introduction to balancing - maintenance · iso 2953 describes balancing machines, and iso 3719 has...

INTRODUCTION TO BALANCING

WILLIAM C. FOILES

1. INTRODUCTION TO BALANCING

This chapter will introduce the fundamentals of balancing. Balancing

will be defined and explained as to what the process encompasses and

the physical principles behind balancing. This chapter examines aspects

of balancing such as balance weights and auxiliary computations.

The most important rule to understand and remember ifone plans to balance rotating equipment safety rules every-thing. If one is not comfortable with a situation then one shouldevaluate whether to continue or not. To balance one must op-erate and make changes to rotating equipment. This can eas-ily lead to harm to people, the environment, and equipment.Safety comes first.

This material can not attempt to cover the many aspects related to a

safe balancing job. High energy will be present near all balancing activi-

ties. Dangers may include rotating parts, temperature extremes, electri-

cal energy, high pressures, hazardous chemicals, enclosed spaces, explo-

sive gases, poisonous gases (including suffocating gases), heights, falling

material, trips and falls, driving (within the facility or to and from the

facility), magnetic fields, and potentially many other location specific

hazards. Balancing equipment itself may present hazards such as strobe

lights if used and fish tails – a seismic device used to measure shaft abso-

lute vibration; it looks like a fish tail and is used in direct contact with the

shaft; both of these if used for balancing can be dangerous, and a number

of facilities have restricted or banned their use.

One always needs familiarity with the local safety practices prior to

starting any job. Good practice includes reviewing the safety aspects of

any job, balancing included.

2. STANDARDS REGARDING ROTOR BALANCING

What is balancing? ISO DIS 19499 E, currently a draft International

Standard introducing the balancing standards, says the following in re-

gard to balancing.

Date: August 14–17, 2006 UVA Rotordynamics Short Course.

1

2 WILLIAM C. FOILES

Balancing is a procedure by which the mass distribution

of a rotor (or part or module) is checked and, if necessary,

adjusted to ensure that balance tolerances are met.

Rotor unbalance may be caused by many factors, in-

cluding material, manufacture and assembly, wear during

operation, debris or an operational event. It is inevitable

that every rotor, even in series production, has an individ-

ual unbalance distribution.

New rotors are commonly balanced by the manufacturer

in specially designed balancing machines before installa-

tion into their operational environment. Following rework

or repair, rotors can be re-balanced in a balancing machine

or, where appropriate facilities are not available, the rotor

can be balanced in situ, see ISO 20806. In the latter case

the rotor is held in its normal service bearings and support

structure and installed within its operational drive train.

The unbalance on the rotor generates centrifugal forces

when it is rotated in a balancing machine or in-situ. These

forces can be directly measured by force gauges mounted

on the structures supporting the bearings or indirectly by

measuring either the motion of the pedestal or the shaft.

From these measurements the unbalance can be calcu-

lated and balancing achieved by adding, removing or shift-

ing of correction masses on the rotor. Depending on the

particular balancing task the corrections are performed in

one, two or more correction planes.

From this it may not be evident what occurs during rotor balancing.

The first paragraph in the above quote relates more to balancing rigid ro-

tors. Generally, the goal to in situ balancing is to reduce vibration levels

to acceptable levels. When one is balancing it is good practice to agree

upon the goal prior to commencing the balance; otherwise, it can be dif-

ficult to know when one has finished the balance. Balancing a real rotor

in the field can consume a great deal of time, and one will have limita-

tions placed on the time spent.

Take time required into consideration when planning a balance job.

Advice the customer or operator how long the job may take and agree

to have the required time, resources, and number of runs ahead of the

balance. When in doubt use a pessimistic estimate.

2006 UVA ROTORDYNAMICS SHORT COURSE 3

Without too great of a diversion, some coverage of relevant standards

will be given. Many countries’ standard organization have their own stan-

dards regarding balancing, and many of these are adopting the ISO stan-

dards and aligning with these. The aforementioned draft document, ISO

19499, will review the balancing standards if accepted.

ISO 1925 contains the vocabulary used in balancing, and ISO 2041 has

the vocabulary for vibration and somewhat for balancing. This is unusual

for standards to have separate places for the terms, and it makes it diffi-

cult to maintain consistency.

2.1. Balance Quality and ISO 1940. This article does not cover low speed

balancing in significant detail. ISO 1940-1 and ISO 1940-2, parts 1 and

2, cover this subject. The balance quality grade, G, is introduced in ISO

1940-1.

ISO 1940 reasons that the permissible residual imbalance of a rigid ro-

tor (Actually, the term concerning rigid rotor has recently changed, adding

some confusion.) is proportional to the mass of the rotor, i.e. Uper ∼ M .

It then defines a specific residual imbalance, similar to mass eccentricity,

as

eper =Uper

MNext, the standard states that experience has shown that eper varies

inversely proportional to rotor angular velocity, Ω.

Thus, this line of reasoning results in the following.

eperΩ= constant

G , the balance quality grade is defined as eperΩ in units of mm/s. For

a thin disc the balance quality would equate to the peak velocity of the

mass eccentricity. Theoretically, on a soft bearing balance machine, with

the disc running well above the resonance, this would be the peak veloc-

ity of the measured vibration.

Suggested balance grades are in ISO 1940-1 for rigid rotors, or rotors

in a rigid state. Generally, balance grades have a separation factor of 2.5.

This gives typical grades of 6.3, 2.5, 1, etc.

To obtain the permissible imbalance in terms of mass–length or weight–

length, one reverses the equations.

(1) Uper =GM

Ω=

MeperΩ

Ω

For consistent units in the above M is in g, Ω is in rad/s, and Uper is in

g-mm. Use of other units such as ounces, pounds, and rpm is possible.

4 WILLIAM C. FOILES

Below Ui noz is the permissible imbalance in ounce-inches, W is the rotor

weight in pounds, and N is the rotor speed in rpm.

(2)

Ui noz =W lb16 oz

1 lb

1 in

25.4 mm

60 sec-per-min

N rpm(2π rad/rev), or

Ui noz = 6.0153GW

N

2.2. API vs. ISO 1940 Balance Quality Grade. One may notice the simi-

larity in form to Equation 2 of the balance tolerance in API 617, American

Petroleum Institute compressor specification, of 4W /N . This API speci-

fication provides a relatively tight equivalent balance quality grade G of

0.665.

API =4W

N, or

G API =4

6.0154= 0.665

2.3. Other Standards. ISO 2953 describes balancing machines, and ISO

3719 has the symbols for balancing machines and their associated in-

strumentation. Balancing machines have inherent safety concerns; ISO

7475 covers balancing machine enclosures and protective measures for

the measurement station.

ISO 8821 pertains to shaft and fitment key conventions. In the Unites

States this has not been too controversial; mostly, half keys have been

used. However, internationally this has not always been the case. In fact,

because the standard called for full shaft keys and no disc key at one time,

some countries or organizations switched their conventions.

Presently, the ISO 8821 calls for the use of half keys. Since this was not

always the case, one may run into rotors or pieces that were balanced in

a different manner. The standard covers this also. Additionally, practical

methods are examined to balancing with keys.

ISO 10814 deals with the susceptibility and sensitivity of machines to

imbalance. It has some interesting points, but I have seldom seen it used.

It may be too theoretically oriented for most people practicing balancing.

It does contain good pertinent information, which may not make a differ-

ence to the low level balancing practitioner trying to balance a low speed

rigid fan.

Methods and criteria for balancing flexible rotors is covered by ISO

11342. If one desires to balance flexible rotors, one should read this stan-

dard.


ISO 20806 deals with in-situ balancing of medium and large rotors, cri-

teria and safeguards. This provides guidance on the type of balancing

covered here. ISO 20806 states the following about in-situ balancing.

Balancing is the process by which the mass distribution

of a rotor is checked and, if necessary, adjusted to ensure

that the residual unbalance or the vibrations of the jour-

nals/bearing supports and/or forces at the bearings are

within specified limits. Many rotors are balanced in spe-

cially designed, balancing facilities prior to installation into

their bearings at site. However, if remedial work is carried

out locally or a balancing machine is not available, it is be-

coming increasingly common to balance the rotor in-situ.

In-situ balancing is the process of balancing a rotor in

its own bearings and support structure, rather than in a

balancing machine....

For in-situ balancing, correction masses are added to

the rotor at a limited number of conveniently engineered

and accessible locations along the rotor. By so doing the

magnitude of shaft and or pedestal vibrations and/or Me-

chanical vibration U In-situ balancing of rotors U Guid-

ance, safeguards and reporting unbalance will be reduced

to within acceptable values so that the machine can oper-

ate safely throughout its whole operating envelope. In cer-

tain cases, machines that are very sensitive to unbalance

may not be successfully balanced over the complete oper-

ating envelope. This usually occurs when a machine is op-

erating at a speed close to a lightly damped system mode,

(see ISO 10814), and has load dependent unbalance.

ISO 20806 may be helpful to the balancer. Also, this document can

provide guidance to someone purchasing a balance job; it gives sample

report features in an appendix. I recommend reading and understanding

this document.

API 684 is a tutorial on rotordynamics and balancing. Understand-

ing rotordynamics, forced response, resonances, and mode shapes, will

greatly improve ones skills at balancing from determining the appropri-

ateness of balancing to placing the trial weights.

API 670 covers machinery protection systems. Other ISO standards re-

garding vibration may also pertain to a particular balance; customers of-

ten ask to meet some standard. Some may find some of the ISO criteria

rather generous, and the balancing goal may be much less than one of the

6 WILLIAM C. FOILES

boundaries – in general one should not provide a balance that leaves the

vibration close to an alarm condition, at least as a permanent solution.

3. THE BALANCING PROCESS AND STANDARD ASSUMPTIONS

This section reviews what occurs during a rotor balance in situ and

what assumptions one uses in general to affect a balance.

The common assumptions used in balancing are listed below.

(1) The mass of the balance weight additions or removals are insignif-

icant in comparison to the overall rotor mass. Additionally, this

means that the system dynamic properties do not change as a

function of the balancing process. Balance weights act as 1× or

synchronous forces applied to the rotor. As such, the balance

weight should not deform the structure of the balance plane, ei-

ther.

(2) Linearity, the rotor system responds linearly, specifically to 1×

synchronous force input (balance weights). Given linearity, ro-

tor system response to balance weights attached to the rotor that

generate a force synchronized to rotational speed or frequency

will occur at a frequency equal to 1×. Response to a time delay

in the input force (i.e. angular positioning of the balance weight)

results in a similar delay in the 1× response. This combines with

the typical linearity above to give a complex valued linearity.

(3) Repeatability, the rotor system exhibits repeatable behavior.

(4) Balancing addresses only the 1× response, generally vibration but

sometimes force.

3.1. Balance Weights. Balance weights consist of eccentric addition or

removal of mass from the rotating element. The actual mass involved in

balance weight addition or removal leaves the total rotating mass rela-

tively unchanged.

For example, on a class of large gas turbine that I used to balance regu-

larly with total weight around 100,000 pounds for the rotor I would typi-

cally use from 0.83 pounds (375 g or 13.23) to 1.38 pounds (625 g or 22.05

oz). These weights did not affect the system dynamics, being such a small

fraction of the overall mass of the rotating element.

So, how does such a relatively small amount of mass help to balance

a rotor? One must remember that locally in the rotor force results from

mass eccentricities spinning about the center of rotation. The proper full

units for a balance weight includes both the mass and the eccentricity,

m–e , where m stands for mass and e for the radius or eccentricity of the

mass.

Force is mass times acceleration. For circular motion acceleration is


a =v 2

e= eω2 v - velocity; ω - angular velocity

For a mass revolving at a circular eccentricity the force is given below.

(3) F = meω2

To use Equation 3 one must use consistent units. A 500 g weight at a 24

inch radius (61 cm) spinning at 3600 rpm generates 9738 pounds of force

(43,345 N). This can be seen by converting 500 g to pounds, 500 g = 1.102

pounds and converting 3600 rpm to rad/sec 3600 rpm = 376.99 rad/sec.

In U.S. customary units pounds represents the force created by the mass

of 500 g at 1 g. 1 g = 386.089 in/sec2. Thus, Equation 3 gives

F = meω2

=1.102

386.089(24)(376.99)2

= 9738 pounds

Calculating the force created by balances weights is good practice. If

there is any question as to the safety of adding the weight, further con-

siderations may be appropriate.

4. LINEARITY

In many ways, linearity is what makes balancing methods work. In

the context of balancing rotor systems linear response to balance weights

implies that the balance weight produces a response at 1× the speed or

frequency of the rotor and only at 1× the frequency of the rotor. This

assumes the use of a linear scale and not one like db’s or other non-linear

measure.

Other measures such as force or strain could be used; although these

have few uses in practice, unless used for a special test. For in situ bal-

ances, vibration values determine the need and quality of the balance;

balance machines may use force measurements. Whatever measure one

uses, it should respond linearly to the force induced by the balance weights;

for example, velocity in mm/s will work for balancing, but V dB’s will not

work in the usual manner, since it is not a linear measure with regard to

the effect balance weights have.

Linearity as used in balancing means two things, generally. First, the

amplitude responds proportionally to the magnitude of the balance weight,

mass-length. Second, the phase of the response changes directly to changes

8 WILLIAM C. FOILES

in the angular location of the balance weights. One may call these ampli-

tude and phase linearity.

The phase linearity corresponds to the response following time delays

to the input force that results from the balance weight angular position-

ing. Both amplitude and phase linearity should depend upon the cir-

cumferential symmetry of the rotor. Many rotors do not have perfectly

symmetric axial cross sections, such as motors and generators, but this

assumption has proved reasonable over the years. Balancing crankshafts

may be another story.

4.1. Linear Response, Influence Coefficients, and the Fundamental Bal-

ance Equation. In this section the following notation will be used.

NOTATION FOR INFLUENCE COEFFICIENTS

H Complex influence coefficient

H Matrix of influence coefficients

h |H |

ω Rotor speed

t time

θ ∠(H) or the lag angle to a unit imbalance at

angle 0

φ Balance weight angle referenced to the bal-

ance plane in lag angles

W Arbitrary balance weight with amplitude, w

and angle φ

R Response due to balance weight

R Multiple response measurements, vector of

measurements

R0 Initial response vector

B Vector of balance weights

Placing a unit balance weight on a balance plane at a particular loca-

tion produces in a linear system a response of amplitude h at the mea-

surement location. We assume that the location on the balance plane

does not matter; the response due to a unit balance weight always pro-

duces an amplitude of h. Of course, this is only the effect due to the bal-

ance weight and does not include other vibration that may be present.

This property comes from linearity and the angular symmetric response

that has been assumed. This response will be sinusoidal of the following

form, h cos(ωt −θ) with the unit balance weight placed at the 0 reference

location for the balance plane.

If one uses an oscilloscope to display the vibration signal and triggers

or starts the display on the shaft trigger signal Figure 1 shows a 1× signal


with peak-to-peak (pp) amplitude of 3 and a phase lag of 45 degrees, and

a second plot with an amplitude of 4 pp with a phase of 270 degrees.

FIGURE 1. Amplitude of 3 pp, phase of 45 – Amplitude 4

pp, phase 270

timeDiv :

5ms/div

Channel A :

1 V/div

Channel B :

1 V/div

XY

OFF

OffsetB

5

OffsetC

0

OffsetA

0

timeDiv :

5ms/div

Channel A :

1 V/div

Channel B :

1 V/div

XY

OFF

OffsetB

0

OffsetC

0

OffsetA

9

The trigger in Figure 1 starts the waveform displays. To measure phase

one starts on the left side, the time origin of an oscilloscope, and mea-

sures phase from this point. If one knows, as in these instances, that the

waveform is 1× filtered then one can measure the time for one cycle and

the time to the first peak. The phase angle in degrees is given by 360 times

the ratio of the time to the peak divided by the time for one cycle.

One cycle in Figure 1 is 20 ms. The time to the peak of the left display

is 2.5 ms. The amplitude is 3 divisions pp or 3 volts – this assumes that 1

sensor unit equals to 1 volt. Thus, the phase for the left display is

θ1 = 3602.5 ms

20 ms= 45

Similarly, the right display has time lag of 15 ms until it reaches its peak.

This gives the phase lag of the right display as

θ1 = 36015 ms

20 ms= 270

To gain familiarity with the definition of phase lag two more figures are

given. Verify the phase lags in Figure 2 yourself. What are the amplitudes?

Use the lag angle convention, the most common, to measure angles

in the balance plane from the reference 0 angle. With this convention

adding a unit weight at angle φ produces a response of h cos(ωt −θ−φ).

10 WILLIAM C. FOILES

FIGURE 2. Phase of 90 – Phase 0

timeDiv :

5ms/div

Channel A :

1 V/div

Channel B :

1 V/div

XY

OFF

OffsetB

5

OffsetC

0

OffsetA

0

timeDiv :

5ms/div

Channel A :

1 V/div

Channel B :

1 V/div

XY

OFF

OffsetB

0

OffsetC

0

OffsetA

9

The complex valued influence coefficient is generally defined in terms

of amplitude and phase as (h,θ), the response with the lag angle, which

is equivalent to a time delay at the given rotor speed.

At this point something strange occurs in general practice. Either one

views the angles as implicitly negative or accepts the following complex

formulism. Usually, one can overlook this fine point but not always. Basi-

cally, we move to a lag angle world at this point.

The complex formulism used to describe balancing uses Euler nota-

tion for complex exponentials. The complex influence coefficient is given

below, with i 2 =−1. Remember, you are now in the world of lags.

(4)H = (h,θ) = he iθ

= h (cos(θ)+ i sin(θ))

The response to a balance weight W = (w,φ) in this notation would be

(5) R = HW as a multiplication using complex arithmetic

Equation 5 will lead to the basic equation used for balancing from a

single plane to multiple planes and speeds. Write this equation in matrix

from as in Equation 6 for multiple balance planes and/or multiple mea-

surement points. The measurement points may include various sensors,

different speeds, or a variety of operating conditions.

The basis for influence coefficient balancing is given in Equation ??,

rewritten below for reference. The goal of any balance effort is to achieve

an acceptable residual response after balancing, R. Thus, some type of


minimization is used to find the balances weights B that minimize this

equation. Here bold face represents matrices and vectors.

(6) R = HB+R0

The response R and the influence coefficient matrix, H have the same

number of rows. Again, the response can include sensor locations and

data from various discrete speeds. The column dimension of H matches

the number of balance weights, i.e. the row dimension of H.

4.2. Real and Complex Valued Influence Coefficients. Most mechanical

balancing has used complex influence coefficients where the amplitude

and phase of the vibration response and the balance weights is viewed as

the polar representation of complex number. Open loop magnetic bal-

ancing has preferred real valued influence coefficients. The linear pro-

gramming formulation of min-max mechanical balancing requires the

use of real valued influence coefficients.

Complex arithmetic handles the phase angle information properly, be-

cause the rotor-bearing system is time invariant. The time invariance

means that knowledge of the response to a balance weight at 0 angular

position implies knowledge at all other phase angles; because other an-

gles simply correspond to delays of the input which are the forces created

by the balance weight.

Getting from a complex influence formulation to a real formulation

can be done with the aid of an algebraic isomorphism of the complex

numbers, C, to a subset of 2×2 real matrices, R2×2.

The isomorphism, Φ, is given in Equation (7).

Φ : −→ R2×2

Φ (Z ) =Φ (x + i y) =

(

x −y

y x

)

for z = x + i y ∈C, and x, y ∈R

(7)

The basic properties of this correspondence of the complex numbers

with a subset of 2×2 real matrices follow.


(1) Φ (z1z2) =Φ (z1)Φ (z2) =Φ (z2)Φ (z1).

For z1 = x1 + i y1, and z2 = x2 + i y2

Φ (z1z2) =Φ(

x1x2 − y1 y2 + i (x1 y2 + y1x2))

=

(

x1x2 − y1y2 −(x1 y2 + y1x2)

(x1y2 + y1x2) x1x2 − y1 y2

)

=

(

x1 −y1

y1 x1

)(

x2 −y2

y2 x2

)

=

(

x2 −y2

y2 x2

)(

x1 −y1

y1 x1

)

(2) Φ (z1 + z2) =Φ (z1)+Φ (z2). This is obvious.

(3) Φ (1/z) =(

Φ (z))−1

.

If z = x + i y then

1

z=

x − i y

x2 + y2=

z

|z|2and

Φ

(

1

z

)

=1

x2 + y2

(

x y

−y x

)

=

(

x −y

y x

)−1

=(

Φ (z))−1

(4) Φ (z) =Φ (z) =(

Φ (z))T

, obvious.

(5) |z|2 = det(

Φ (z))

This isomorphism can be extended to apply to complex valued influ-

ence coefficient matrices, H ∈Cn×m , and the image is a 2n×2m matrix in

R2n×2m . For such a complex influence coefficient matrixΦ can be defined

in the following manner.

(8) Φ(H) =

(

ℜ(H) −ℑ(H)

ℑ(H) ℜ(H)

)

The real number valued balance Equation (??) can be completed by

transforming the imbalance vector, B, and the vibration response vector,

R. However, in these cases only one column is required.

B −→

(

ℜ(B)

ℑ(B)

)

and

R −→

(

ℜ(R)

ℑ(R)

)

Whereas the complex version of B ∈Cm×1 the real version is inR

2m×1, and

the real version of R ∈Cn×1 is in R

2n×1. In each case the second column is

not required for the matrix algebra to function properly.


For fixed rotational speeds and operating conditions a rotor system is

a linear time invariant system, LTI. The real valued influence coefficient

matrix for a LTI rotor system for either mechanical or magnetic open-

loop balancing has the following form.

(9) Φ(H) =

(

Hc −Hs

Hs Hc

)

The c and s subscripts relate to cosines and sines or the corresponding

in-phase and quadrature components of the influence coefficient matrix.

Because LTI rotor systems have this form of influence coefficient matrix,

they lie in the image of the isomorphism, Φ. Thus the real influence coef-

ficient matrix can be transformed to a complex valued one which at time

may be more convenient.

4.3. Generating Influence Coefficients. One applies trial weights to gen-

erate the influence coefficient matrix. Traditionally this has been done

by adding an individual trial weight to balance plane j and measuring

the response due to the balance weight. The same can be done for modal

style balance weights.

The influence coefficient at measurement location i can be calculated

from Equation 6 as follows.

Hi j =Ri −R0

B jusing complex arithmetic

Here Ri indicates measurement at location i from the response vector

of measurements with a single trial weight at balance plane j . R0 is the

reference measurement prior to adding B j , the balance weight in plane

j . One can define modal influence coefficients similarly when the trial

weight has a modal distribution, notably the so-called static and couple

combinations, which are frequently used to balance turbine generators.

More generally Equation 6 can be used to generate influence coeffi-

cients while treating each balance shot as a statistical trial. If one has at

least a linearly independent set of trial weights then the influence coeffi-

cients, H can be thought of as the best fit to Equation 10 that minimizes

the error vector, e, in some fashion.

(10) [R1R2 . . .Rn] = H [B1B2 . . .Bn]+[

R10R2

0 . . . Rn0

]

+e

In Equation 10, responses Ri (a vector of responses at the various mea-

surement points) occurs upon adding balance weights Bi , a vector of the

applied balance weights. Each Ri0 corresponds to the reference run prior

to adding the balance weights; since balance weights may be left in from


run to run, or the balancing may take place on different rotors this nota-

tion is necessary.

If one minimizes the 2-norm of the residuals, e, this produces a re-

gression fit for H. Using the superscript + to denote a pseudo-inverse,

Equation 11 produces the proper estimate for the influence coefficient

matrix.

(11) H =(

[R1R2 . . .Rn]−[

R10R2

0 . . .Rn0

])

[B1B2 . . .Bn]+

Since most of today’s balancing uses computer programs Equation 11

would be easy to implement. If one balances several machines of the

same type or the same machine often, this equation provides a best fit for

the influence coefficients. Some have portrayed the calculation of influ-

ence coefficients as a calibration of the rotor system. However, at best

one can approximate the influence coefficients and not precisely know

them. Thus, using a best fit approach often has advantages.

The formulation in Equation 11 also provides a means to compute the

influence coefficients with various combinations of trial weights, instead

of having to use individual planes. Using individual planes can result in

high vibration; often modal sets of trial weights are used to avoid this.

Superposition. Because of linearity superposition principles can and should

be used. Uses of superposition include the following.

: On an individual balance plane The effect of all the balance weights

on a single balance plane is equivalent to a single balance weight

equal to the vector sum of the individual balance weights on that

plane or conversely achieve the required effect of a balance weight

by using two or more weights in the same balance plane.

: Among two or more balance planes, speeds, or conditions The ba-

sic equation for multiplane balancing results from superposition

or linear response along the rotor and at different speeds or con-

ditions. Conditions could include loadings or peaks of thermal

vectors, as in a turbine or generator.

: Balancing multiple forces in the rotor system

BALANCE WEIGHT CONSOLIDATION

Over the course of years (or perhaps shorter if the balancer does not

take care), an individual balance plane can substantially fill with balance

weights. Some of these weights may effectively cancel the effects of other

weights. When this occurs further balancing can be difficult or impossi-

ble.


To remedy this one cleans the balance plane by balance weight consol-

idation. Generally, this is easiest with the rotor out in a shop, and some-

times shops may consolidate balance weights. Balance weight consoli-

dation is not the same as stripping the weights in a shop and performing

a new balance. If the balance weights were placed either in situ or in

a high speed balance facility, stripping the weights and replacing them

based on a low speed balance can make life interesting. Balance weight

consolidation simply reduces the number of balance weights while pro-

ducing the same force on the individual balance plane.

To perform balance weight consolidation one establishes a phase ref-

erence system for the balance plane, measures the angular location of

each of the balance weights, and measures the balance weight magni-

tude, both mass and radial distance from the center. Detailed accurate

records must be maintained as with other maintenance work on the ro-

tor. The object is to add a balance weight or balance weights equal to the

vector sum of the original balance weight vectors just measured. It may

not be possible to add a single balance weight to equal the vector sum.

The next section on splitting balance weights describes how to find a

combination of balance weights that equal a desired balance weight.

5. BALANCE WEIGHT SPLITTING

This section is based on an article in Vibrations magazine published by

the Vibration Institute by William C. Foiles in 2005.

5.1. Introduction. Balance weight splitting is the process of installing

more than one balance weight in a balance plane whose sum equals or

approximates the required balance weight. A number of balance pro-

grams and auxiliary programs include this for the user today for a two

weight split.

The necessity of splitting balance weights at locations occurs frequently.

Specific instances requiring weight splitting include occasions:

(1) When the location required to place a balance weight do not exist,

the balance weights need splitting between two or more angular

locations to achieve the required effective balance weight.

(2) When the required balance weight is larger than can be applied at

a single location, or when heavy balance weights are not available

or should not be used in the balance plane.

(3) When the required location has damage or has a weight already

installed.

This section is not meant to be a complete tutorial on the basics of

weight splitting or balancing, nor does this paper cover any safety aspects


necessary to work with rotating equipment when doing a balance job.

The methods shown are for the enjoyment of the readers.

One can not split weights 180 degrees apart to get a solution. Splitting

weights on one side of the desired weights will require a weight removal

on one of the weights; for practical considerations the split angles must

be on either side of the desired weight location and less than 180 degrees

apart.

5.1.1. Two Weight Splits. Sometimes today as in the past the weights are

split graphically. Figure 1 shows the geometry involved in a weight split.

Typically the graphing is done on polar paper.

(1) First one plots the desired balance weight vector, amplitude and

angle.

(2) Next lines at the desired angles to add the split weights are added

to the graph.

(3) Parallel lines to these lines at the desired angle, one-at-a-time, are

constructed through the ends of the desired balance weight vec-

tor to form the required split weight vectors (See Figure 3.). These

parallel lines can be commonly constructed using either a set of

parallels or two triangles.

(4) Usually, although not absolutely required, the parallelogram is com-

pleted by sliding the other weight add vector to the origin as in

Figure 4.

NOMENCLATURE FOR BALANCE WEIGHT SPLIT

~d Desired balance weight addition

α ∠~d

α1 Angle of first balance weight of the split

α2 Angle of second balance weight of the split−→W1 Balance weight vector at angle α1−→W2 Balance weight vector at angle α2

The purpose of the balance weight split is to find−→W1 and

−→W2 such that

−→d =

−→W1 +

−→W2

This two weight add problem has a simple analytical solution using

the. The Law of Sines formula applied to the geometry in Figure 4 gives

the following derivation.

(12)|−→W2|

sin(|α−α1|)=

|−→W1|

sin(|α−α2|)=

|−→d |

sin(180−|α2 −α1|)


34

5330

315

300285270255

240

225

210

19

51

80

16

515

0

135

120105 90 75

60

4530

15

0

0

5

10

15

0 5 10 15

CW

α1

α 2

~d

FIGURE 3. Balance weight split – Desired balance weight

and angles for the split

The absolute value sign takes care of odd cases. The above equation

leads directly to a solution for the weight splits at the desired angles.

(13) |−→W1| = |

−→d |

sin(|α2 −α|)

sin(180−|α2 −α1|)= |

−→d |

sin(|α2 −α|)

sin(|α2 −α1|)

And

(14) |−→W2| = |

−→d |

sin(|α1 −α|)

sin(180−|α2 −α1|)= |

−→d |

sin(|α1 −α|)

sin(|α2 −α1|)

The was the simple way to get to the formula for weight splitting into

two angles. One can break the equations up into the Cartesian compo-

nents, x and y, and use algebra and trigonometry to arrive at the above


34

5330

315

300285270255

240

225

210

19

51

80

16

515

0

135

120105 90 75

60

4530

15

0

0

5

10

15

0 5 10 15

CW

α1

α 2

~d

−→W1

−→ W2

FIGURE 4. Balance weight split – Geometric Solution

equations or something equivalent. I have not intention of duplicating

the easy solution by the more difficult solution here, but with a little pa-

tience or one of the symbolic computation engines available today one

can get there.

Example for split in two locations. Following the example in Figure 3,

let ~d = (13∡55). This weight is to be split at the two angles α1 = 20

and α2 = 75. Figure 4 shows the graphical solution; below the analytic

solution is followed as above.


|−→W1| = 13

sin(|75−55|)

sin(|75−20|)= 5.428

|−→W2| = 13

sin(|55−20|)

sin(|75−20|)= 9.103

5.428 (with proper units, which may include the radius) is placed at

20, and 9.103 is placed at 75. One can check that the vectorial sum of

these weights is as desired.

5.1.2. Spitting weights at n angles. For this section on splitting a balance

weight into several locations, the Cartesian coordinate formulation of

the two weight problem will help, the algebra and trigonometry to show

equivalence with the above Law of Sines formulation is left for the curious

reader.

The two equations for the x and y coordinates from Figure 4 result in

Equation 15.

(15)|−→W1|cos(α1)+|

−→W2|cos(α2) = |

−→d |cos(α)

|−→W1|sin(α1)+|

−→W2|sin(α2) = |

−→d |sin(α)

These equations in the two unknowns for the desired split magnitudes,

|−→W1| and |

−→W2| can be solved as usual. One can check that the two weight

split example above solves the Equation 15 above.

The above framework for writing the Cartesian coordinates generalizes

to splitting at more than two angles.

NOMENCLATURE FOR BALANCE WEIGHT SPLIT AT n ANGLES

~d Desired balance weight addition

α ∠~d

α2 Angle of second balance weight of the split−→Wi Balance weight vector at angle αi

Wi |−→Wi | for simplicity, i = 1, . . .n

Li Lower bounds on balance weights, i = 1, . . .n

Ui Upper bounds on balance weights,i = 1, . . .nFor splitting among more than 2 angles, one has the following basic

equations.


(16)

n∑

i=1

|−→Wi |cos(αi ) = |

−→d |cos(α)

n∑

i=1

|−→Wi |sin(αi ) = |

−→d |sin(α)

The equations above, Equation 16, in general have an infinite number

of solutions. Again, the split angles should be on either side of the de-

sired angle, and generally the angles should spread less than 180 degrees.

Without something else, this doesn’t help much.

Adding a goal or optimization objective,such as Equation 17 will lead to

finding solutions. One practical solution is to minimize the total amount

of added weight as below.

(17) minn∑

i=1

Wi

To the above assume all the weights are additions or positive magni-

tudes. The weight removal procedure can be processed by adding weights

at the opposite angles. These equations form a linear programming prob-

lem that can be solved.

On top of the equalities, one may desire to limit the magnitudes of the

weights that can be added. Such extra constraints would give the fol-

lowing equations, Equation 18. Without some constraints the solution,

which minimizes the sum of the weights, only uses the two positions clos-

est to the desired balance weight position.

(18) 0 ≥ Li ≥Wi ≤Ui , i = 1, . . .n

The following linear program incorporates these aspects and will split

a desired balance weight at n desired angles while minimizing the sum

of the split weights, provided a feasible solution exists. Weight bounds,


both upper and lower, may be applied to the split weights.

minn∑

i=1

Wi such that

n∑

i=1

|−→Wi |cos(αi ) = |

−→d |cos(α)

n∑

i=1

|−→Wi |sin(αi ) = |

−→d |sin(α)

with

0≥ Li ≥Wi ≤Ui , i = 1, . . .n

(19)

The unknowns in the Linear Program 19 are the balance weights, Wi .

A sample routine written for Scilab©

was included in the original Vibra-

tions article.

Examples of multi-angle weight splits. The following example use the

sample Scilab program. The desired balance weight is 450∡100, and

the angles for the split are 75, 90, 105, and 120. Upper bounds were

placed of 250, 100, 100, and 250 were placed respectively on the angular

locations with lower bounds of 0 for each angle. The solution that mini-

mizes the total weight addition is to add 110.35∡75, 100∡90, 100∡105,

and 161.64∡120,

-->bwt=[450 100];

-->angles=[75;90;105;120];

-->lower=[0;0;0;0];

-->upper=[250;100;100;250];

-->split(bwt,angles,lower,upper);

! 110.34707 !

! 100. !

! 100. !

! 161.6394 !

The above example would not have been too difficult to solve manu-

ally, because one always tries to add the most weight closest to the de-

sired angle as possible.

Copyright ©1989-2005. INRIA ENPC. and Scilab is a trademark of INRIA,

www.scilab.org

www.scilab.org


A more difficult example with both upper and lower bounds follows.

-->ang=[90;120;150;180];

-->lower=[100;100;100;100];

-->upper=[500;250;500;500];

-->split(balwt,ang,lower,upper);

! 145.01279 !

! 250. !

! 372.97723 !

! 100. !

E-mail address: [email protected]

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