3 - Rigid Rotor Ballancing

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Chapter 3Rigid rotor balancing

By Danmei Xie


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2012/9/25Wuhan University- Dr. Danmei Xie

The main purposes of this course

Basic theory of vibration

Methods of rotor balancingReasons and features of vibration

increase understanding ofrotor vibration phenomena

provide a means for controllingor eliminating these vibrations


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Lateral vibration


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Rigid rotor

a rotor which operates substantially below its firstbending critical speed.

A rigid rotor can be brought into, and will remain in,a state of satisfactory balance at all operating speedswhen balanced on any two arbitrarily selectedcorrection planes//


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e1

F1

F

2F

e2

e1

1F

2F

e2

2F

Unbalance

Static unbalance Dynamic unbalance

Combined unbalance


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3.1 Two terms & conditions of rigid rotor balancing

For a single degree-of-freedom forced vibration system, if the

damping is given, then the amplitude and the phase of the systemunder forced vibration should be expressed as followings

2

22

2

2 4)1(

nn

c

K

FA

22

2arctan

n

Where, is the coefficient of resistance

m

c

K is the coefficient of stiffness

c is the coefficient of damping

is the static displacement//stc yk

F


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2222

2

2222

2

)/(4)/1(

)/(

)(nn

na

bmk

maA

222/1

/2arctanarctan

n

n

mk

b


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Loose or soft bearings tight or hard bearings


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a) Parallel Eccentricity b) Conical Eccentricity

c) Self-Canceling Eccentricity d) Total Eccentricity

Figure 4.3 Distribution of Mass Centroidal Axis Eccentricity and the Effective Components in Terms ofRigid Rotor Response


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For rigid rotor, according to the kind of unbalance, balancing

Static balancing refers to single-plane balancing Dynamic balancing refers to two-plane balancing,

subdivided as

low speedbalancing and high speedbalancing//


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Static balancing rig

Parallel rail

rail

shaft

roller


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(a)(b)

1234

567

Low speed balancing rig


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12


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3.2 Methods of rigid rotor balancing

3.2.1 Two trial runs (steps) (low speed balancing)

Procedure Measure the initial rotor vibration A0 at a speed firstly (e.g.

balance speed), as uncorrected rotor data

Install a trial mass P at a position (usu. zero position ), and

measure rotor vibration A1 at the same speed

Shift the trial mass to the second position (e.g. 1800) , and

measure rotor vibration A2 at the same speed

Draw a geometric figure//


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Make OMDOM:OD:DM=A0:A1/2:A2/2

O

M

N

D

C

A0

A1

A2

S

Ap1

Ap2

prolong MD to MC, and make MD=CD

prolong OD to ON, and make OD=DN

Link OC and MN

Make a circle, its radius is OC

Measure the angleSOC

Vector analysis A0Fc

A1 FcP A1 = A0Ap1 A2 FcP A1 = A0Ap2


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Calculate the correction mass Q1

0

pA

APQ

krAP /0 For rotors balanced on balance rig :

r is the correction mass radiusk is the coefficient of sensitivity

sr

MgAP

2

0

For rotors balanced on bearings :

rAP /60 0

s is the sensitivity

2) On where should the correction mass or trial mass be placed?

1) How to choose the suitable trial mass P?


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Correction mass

Mass groove


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3.2.2 Three trial runs (steps) Procedure

Shift the trial mass to the second position (1200) , and

measure the rotor vibration A2 at the same speed

Shift the trial mass to the third position (2400) , and

measure the rotor vibration A3 at the same speed

Plot a geometric figure

Measure initial rotor vibration A0 at a speed

Install a trial mass P at a position (zero position), andmeasure the rotor vibration A1 at the same speed


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A1 A2 A3

1

3

2 C

Plot three semicircles with the radius of A1,A2,A3 respectively

O

A0

Find three points 1,2,3 on the semicircles (equilateral triangle123)Find the geometrical point C in the triangle, make a circleonwhich point 1, 2, and 3 pass simultaneouslyO1= A1 , O2= A2, O3= A3, OC= A0,

Then C1= Ap1, C2= Ap2, C3= Ap3,

Measure the angleOC1 1

0

pA

A

PQ


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3.2.3 The polar plot method Measure the initial rotor vibration A0 at first,

referred to as uncorrected rotor data.

Install a trial mass of known size at a

predetermined angular locationthen measurethe rotor vibration A1 at the same speed, referredto as trial mass data.

Use this rotor vibration data and fairly simple polar

plotting techniques, calculate the appropriateunbalance compensation, or correction mass


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-A0

A0

Ap

Aptrial mass data

uncorrected rotor

subtraction of vector

A1

1

0

pA

APQ


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Known A00=275m40, ( single balance plane)

Calculate trial mass: P=60275/R=658670 g

Add the trial mass on the rotor, measure A11=290m80

Calculate App= 216.6m149

Measure71.1

Then Q=P A0 / Ap 670 275 / 216.6=850.7 g

Example 1


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Figure 4.4 Illustration of Polar Plot Calculation Procedure for SinglePlane Rotor Balancing

U

U the uncorrected rotor data

T the trial mass data

UD the subtraction of vector from vectorT


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Application of single plane balance

Rotor of fans, and pumps etc

Couplings of steam turbine Shaft of main oil pump for ST


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Sample: Machine to be balanced

Impeller Parameter:

Diameter: 1400mm Thickness: 500mm

Blade Number: 12

Material:Fiberglass-Reinforced

Plastics

RPM: 1825 r/min

Bearing Model: ?

Motor Parameter:

Power: 75kW

RPM: 1500 r/minOthers:

Belt transmission

Spring base

Manufacture: LG

Impeller

Bearing1 Bearing2

Motor


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Influence Coefficient - A complex value representing

the effect of the addition of a unit trial mass in a

specific balancing plane on the rotor response at a

particular measurement plane. Influence Coefficient Balancing- An entirely empirical,

flexible rotor balancing method which uses known trial

masses to experimentally determine the sensitivity of a

rotor; and subsequently uses this sensitivity informationto determine a set of discrete correction masses that

will minimize synchronous vibrational amplitudes

3.4 Influence Coefficient Balancing


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ij are the influence coefficients relating the rotor responsefor the specified sensors and speeds to the balancing

In the simplest case, a single trial mass is used for each plane,one plane at a time, and

j

iij

ijT

xx0

where xi0 is the ith vibration reading with no trial masses

installed,

xij is the i th vibration reading with a trial mass installed in

the jth

balancing plane,and Tj is a complex value representing the amplitude andangular location, in rotating coordinates, of this trial mass .


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Known A0

0=275m 40, ( single balance plane)

Calculate trial mass: P=60275/R=658670 g

Add the trial mass on the rotor, measure A11= 290m80

Calculate App= 216.6m 149 Calculate influence coefficientij

Example 2

kgm /)(1493.323067.0

1496.216

QA

QA

1493.323220275402750

0

71)(851.01493.323

2202750

kg

AQ


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www.sendig.com36

Basic PrincipleOf 1 Plane Balancing

Q

AA01

0AP

1 select a plane to fix trial mass and a point

to measure, draw scale of phase and sign of

0ophase

2 measure initial vibration A0(phase and

amplitude)

3 fix a trial mass Qon the plane, measure

vibration A1

4 calculate influence coefficients:

5 calculate balancing mass P:

amplitude phase RPM


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Using simple statics, but with complex valued forces, we have

02

22

2

11 RL RrUrURF

0)( 22

221

2

11 RRLLL lRlrUlrULRM

)](/[)]()([ 122

1221 llrllRllRU LLRR

)](/[)]()([ 122

2112 llrllRllRU RRLL

whereF represents the sum of the forces on the rotor

(M )L represents the sum of the moments about the left end of the rotor

RL, RRare the bearing reactions at the left and right rotor supports,

respectively

U1, U2 are the unknown equivalent discrete unbalances at axial locations 1and 2 in Figure 5.

r1, r2are the corresponding radii for application of the correction masses

is the speed of rotation in radians per second//

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For two balance planes

Measure the amplitudes of the two bearings 00 BA

T GEN

2 1

AB

brg brg brgbrg

0202 BA

bP

Add the trial mass on the B end of the rotor,measure the amplitudes of the two bearingsCalculate influence coefficients//

aP

Add the trial mass on the A end of the rotor,measure the amplitudes of the two bearings 0101 BA


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influence coefficient caused by trial mass At A end

At B end

b

a PA

/22

b

b

b PB

/2

bP

Amplitude caused by trial mass At A end

At B end

aP

0011AAA

0011 BBB

bPAmplitude caused by trial mass

At A end

At B end0022

AAA

0022 BBB

aP

influence coefficient caused by trial mass

At A end

At B enda

a PA

/11

a

b PB

/11

A0A01

A1

A2

A02

B0B01B02

B1B2


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Calculate the correction masses

baab

ab

a

BAQ

2121

2020

At A end

baab

ba

bABQ

2121

1010

At B end


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3.3 Features of rigid rotor balancing

no more than two balancing planes are required

for complete balancing of a rigid rotor

Their balance can be accomplished at any speed,i.e. it is not related with balance speed//


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Typically, the balancing speed for hard-bearing machines isbetween 600 and 1800 rpm

Generally, higher speeds are used for lighter rotors or where tightbalancing tolerances are encountered

The bearing forces, and thus the sensitivity to unbalance, is

proportional to the square of the speed of rotation Therefore, when more sensitivity is required, higher balancing

speeds can be used

The balancing speed is, however, limited by the flexibility of therotor and supports in that it must remain well below the lowest

rotor critical speed Sometimes, particularly in the case of very light rotors, it is not

possible to attain sufficient sensitivity at allowable balancingspeeds. In this case, it may be necessary to use a soft-bearing

balancing machine//


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History of rotor balance Jeffcott demonstrated the necessity of rotor balancing in his

classic paper in 1919

The first significant contributions to the rotor balancing

literature did not appear until about 1930 Prior to the 1950s, the balancing literature was concerned with

the balancing of rigid rotors and, in a few cases, very simple

flexible ones

The first flexible rotors of significance to be built were steamturbine rotors

Initially, these rotors were balanced using simple, rigid-rotor

procedures//



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problems associated with use of rigid rotor balancingmachines

These problems are generally a result of improper or inappropriate use of rigid rotor machines and can be avoided ifthe supervisory engineer is aware of them

It is essential that rigid rotor machines not be used for balancing rotors which are, in fact, flexible

Rotors should have the same centers of rotation on thebalancing machine as they do in operation. For example, if arotor is to be supported by rolling-element bearings, it should,if possible be balanced mounted in these same bearings. If thesame center of rotation is not used, substantial unbalance can

be introduced which may have a detrimental effect on bearingand rotor life

A third source of potential problems with rigid rotor balancing machines occurs when a rotor stack-up must bedisassembled after balancing in order to be installed//


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Flexible rotor balancing procedures can generally be dividedinto two groups

modal balancing, in 1953 by Grobel-a trial mass procedure

and influence coefficient balancing, in the early 1960s

As very limited instrumentation and computational tools were

available at that time, a balancing method was needed that did

not depend heavily on such tools

Modal balancing fit naturally into these requirements as only

simple calculations are required and operator insight is theprimary ingredient, rather than large quantities, and quality, of

vibration data//



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Most mechanical engineering handbooks and general

references include some mention of rotor balancing. In most

cases, this mention is strictly limited to rigid rotor balancing.

A number of technical papers concerned with general and

rigid rotor balancing have also been published

More recently, similar discussions were presented in both

editions ofThe Shock and Vibration Handbook, in 1961 and

1976.

While the two editions presented slightly different

discussions, the content was basically the same

The primary emphasis was on an updated review of machines

and methods for balancing rigid rotors//



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Influence coefficient balancing was developed some yearslater, made possible by improvements in instrumentation and

the introduction of the digital computer

Consequently, the use of large quantities of high quality data

was substituted for operator insight as the central componentin the balancing Procedure

Subsequently, the Unified Balancing Approach was

developed as an empirical method, in the mold of influence

coefficient balancing

It was designed to take advantage of the modal nature of rotor

response, so as to avoid some of the difficulties of influence

coefficient balancing//



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In the early 1940s, Kroon published two papers on rotorbalancing, which were apparently intended as a design guide

In the first of these papers, Kroon described the theory behind

synchronous rotor vibration and the need for balancing of both

rigid and flexible rotors

In the second paper, he discussed a number of specific rotor

balancing machines and methods

While this discussion was primarily concerned with rigid rotor

balancing, a graphical method was described for two plane

balancing of flexible rotors

He also presented a brief, practically oriented discussion of

field balancing//



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A number of other papers concerned with general rotor

balancing have been published, including papers by Muster

and Flores, Jackson, and Stadelbauer

Muster and Flores compiled rigid rotor balancing criteria from

a variety of sources and compared these criteria with the

actual criteria used in American industry at the time (1969)

Jackson described a procedure for single plane field

balancing of rigid or flexible rotors using an oscilloscope

lissajous pattern of the rotor orbit

Van de Vegte and Lake proposed a procedure for balancing

rigid rotors during operation which utilizes actively

controllable, eccentric disks

They indicated the potential adaptation of such a mechanism

to modal balancing of flexible rotors, but provided no details//2012/9/25Wuhan University- Dr. Danmei Xie


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Bishop proposed to use this same balancing head design for

balancing flexible rotors

This would be done with a single head located, axially, as far

as possible from all mode shape nodes

Then, the head would be readjusted, using a simplified

procedure also proposed by Bishop, in the vicinity of each

critical speed during each run up and run down of the rotor

Gosiewski also promoted the use of balancing heads for

balancing flexible rotors

Unlike Bishop, he proposed using multiple heads in a

procedure which might be described as automatic, and

continuous, influence coefficient balancing//



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Rigid rotor: the rotor being balanced does not elastically

deform at any speed up to its maximum design speed Two terms & conditions of rigid rotor balancing

2

22

2

2 4)1(

nn

c

K

FA

22

2arctan

n

Summary

two- or three-steps method

Influence Coefficient Balancing

Features of rigid rotor balancing

no more than two balancing planes are required for complete

balancing of a rigid rotor.

Their balance can be accomplished at any speed, i.e. it is not

related with balance speed//


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Questions

Can you distinguish between the trial mass and the correctionmass in rotor balancing ?

Please list the procedures of three steps balancing of rigid rotor?

Please explain the inf luence coeff icient. Can you balance a

rigid rotor by using influence coefficient balancing? And how? The rigid rotor balancing, is based on two important

assumptions, what are they?

Please explain the least-squaresmethod used in rotor balance?

Please explain why a rigid rotor can always be balanced in twoplanes?

Try to describe the development of rigid rotor balance (from asingle trial mass run, two trial mass runs, three trial mass runs,

polar plot, influence coefficient method )//

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3 - Rigid Rotor Ballancing