points, which tend to be corrections associated with actual unbalance. Do you agree? Or am I off-base?

==================
By the way, if you want to update this spreadsheet with your own data, enter the input data into the green cells (original vib, trial weight plane A magnitude/position, trial A vibration, trial B magnitude/position and trial B vibration). If you have less than 6 measurement points, enter 0's for points you don't need.

The pairs of standard 2-plane balance solutions at the bottom of the spreadsheet will automatically udpate as you put in your data (you don’t even need solver add-in). To refresh the sum-of-residual-squared solution, you would need to re-run the solver after you enter your input data.

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

TwoPlaneBalanceMultipleVibPointsAddIndividualPairs.xls (176 Kb, 17 downloads)

quote:

Originally posted by electricpete:
... Your example provided a good demonstrationt that no amount of calculation will solve a problem using single-plane solution if there exists a 2-plane unbalance. (The two plane worked better than the 1-plane-minimized-sum-of residuals-squared)....

I want to clarify that the two-plane balance was performed using the least-square methodology. I think your post on 11/07/05 clearly indentifies that this method works well at reducing the amplitude of the largest measurement without "hurting" the other points.

FYI - I ran your numbers for the two-plane balance and got similar answers (2.12 @ 216 and 1.57 @ 269).