Single Plane and Multi-Plane Rotor Balancing Using Only Amplitude

W. C. Foiles, BP America P. E. Allaire, University of Virginia

26 September 2006

Introduction

• Deals with balancing rotors using only the amplitude information

• History – old technique with equal balance weights

• Define relative influence coefficients

• Present a matrix formulation to the problem using 3 or more balance weights to calculate the relative influence coefficients

• Single plane and multi-plane balancing possible

• Graphical solution technique for unequal trial balance weights

Turbine Generator Balancing - 1928

G. B. Karelitz ((7)), in the Research Department of Westinghouse Electric &Manufacturing Company, used a three trial weight method to balance turbine generators. This graphical technique used an unbalance finder to locate the mass imbalance; the unbalance finder consisted of four transparent strips held together with a pivot at one end. The method could be used with trial weights of unequal magnitudes.

[7] G. B. Karelitz (1928). Field balancing rotors at operating speed. Power, 67(7):286–289, 1928.

Other Early Work

F. Ribary (8) presented a graphical construction that balanced using only the amplitude taken from an initial run and three trial weight runs. I. J. Somervaille (9) considerably simplified the graphical construction of Ribary (8). Somervaille’s construction is the four circle method generally used today.

[8] F. Ribary (1936). The balancing of masses in rotating bodies. Brown Boveri Review, 23:186–192, 1936. [9] I. J. Somervaille (1954). Balancing a rotating disc, simple graphical construction. Engineering, pages 241–242, 1954.

K. R. Hopkirk

[3] K. R. Hopkirk (1932). Problèmes se rattachant à l’Èquilibrage mécanique des machines Èlectriques. Comptes Rendus du Congrès International d’Èlectricté, 4:403–428, 1932. Paris. [4] K. R. Hopkirk (1940). Notes on methods of balancing. The Engineer, 170:38–39, 1940.

K. R. Hopkirk (3, 4) derived an analytical method using only the amplitude information from the response to solve for the required balance weights. Hopkirk’s method required three trial weights; he, as other authors have, assumed that the trial weights would all have the same magnitude. His solution produces the vector that represents the effect of the initial trial weight referenced to the original vibration whose angle remains unknown. These papers (3, 4) presented a two plane balance solution using only the amplitude of the vibration— no vibration phase was required for a two point exact-point balance. He also formulated multi-plane balancing.

Nomenclature

Geometry of a Balance

Equations for Amplitudes of Vibration

(as an exterior angle, hence the +2 … in the Law of Cosines)

Using the Law of Cosines

Matrix Formulation

Where

- Magnitude of trial weight i

Apply angle addition formula from trigonometry

Note the relation between A, B, and C for a linear noise free system

Greater than 3 Trial Weights

Solve Using Pseudo-Inverse

Analytical Solution – 3 Trial Weights

With

Single Plane Balancing

1. Determine h - 2 2tan , or B h C A B

Aφ = = = +

2. Determine Balance Weight

Complex arithmetic gives magnitude and angle for correction

Example from Barrett and Gunter

Put in Matrix Formulation

Trial Weight magnitude 0.17g per disc – a couple pair used

Solve Example

2 2A B C+ ≠because the three circles do not meet in a single point.

Yielding,

Correction Weight Calculation

Solution Reported in Original Paper †

† The authors used a couple balance wt, referenced to disc 1 disc 2 had an equal weight by opposite in phase

Multiplane Balancing

• Use amplitude only method to calculate relative influence coefficients including cross terms.

• Use amplitudes for initial vibration and complex relative influence coefficients to calculate a balance (initial vibration has a reference angle of 0 – the influence coefficients are relative to these unknown angles.)

• The paper sketches an example

Geometry of a balance – Full information

u – Original Vibration u1 – with 10∡0 u2 – with 15∡120 u3 – with 12∡240 x1 – Effect of 10∡0 x2 – Effect of 15∡120 x3 – Effect of 12∡240

Geometry

Effects vectors length proportional to trial weight magnitude Angle of effects relate to angle of trial weight Effect Vector is an unknown – need it to ‘cancel’ vibration

Geometric Reasoning

• Need to find the effect vector, x1, to cancel vibration, u

• Just need to find effect relative to u or one of the ui

• Know relative angles of effects

• Know relative sizes of effects

• Scale and rotate triangles to line up the effects – will make it easy to geometrically compute x1

Scaled and Rotated Geometry

Origin rotated for two triangles O’ and O’’.

Triangle scaled by tw1/twi.

Geometry rotated to be able to compute the effect vector relative to the original vibration.

Graphical Solution

• Locate centers at

1. Orig. vib. ∡tw1

2. Scaled orig. vib. ∡tw2

3. Scaled orig. vib. ∡tw3

• Draw arcs of scaled vibration

• Intersection determines relative effect, x

• Correction weight = abs(u1/x) tw1

• Correction ∡ is θ

Scaling by abs(tw_1/tw_i)

tw1

tw2

tw3

Reference data for Example

• o=10 ∡ 0

• IC = .5 ∡ 30 - used to generate data

• tw1 = 10 ∡ 0 – u1 = 6.1965684 ∡ 336.20602

• tw2 = 15 ∡ 120 – u2 = 16.916081 ∡ 347.19212

• tw3 = 12 ∡ 240 – u3 = 11.661904 ∡ 30.963757

Conclusions

• Extends amplitude only balancing to allow for unequal trial weights

• Matrix formulation for 3 or more trial weights to calculate relative influence coefficients

• Relative influence coefficients can be used similar to conventional influence coefficients for balancing, single plane or multi-plane

• Relative influence coefficients reference balance weight effects to the unknown original vibration phase

− Thus, relative influence coefficients can not be reused for trim balancing – a limitation of amplitude only balancing

• Graphical technique for single plane amplitude only balancing extended to using unequal balance weights with a graphical derivation

• Amplitude only balancing should not be everyday balance method