Does anyone have any notes or some sort of discussion on performing 2-plane balancing manually? I'd appreciate if anyone can share it with me. Kinda wondering on how to do it using vector calculation and iteration method.

another thing, is it possible to do 2-plane balancing with just 1 trial run? I've heard it can be done. If can, it'll be helpful if anyone can share it. Thanks in advance

It takes 3 equations (initial run + 2 trial runs) to solve for 2 unknown quantities (left and right plane unbalance)

Are you talking about static/couple method or full two-plane balance?

Do you mean manual as in excel spreadsheet, or manual as in using graph paper?

For static/couple method, graph paper method is manageable and understandable.

For full 2-plane balance, there is a step-by-step cookbook procedure which can be followed to solve the calculation using graph paper, but it doesn't add much intuition imo. It gets pretty messy and imo focuses us on smaller calculational details while distracting us from the big picture of the underlying math problem. As Danny said, the math problem amounts to solving two vector equations in two vector unknowns. The equations listed as equation1 and equation2 on the middle of page 2 of the attachment. The general solution is listed just below those - a little messy, but it is the solution that goes with those two equations. . In summary, I would say it is good to understand what math problem is being solved, but the actual solution of a full 2-plane balance is best left to computer imo. I can provide a spreadsheet if you are interested.

TwoPlaneBalanceTrialATrialB_Maple.PDF (89 Kb, 75 downloads)

There are companies that use (very) good knowledge of the rotor dynamics of a specific rotor type/machine and from that can extract the information eg. the equations from logging the run up or down vector data. Similar claims go for using 1 transducer for 2-plane balancing, placed "anywhere". I would guess some specific rotor dynamic criteria must be fulfilled. I doubt it will work in all and general cases, but maybe most rotors are such that it is possible? Anyway if you pass at least one resonance? Anyway, this procedure is in no way less mathematical than the classical "simple" vector solution. For graphic solution it would need like minimum 4 vector plots or maybe 6 to keep track of the data? I don´t think it´s practical. Olov

I’ll leave the math to Pete. But two planes balancing with one trial run is possible as long as you had saved one full balancing task on that specific machine (With two trials runs) before. Assuming that the system reaction remain the same it is possible to use prior calculation from the computer to do two planes-one shot balancing jobs. You won’t probably get right on the money the first time but a second trim will get you there. It just depends on your system, CSI works.
Best regard, Marcel

Basically the computer calculation is the way to go. From IRD back in the '70's (they had it right and a good program) to Bently to almost all I guess (haven't used all). But as mentioned if you store your data you can recall it later and do a 'trim' balance speeding up the process. I have taken like rotors and performed a trim on one I hadn't previously balanced.

But doing it by hand just for your edification and practice - why not use a test rotor in the shop?

Although the procedure may have been around prior to the 1930's, E. L. Thearle of General Electric wrote a paper on the method I think in 1934.

The method is documented in the attached, which covers large hydro balancing.

There is a possibility that Bill Foiles can contribute the original Thearle paper. I know he has written on various balancing methods, including the Thearle method.

John from PA

Hydro_balancing.pdf (1,569 Kb, 67 downloads)

The Vibration Institute still teaches the semi-graphical 2-plane balance. It is in some of their literature and well explained with a step-by-step procedure.

I do not have a digital copy of Thearle's paper (copyright expired?).

In Rathbone's paper from 1929 (Turbine vibration and balancing in Trans. Amer. Society of Mech. Eng, 51 Part 1 pages 267-284) he mentions in a footnote that an undergraduate at the University of Copenhagen named Nils O. Myklestad had worked out a solution to the two plane problem, but Rathbone neglected to show the details. Some may know Mykestad as getting co-credit for the transfer matrix method with GE's Prohl. Rathbone did present an iterative method.

If you are looking Thearle's work was:

E. L. Thearle, Dynamic balancing of rotating machinery in the field, Trans. of the American Society of Mech. Eng, Journal of Applied Mechanics, 56 (1934), pp 745-753.