Balancing of rigid rotors _Integrated course - ISET Gafsa

Higher Institute of Technological Studies of Gafsa-Tunisia

Department of Mechanical Engineering

Professional Master in Mechanics of Mining Industry Systems

Course & Tutorials: Balancing of rigid rotors

Prepared by: Rached Nciri

Technologist, Doctor Engineer

Level 2- Professional Master in Mechanics of Mining Industry Systems

Academic year: 2020-2021

Foreword

The content of this material consists of a course and some tutorials about

the balancing of rigid rotors. It is addressed to the students preparing the

professional master in Mechanics of Mining Industry Systems, taught in the

Department of Mechanical Engineering of the Higher Institute of Technological

Studies of Gafsa-Tunisia. More globally, all students who receive a

technological education in mechanical engineering can benefit from this

material. This is a first version that can be improved and updated whenever

required.

BALANCING OF RIGID ROTORS

COURSE & TUTORIALS

• Domain : Mechanical Engineering

• Level : M2-MSIM-S3

• Prerequisites: Applied License in Mechanical Engineering.

• Number of hours: 21 hours.

• Goals :

- Understanding the different types of unbalances that may

occur on rotors.

- Understanding the different types of balancing that permit to

correct unbalances on rotors.

- Knowing the specific equipment for rotor balancing.

- Understanding the principal rules associated to the balancing

of rigid rotors (Standards ISO 21940-11:2016).

- Master the application of static balancing.

- Master the application of dynamic balancing.

Table of contents

CHAPTER 1. ROTORS UNBALANCES AND BALANCING: TECHNOLOGICAL

OVERVIEW. 5

1. Context............................................................................................................................... 5

2. Principle ............................................................................................................................. 5

3. Types of unbalance ............................................................................................................ 5

3.1. Static unbalance .......................................................................................................... 5

3.2. Couple unbalance ........................................................................................................ 6

3.3. Quasi-static unbalance ................................................................................................ 6

3.4. Dynamic unbalance ..................................................................................................... 7

4. Types of balancing ............................................................................................................. 8

5. References ......................................................................................................................... 9

CHAPTER 2. STATIC BALANCING (SINGLE PLANE BALANCING). ............... 10

1. Static balancing setup ...................................................................................................... 10

2. Static balancing methodology ......................................................................................... 10

3. Static balancing techniques ............................................................................................. 11

4. References ....................................................................................................................... 14

CHAPTER 3. DYNAMIC BALANCING (TWO PLANES BALANCING). .............. 15

1. Dynamic balancing setup ................................................................................................. 15

2. Static balancing methodology ......................................................................................... 15

3. Dynamic balancing techniques ........................................................................................ 17

4. References ....................................................................................................................... 18

CHAPTER 4. SPECIFIC EQUIPMENT FOR ROTOR BALANCING. .................... 19

1. Balancing methods .......................................................................................................... 19

1.1. Balancing in workshop ............................................................................................. 19

1.2. In-situ balancing........................................................................................................ 20

2. Devices for balancing setup ............................................................................................. 20

3. References ....................................................................................................................... 23

CHAPTER 5. USEFUL STANDARDS FOR THE BALANCING OF RIGID

ROTORS. 25

1. ISO balancing grades ....................................................................................................... 25

2. Permissible Residual Unbalance ..................................................................................... 26

3. Maximum Residual Mass and Trial mass........................................................................ 29

4. Mass and phase correction ............................................................................................... 29

5. Admissible residual vibration .......................................................................................... 29

6. References ....................................................................................................................... 30

CHAPTER 6. STATIC BALANCING TUTORIAL. .................................................... 31

1. Problem statement ........................................................................................................... 31

2. Technological answers .................................................................................................... 37

CHAPTER 7. DYNAMIC BALANCING TUTORIAL. ............................................... 42

1. Problem statement ........................................................................................................... 42

2. Technological answers .................................................................................................... 43

Higher Institute of Technological Studies_Gafsa-Tunisia Balancing of rigid rotors

Department of Mechanical Engineering Rotors unbalances and balancing: Technological overview

5 Rached Nciri; Technologist, D. Eng.

Chapter 1. Rotors unbalances and balancing: Technological overview.

1. Context

The balancing aims to correct a mass unbalance in a rotor of rotating machines (rotor of

electric machines, rotor of turbines, rotor of propeller, rotor of pumps…etc.). This allows the

vibration levels to be reduced at the rotational speed of the rotor. The vibration level is an

important indicator reflecting the condition of the machine in terms of condition-based

maintenance. The high vibration level accelerates the wear of the rotor, decreases the life of

the bearings, generates an annoying noise and makes the work conditions unsafe. In this

context, the balancing of rotating machines takes on all its interest [1.1-1.4]

2. Principle

The rotor of a rotating machine naturally has a mass distribution that is not perfectly

homogeneous in its volume. This non-homogeneity is called unbalance. The axis of rotation is

then not coincident with the axis of inertia. This unbalance is equivalent to a point unbalance

mass, located at a given distance from the axis of rotation, and it generates a rotating force,

under the form of a centrifugal force [1.5]:

{ ( )

[ ][ ]

[ ][ ]s/rd

mR

kgm

NF

tmRF

centrifuge

lcentrifuga

locity Angular ve :

radius Unbalance:

mass punctual Unbalance:

force lCentrifuga :

withsin2

unbalance Rotating

ω

ωω=

3. Types of unbalance

Four types of unbalance are distinguished [1.6]: static unbalance, couple unbalance, quasi-

static unbalance and dynamic unbalance.

3.1. Static unbalance

Figure 1.1 models a static unbalance in terms of equivalent unbalance punctual mass

(hatching), inertia axis (solid line) and geometry axis (dashed line). Static unbalance is

characterized by an inertia axis offset parallel to the geometry axis. The center of inertia,

noted I, is then distinct from the geometric center, noted G.




3.2. Couple unbalance

Figure 1.2 models a couple unbalance in terms of equivalent unbalance punctual mass

(hatching), inertia axis (solid line) and geometry axis (dashed line). Couple unbalance is

characterized by an inertia axis secant to the geometry axis at the geometric center. The center

of inertia, noted I, and the geometric center, noted G, are then merged.

Figure 1.1. Static unbalance model.

Figure 1.2. Couple unbalance model.

3.3. Quasi-static unbalance

Figure 1.3 models a quasi-static unbalance in terms of equivalent unbalance punctual mass

(hatching), inertia axis (solid line) and geometry axis (dashed line). Quasi-static unbalance

represents the superposition of static and couple unbalances. This type of unbalance is

characterized by an inertia axis secant to the geometry axis at a point different from the

G

I

I

G




geometric center. The center of inertia, noted I, and the geometric center, noted G, are

therefore distinct.

Figure 1.3. Quasi-static unbalance model.

3.4. Dynamic unbalance

Figure 1.4 models a dynamic unbalance in terms of equivalent unbalance punctual mass

(hatching), inertia axis (solid line) and geometry axis (dashed line). Dynamic unbalance is

similar to quasi-static unbalance except that the inertia and the geometry axes are not secant

(do not belong to the same plane). This unbalance is not necessarily a superposition of static

and couple unbalances. It is rather a random unbalance.

Figure 1.4. Dynamic unbalance model.

I

G

I

G




4. Types of balancing

Balancing makes it possible to reduce or even cancel the centrifugal force, by acting on the

reduction of unbalance. For rigid rotors, there are 2 types of balancing [1.7]:

• Static balancing (single plane): It consists in bringing the center of inertia of the

rotor back to the axis of rotation. Therefore, gravity will not move the rotor regardless

of its angular position. The rotor is said, then, to be statically balanced. Chapter 2 will

be devoted to static balancing. The technical procedure and technological details of

such balancing will be provided.

• Dynamic balancing (two planes): It consists in making one of the main inertia axes

of the rotor coincide with the axis of rotation. This is possible by a well-studied

addition or removal of material. Chapter 3 will be devoted to dynamic balancing. The

technical procedure and technological details of such balancing will be provided.

The choice of the correct balancing: static or dynamic depends mainly on two parameters:

the size and the rotational speed of the rotor (Figure 1.5 and Table 1.1) [1.8]

Table 1.1.

Figure 1.5. Rotor size.

Table 1.1. Choice of balancing type for rigid rotors.

Size Rotational speed Balancing type

5.0≤D

L

rpm 1000≤N Static (Single plane)

rpm 1000>N Dynamic (Two plane)

5.0>D

L

rpm 150≤N Static(Single plane)

rpm 150>N Dynamic (Two plane)

L

D




5. References

[1.1] Derek Norfield, Practical Balancing of Rotating Machinery, Elsevier Science 2005,

ISBN 978-1-85617-465-7, https://doi.org/10.1016/B978-1-85617-465-7.X5000-1

[1.2] Michael I. Friswell, John E. T. Penny, Seamus D. Garvey

and Arthur W. Lees, Dynamics of Rotating Machines, Cambridge University Press 2010,

ISBN: 9780511780509, https://doi.org/10.1017/CBO9780511780509

[1.3] Robert B. and McMillan, Rotating Machinery: Practical Solutions to Unbalance and

Misalignment, River Publishers 1st Ed. 2003, ISBN: 9780824750527

[1.4] J B Wilcox, Dynamic Balancing of Rotating Machinery, Pitman 1967, ISBN-13: 978-

0273429593

[1.5] Djaidir Benrabeh, Ahmed Hafaifa and Kouzou Abdellah, Faults detection in gas turbine

rotor using vibration analysis under varying conditions, Journal of Theoretical and Applied

Mechanics 55(02) (2017) 393-406, https://doi.org/10.15632/jtam-pl.55.2.393

[1.6] Balancing: Static, Coupled and Dynamic,

https://community.sw.siemens.com/s/article/balancing-static-coupled-and-dynamic, Access

date: December, 16 2020 14:27 GMT

[1.7] John Vaughan, Static and Dynamic balancing, Bruel & Kjaer application notes, 2nd

Ed.

https://pearl-

hifi.com/06_Lit_Archive/15_Mfrs_Publications/10_Bruel_Kjaer/04_App_Notes/Static_and_

Dynamic_Balancing_2nd_Edn.pdf, Access date: December, 16 2020 14:50 GMT

[1.8] Jaafar Alsalaet, Dynamic Balancing and Shaft Alignment, College of Engineering-

University of Basrah 1st

Ed. 2015


Department of Mechanical Engineering Static balancing (single plane balancing)


Chapter 2. Static balancing (single plane balancing).

1. Static balancing setup

The static balancing setup [2.1] permits to measure the amplitude V and the phase φ of the

vibration on a rotor bearing. Figure 2.1 depicts the main devices connected on this setup.

Figure 2.1. Static balancing setup.

2. Static balancing methodology

The static balancing (single plane balancing) procedure [2.2] can be presented by detailing

the general balancing process:

Establishment of the initial conditions of the

unbalance (V0 and φ0)

Calculation, application of the trial mass and

establishment of the conditions for

{unbalance + trial mass Mt}

Place the accelerometer sensor on the

vibrating bearing in order to get the

sinusoidal function of the vibration

(amplitude V0)

Place the rotation sensor and get the pulse

function of the rotation

Determine the angular phase shift φ0 between

the vibration function and the rotation


vibrating bearing and get the sinusoidal

function of the vibration (amplitude V1)


function of the rotation.



Calculate [2.3] and apply the trial mass Mt on

the balancing plane on an arbitrary radial

position Rt.

Accelerometer

sensor

Rotation

sensor

Amplitude-phase

analyzer

Rotor

V ; φ




3. Static balancing techniques

The static balancing (single plane) can be technically achieved by a vector construction.

There are 3 main steps [2.1]:

1/ Model the initial vibration vector 0V by its amplitude V0 and its phase φ0.

Calculation and application of the correction

mass Mc and the correction phase φc

Remove the trial mass Mt. Apply the

correction mass Mc on the balancing plane on

the same radial position Rt. For a positive φc, go in the direction of rotation. For a negative

φc , go in the direction anti-rotation.

Calculation of the admissible residual

vibration

Satisfaction of the residual vibration standard

Yes

End

No

Calculate [2.3] the correction mass Mc.

Calculate [2.3] the correction phase cϕ .

Standards: ISO 21940-11:2016 [2.3]

0°

90° 0V

φ0

Standards: ISO 21940-11:2016 [2.3]

180°

270°




2/a/ Place an appropriate trial mass Mt on an arbitrary radial position Rt (Figure 2.2).

(t

tR

Mrotor theof Mass UnblanceResidual ePermissibl

10××= [2.3])

Figure 2.2. Trial mass placed on an arbitrary radial position Rt.

2/b/ Model the vector vibration with trial mass 1V by its amplitude V1 et its phase φ1.

3/ Deduce graphically by a Chasles relation the vibration vector induced by the trial mass

alone tV ( 01 VVVt −= ). Graphically determine the amplitude Vt and the phase φt of the vector

tV .

0°

90° 0V

φ0

φ1

0°

90° 0V

φ0

φ1

tV

1V

tV

1V

Mt

Rt Vibrating bearing

Balancing plane

180°

270°

180°

270°




3/a/ Knowing the value of φ0 and φt, deduce the angular position φc of the correction mass Mc

in relation to the angular position of the trial mass Mt. ( tc ϕϕϕ −+= 1800 [2.3]).

3/b/ Knowing the value of the trial mass Mt, the amplitude of the initial vibration V0 and the

amplitude of the vibration vector induced by the trial mass alone Vt, deduce the value of the

correction mass Mc. (t

tcV

VMM 0= [2.3]).

3/c/ Remove the trial mass Mt. Place the correction mass Mc on the radial position Rt and its

angular position φc. The angular position φc is counted from the angular position of the

removed trial mass as it is depicted by Figure 2.3. For a positive φc, go in the direction of the

rotation of the rotor. For a negative φc, go in the direction anti-rotation of the rotor.

Figure 2.3. Appropriate radial and angular positions of a correction mass.

0°

90° 0V

φ0

φ1

tV

tV

1V

cV

φc

Mt

Rt

Mc

Rc=Rt

φc

Removed trial mass

180°

270°




4. References


Ed.

https://pearl-



[2.2] VibroMetra System, Application note: Basics of balancing,

https://www.mmf.de/pdf/an31e-vm-bal.pdf, Access date: December, 23 2020 11:04 GMT

[2.3] International Organization for Standardization, ISO 21940-11:2016, Mechanical

vibration — Rotor balancing — Part 11: Procedures and tolerances for rotors with rigid

behaviour, September 2016


Department of Mechanical Engineering Dynamic balancing (Two planes balancing)


Chapter 3. Dynamic balancing (Two planes balancing).

1. Dynamic balancing setup

The dynamic balancing setup [3.1] permits to measure the amplitudes V and the phases φ

of the vibrations on the two rotor bearings. Figure 3.1 depicts the main devices connected on

this setup.

Figure 2.4. Dynamic balancing setup.

2. Static balancing methodology

The dynamic balancing procedure can be presented by detailing the general balancing

process:


unbalance (V01 and φ01) for the balancing

plane 1


vibrating bearing 1 in order to get the

sinusoidal function of the vibration (amplitude

V01).






unbalance (V02 and φ02) for the balancing

plane 2


vibrating bearing 2 in order to get the

sinusoidal function of the vibration (amplitude

V02).





Accelerometer

sensor 1

Rotation sensor

Amplitude-phase analyzer

Rotor

V1 ; φ1

V2 ; φ2

Accelerometer

sensor 2






{unbalance + trial mass Mt} for the balancing

plane 1


vibrating bearing 1 and get the sinusoidal






Calculate [3.2] and apply the trial mass Mt on

the balancing plane 1 on an arbitrary radial

position Rt.










{unbalance + trial mass Mt} for the balancing

plane 2








Remove the trial mass Mt from the balancing

plane 1 and apply it on the balancing plane 2

on the same arbitrary radial position Rt.











3. Dynamic balancing techniques

The principle of the dynamic balancing [3.3] technique is based on the application of the

static balancing technique on the two balancing planes. However, we must take into account

the effect of the unbalance of the plane 1 on the vibration of the bearing 2, and the unbalance

of the plane 2 on the vibration of the bearing 1. The calculation is then more complicated.

(For more details, see Tutorial 2_Dynamic balancing).

Removing of the trial mass Mt, calculation

and application of the correction mass Mc1

and Mc2 and the correction phase φc1 and φc2

on the balancing planes 1 and 2 respectively.

Adjust the correction radius.

Measurement of the residual vibration Vr

Calculation of the admissible residual

vibration admrV ,

Satisfaction of the residual vibration

standards

Yes

End

No

Standards [3.2]

Standards [3.2]

Standards [3.2]




4. References


Ed.

https://pearl-






[3.3] VibroMetra System, Application note: Basics of balancing,

https://www.mmf.de/pdf/an31e-vm-bal.pdf, Access date: December, 23 2020 11:04 GMT


Department of Mechanical Engineering Specific equipment for rotor balancing


Chapter 4. Specific equipment for rotor balancing.

1. Balancing methods

There are two balancing methods: in workshop and in-situ (known also as field balancing)

[4.1].

1.1. Balancing in workshop

The rotor is dismantled and then installed on a balancing machine located in a workshop.

This balancing method is well suited to serial work (serial balancing of rotors). Figure 4.1

shows a balancing machine with an axial drive. Figure 4.2 shows a balancing machine with a

radial drive.

Figure 3.1. Balancing machine with axial drive [4.2].

Figure 3.2. Balancing machine with radial drive [4.3].




1.2. In-situ balancing

In-situ balancing consists of balancing the rotor without dismantling it (while remaining on

its own bearings). This in-situ balancing is the most widely practiced since it essentially

allows:

• Take into account the actual operating conditions of the rotor to be balanced (speed,

load, gap, etc…).

• Avoid dismantling, reassembly and realignment of the rotors to be balanced.

Figure 4.3 shows an in-situ balancing.

Figure 3.3. In-situ balancing [4.4].

2. Devices for balancing setup

Static or dynamic balancing is carried out using a set of specific devices connected as

shown in Figure 4.4 and Figure 4.5.




Figure 3.4. Static balancing equipment [4.5].

Figure 3.5. Dynamic balancing equipment [4.6].

Shaft

Accelerometer

sensor

Rotation

sensor

Balancing

software

Balancing plane

Rotation sensor

Accelerometer

sensors




• Accelerometer sensor: to measure the vibration function (Figure 4.6).

Figure 3.6. Accelerometer sensor [4.7].

• Rotation sensor : to measure the rotation function

Figure 3.7. Rotation sensor [4.8].




• Amplitude-phase analyzer: to analyze the data sent by the accelerometer and the

rotation sensors and then determine the vibration amplitude and calculate the phase

shift between the vibration function and the rotation function.

Figure 3.8. Amplitude-phase analyzer [4.9].

3. References

[4.1] Jaafar Alsalaet, Dynamic Balancing and Shaft Alignment, College of Engineering-

University of Basrah 1st

Ed. 2015

[4.2] CEMB, Horizontal balancing machine Z50000-G-GV,

https://www.directindustry.fr/prod/cemb/product-7363-439463.html, Access date: December,

23 2020 17:54 GMT

[4.3] Vneshtorg--Micron ltd., Horizontal balancing machine 9K716,

https://www.directindustry.com/prod/vneshtorg-micron-ltd/product-63357-1575868.html,

Access date: December, 23 2020 18:09 GMT

[4.4] PRUFTECHNIK, Equilibrage sur site,

https://www.usinenouvelle.com/expo/equilibrage-sur-site-p79071553.html, Access date:

December, 23 2020 18:28 GMT




[4.5] SIEMENS, Physical setup for single plane balancing,

https://community.sw.siemens.com/s/article/single-plane-balancing-in-

simcenter-testlab, Access date: December, 23 2020 18:36 GMT

[4.6] Spectrum, https://spectrumscientific.com.ph/product/rpm-sensors-keyphasers/, Access

date: December, 24 2020 08:06 GMT

[4.7] Starmeter, Vibration Meter VM-6360, http://www.starmeter.cn/Vibration-Meter-VM-

6360-pd6672544.html, Access date: December, 24 2020 08:46 GMT

[4.8]NEXUS AUTOMECH, Cylindrical photoelectric Sensor / LASER / IP67 / IP69K –

S186ELD Plastic barrel LASER emitter,

https://www.nexusautomech.com/product/cylindrical-photoelectric-sensor-laser-ip67-ip69k-

s186eld-plastic-barrel-laser-emitter/, Access date: December, 24 2020 09:20 GMT

[4.9] CEMB, Predictive maintenance vibration analyzer N600,

https://www.directindustry.com/prod/cemb/product-7363-1533891.html, Access date:

December, 24 2020 09:31 GMT


Department of Mechanical Engineering Useful Standards for the balancing of rigid rotors


Chapter 5. Useful standards for the balancing of rigid rotors.

1. ISO balancing grades

Depending on technological applications, the ISO 21940-11:2016 Standards [5.1] propose

several balancing grades (noted G), as it is detailed by Table 5.1.

Table 1.2. ISO balancing grades.

ISO balancing grades Technological applications

G4000 • Crankshaft drive of marine diesel

engines at slow speed, rigid

mounting, with an odd number of

cylinders

G1600 • Crankshaft drive, rigid mounting,

large two-stroke engines

G530 • Crankshaft drive, rigid mount, large

four-stroke engines

• Crankshaft drive, elastic mounting,

marine diesel engines

G250 • Crankshaft drive, rigid mounting, fast

four-cylinder diesel engines

G100 • Crankshaft drive of fast diesel

engines with six or more cylinders

• Complete engines (gasoline or diesel)

for cars, trucks and locomotives

G40 • Car wheels, wheel rims, wheel set,

drive shafts

• Crankshaft drive, elastic mounting of

fast four-stroke engines (petrol or

diesel) with six or more cylinders

• Crankshaft drive of engines of cars,

trucks and locomotives

G16 • Drive shafts (propeller shafts, cardan

shafts) with special requirements

• Grinding machine parts

• Agricultural machinery parts

• Spare parts for engines (petrol or

diesel) for cars, trucks and

locomotives

• Crankshaft drive of engines with six

or more cylinders under special

conditions

G6,3 • Parts of processing machines

• Main marine turbine gears (merchant

marine)

• Centrifugal drums

• Rolls of paper machines; printing

machine rolls




• Fans

• Assembly of rotors with gas turbines

for aeronautics

• Steering wheels

• Pump impellers

• Machine tools and common machine

parts

• Large and medium electrical

armatures (of electric motors with at

least a rod height of 80 mm) without

special requirements

• Small electrical fittings often

produced in series, for applications

insensitive to vibrations and / or with

insulating supports against vibrations

• Spare parts for engines with special

requirements

G2,5 • Gas and steam turbines including

main marine turbines (merchant

marine)

• Rigid turbogenerator rotors

• Computer memory drums and disks

• Turbochargers

• Machine tool drives

• Large and medium electrical fittings

with special requirements

• Small electrical fittings that do not

qualify for one or both of the

conditions specified for small

electrical fittings of balance quality

grade G6.3

• Turbine driven pumps

G1 • Tape and phonograph (gramophone)

training

• Grinding wheel drive

• Small electrical fittings with special

requirements

G0,4 • Pecision grinding spindles, discs and

armatures

• Gyroscopes

2. Permissible Residual Unbalance

The Permissible Residual Unbalance (PRU) is determined based on the ISO balancing

grades and the rotation speed, using the chart depicted by Figure 5.1 [5.1].




Figure 4.1. Permissible Residual Unbalance chart.




Application example

A propeller driveshaft (G16) rotating at 1000 rpm will have a Permissible Residual

Unbalance (PRU) in the order of PRU=150 g.mm/kg as it is depicted by Figure 5.2.

Figure 4.2. Determination of Permissible Residual Unbalance.




3. Maximum Residual Mass and Trial mass

The Maximum Residual Mass (MMR) is determined based on the Permissible Residual

Unbalance (PRU), the total mass of the rotor Mrotor and the radial position Rt of the trial mass

based on the following formula [5.1]:

t

rotor

R

MPRUMMR

×=

The trial mass Mt and the Maximum Residual Mass MMR are related by the following

inequality [5.1]:

MMRMMMR t ×≤≤× 105

Preferably,

MMRM t ×=10

4. Mass and phase correction

The correction mass mc is determined on the basis of the trial mass me, the initial vibration

V0 and the vibration Vt generated to the trial mass alone, using the following formula [5.1]:

t

tcV

VMM 0×=

The correction phase cϕ is determined on the basis of the initial vibration phase 0ϕ and

that of the vibration generated by the trial mass alone tϕ [5.1]:

tc ϕϕϕ −+= 1800

5. Admissible residual vibration

It is obvious that the residual vibration Vr, measured after the rotor balancing, must be less

than an admissible residual vibration Vr,adm [5.1]:

admrr VV ,≤




The admissible residual vibration Vr,adm is determined as a function of the vibration

generated by the trial mass alone Vt, the Maximum Residual Mass MMR and the trial mass Mt

[5.1]:

t

tadmrM

MMRVV ×=,

Since it is preferably that [5.1]:

MMRM t ×=10

The admissible residual vibration Vr,adm becomes [5.1]:

tadmr VV ×= 1.0,

6. References





Department of Mechanical Engineering Static balancing tutorial


Chapter 6. Static balancing tutorial.

1. Problem statement

A propeller drive shaft has a mass Mrotor=11 kg and rotates at a rotation speed S=900 rpm

counterclockwise. An initial vibration of amplitude mm/s 80 =V and phase °=1200ϕ is

recorded. Figure 6.1 presents the principal dimensions of this rotor.

Figure 5.1. Rotor principal dimensions.

Q1/ Determine the appropriate type of balancing that we should carry out.

Q2/ Explain with an explanatory diagram the experimental setup permitting to measure the

vibrations of the rotor in order to carry out the considered balancing.

Q3/ Determine the balancing grade based on the standard ISO 21940-11:2016.

Q4/ Determine the permissible residual unbalance PRU.

Q5/ We want to place a trial mass Mt at a radial position Rt=190 mm. Calculate the maximum

residual mass MMR. Deduce the trial mass Mt.

Q6/ After placing the trial mass, the vibration becomes mm/s 101 =V and °= 1501ϕ .

Determine graphically (vector representation) the amplitude tV and the phase tϕ of the vector

of vibration generated by the trial mass alone.

Q7/ Calculate the correction mass cM and the correction phase cϕ .

Q8/ Indicate with an explanatory diagram how the correction mass will be placed on the

balancing plane.

L=150 mm

D=460 mm




Useful tables and charts

Table 4.1. Balancing type in function of rotor size and rotation speed.

Rotor size Rotation speed Balancing type

5.0≤D

L

rpm 1000≤S Static (single plane)

rpm 1000>S Dynamic (2-planes)

5.0>D

L



Table 4.2. ISO balancing grades.

ISO balancing grades Technological applications

G4000 • Crankshaft drive of marine diesel

engines at slow speed, rigid

mounting, with an odd number of

cylinders

G1600 • Crankshaft drive, rigid mounting,

large two-stroke engines

G530 • Crankshaft drive, rigid mount, large

four-stroke engines

• Crankshaft drive, elastic mounting,

marine diesel engines

G250 • Crankshaft drive, rigid mounting, fast

four-cylinder diesel engines

G100 • Crankshaft drive of fast diesel

engines with six or more cylinders

• Complete engines (gasoline or diesel)

for cars, trucks and locomotives

G40 • Car wheels, wheel rims, wheel set,

drive shafts

• Crankshaft drive, elastic mounting of

fast four-stroke engines (petrol or

diesel) with six or more cylinders

• Crankshaft drive of engines of cars,

trucks and locomotives

G16 • Drive shafts (propeller shafts, cardan

shafts) with special requirements

• Grinding machine parts

• Agricultural machinery parts

• Spare parts for engines (petrol or

diesel) for cars, trucks and




locomotives

• Crankshaft drive of engines with six

or more cylinders under special

conditions

G6,3 • Parts of processing machines

• Main marine turbine gears (merchant

marine)

• Centrifugal drums

• Rolls of paper machines; printing

machine rolls

• Fans

• Assembly of rotors with gas turbines

for aeronautics

• Steering wheels

• Pump impellers

• Machine tools and common machine

parts

• Large and medium electrical

armatures (of electric motors with at

least a rod height of 80 mm) without

special requirements

• Small electrical fittings often

produced in series, for applications

insensitive to vibrations and / or with

insulating supports against vibrations

• Spare parts for engines with special

requirements

G2,5 • Gas and steam turbines including

main marine turbines (merchant

marine)

• Rigid turbogenerator rotors

• Computer memory drums and disks

• Turbochargers

• Machine tool drives

• Large and medium electrical fittings

with special requirements

• Small electrical fittings that do not

qualify for one or both of the

conditions specified for small

electrical fittings of balance quality

grade G6.3

• Turbine driven pumps

G1 • Tape and phonograph (gramophone)

training

• Grinding wheel drive

• Small electrical fittings with special




requirements

G0,4 • Pecision grinding spindles, discs and

armatures

• Gyroscopes




Figure 5.2. Permissible residual unbalance chart.




Figure 5.3. Polar chart.




2. Technological answers

A1/ The appropriate type of balancing is determined based on the following methodology:

A2/ The explanatory diagram, depicted by Figure 6.4, depicts the experimental setup

permitting to measure the vibrations of the rotor in order to carry out a static balancing.

Figure 5.4. Static balancing setup.

A3/ The balancing grade is determined based on the Standard ISO 21940-11:2016 (depending

on the technological application). Our rotor is a propeller drive shaft. Based on the Table 1.1,

the grade of quality must be G16 (Drive shafts: propeller shafts, cardan shafts with special

requirements).

A4/ The permissible residual unbalance PRU is determined according to the rotation speed of

the propeller shaft, the balancing grade and based on the permissible residual unbalance chart

(Figure 6.5):


5.0≤D

L



5.0>D

L



=

==

rpm 900

330460

150

S

.D

L

ϕ V




Figure 5.5. Determination of the permissible residual unbalance.

180




According to the chart, a propeller drive shaft (balancing grade G16) rotating at 900 rpm

will have a PRU in the order of 180 g.mm/kg.

A5/ The maximum residual mass MMR is calculated using the following formula:

g 42.10190

11180 =×=×= MMRR

MPRUMMR

t

rotor

Or,

MMRMMMR t ×≤≤× 105

Preferably,

g 2.10410 =×= tt MMMRM

A6/ The following vibration vectors are represented on the polar chart (Figure 6.6):

• The initial vibration vector:

°==

=120

mm/s 8

0

0

0 ϕV

V

• The vibration with trial mass vector:

°==

=150

mm/s 10

1

1

1 ϕV

V

Then, we can graphically deduce the amplitude tV and the phase tϕ of the vibration vector

generated by the trial mass alone.

0101 VVVVVV tt −=+=

Based on the vector representation:

°==

=202

mm/s 5

t

t

t

VV

ϕ




Figure 5.6. Vibration vectors representation.

A7/ The correction mass Mc is determined on the basis of the trial mass Mt, the amplitude of

the initial vibration V0 and the amplitude of the vibration generated by the trial mass alone Vt,

based on the following formula:

g 72.1665

82,1040 =×=×= c

t

tc mV

VMM

The correction phase cϕ is determined based on the initial vibration phase 0ϕ and the phase

of the vibration generated by the trial mass alone tϕ .

°=−+=−+= 982021801201800 ctc ϕϕϕϕ

A8/ Based on the fact that the correction phase is positive and the direction of rotation of the

rotor is counterclockwise, the correction phase will be in the counterclockwise direction. The




correction mass will be placed on the balancing plane as it is indicated by Figure 6.7. It is to

note that the trial mass must be removed before placing the correction mass.

Figure 5.7. Location of the correction mass in relation to the location of the trial mass (trial

mass must be removed).


Department of Mechanical Engineering Dynamic balancing tutorial


Chapter 7. Dynamic balancing tutorial.

1. Problem statement

We want to achieve dynamic balancing on a rigid rotor of a crusher (balancing grade G16).

The rotor has the following technical specifications (Figure 7.1):

Figure 6.1. Rotor specifications.

Q1/ What type of balancing should we choose.

Q2/ Explain with an explanatory diagram the experimental setup permitting to measure the

vibrations of the rotor in order to carry out the considered balancing.

The vibration measurements are carried out on the 2 rotor bearings. Table 7.1 summarizes

the recorded measurements:

Table 7.1. Vibration measurements

Position of the trial mass Vibration measuring (Amplitude and phase)

Bearing 1 Bearing 2

Without trial mass V10 7 mm/s φ10 220° V20 12 mm/s φ20 280°

Trial mass on the balancing plane 1 V11 5 mm/s φ11 100° V21 9 mm/s φ21 340°

Trial mass on the balancing plane 2 V12 4 mm/s φ12 80° V22 11 mm/s φ22 270°

Q3/a/ Model, on polar charts, the vibrations measured on the two rotor bearings.

Q3/b/ Based on the synthesis vector representation, explain the principle of the considered

balancing.

Q4/ Mathematically formulate and solve the balancing problem.

Q5/ Calculate the correction masses and phases.

L=460

D=400 600 rpm

Mrotor=450 kg




2. Technological answers

A1/ The dimensional specifications give a ratio 5.015.1 >=D

L

The rotor rotates at a rotational speed rpmS 600= .

Based on the table 7.2, we opt for a dynamic balancing.



5.0≤D

L



5.0>D

L



A2/ Figure 7.2 presents an explanatory diagram depicting the experimental setup that permits

to measure the vibrations of the rotor in order to carry out a dynamic balancing.

Figure 6.2. Dynamic balancing setup.

It should be noted that the above setup (Figure 7.2) presents the most conventional system.

The balancing engineer must himself analyze the data (curves) displayed by the oscilloscope

to determine the amplitude and phase of each vibration. Nowadays, it is not necessary to

provide a channel selector coupled with an oscilloscope to analyze the amplitude and phase of

the vibration. It is possible to connect the 2 accelerometer sensors and the rotation sensor

directly to an automatic amplitude-phase analyzer (computer or programmed tablet for

example). This amplitude-phase analyzer can directly give the amplitude of the phase of each

vibration.

Accelerometer sensor 2

Rotation sensor

Switcher

Channel 1/Channel 2

Vibration

analyser Accelerometer sensor 1




A3/a/ The vibrations measured on the two rotor bearings are modeled on polar charts as it is

depicted by Figure 7.3.

Bearing 1

Bearing 2

Figure 6.3. Vibrations representation on polar charts.

A3/b/ The basic idea of dynamic balancing consists in opposing the initial vibration on each

vibrating bearing by a combination of the two vibrations generated by the trial mass alone

(installed separately on the balancing planes 1 and 2). This will make it possible to neutralize

the initial vibrations.

10Vuur

11Vuur

11 10V V−uur uur

10Vuur

12Vuur

10Vuur

1mm/s 1mm/s 1mm/s

20Vuur

20Vuur

20Vuur

21Vuur

22Vuur

21 20V V−uur uur

22 20V V−uur uur

21 20V V−uur uur

22 20V V−uur uur

2mm/s 2mm/s

2mm/s

1012 VV −

11 10V V−uur uur

1012 VV −

Trial mass on the balancing plane 1 Trial mass on the balancing plane 2 Synthesis

Trial mass on the balancing plane 1 Trial mass on the balancing plane 2 Synthesis




A4/ The balancing problem is mathematically formulated as a 2 complex equations system.

Each equation reflects the idea of the balancing on a vibrating bearing (explained in § A3/b).

( ) ( )( ) ( ) 202022220211

101012210111

VVVQVVQ

VVVQVVQ

−=−+−

−=−+−

With:


Vibration Polar coordinates Complex notation

Amplitude Phase Real Imaginary

10V 7 220° -5,36 -4,50j

11V 5 100° -0,87 4,92 j

12V 4 80° 0,69 3,94 j

20V 12 280° 2,08 -11,82 j

21V 9 340° 8,46 -3,08 j

22V 11 270° 0,00 -11,00 j

( )1011 VV − 4.49 9.42 j

( )1012 VV − 6.05 8.44 j

( )2021 VV − 6.38 8.74 j

( )2022 VV − -2.08 0.82 j

Solving this system of equations gives:

( )( )

( ) ( )( )( ) ( )( )

1012202110112022

2021101011202

1011

10122101

VVVVVVVV

VVVVVVQ

VV

VVQVQ

−−−−−−+−−=

−−−−=

( )( ) ( )( )( )( ) ( )( )

( ) ( )( )( )

2

1

2.08 11.82 4.49 9.42 5.36 4.5 6.38 8.74

2.08 0.82 4.49 9.42 6.38 8.74 6.05 8.44

0.2 0.97

5.36 4.5 0.2 0.97 6.05 8.44

4.49 9.42

0.58 0.71

j j j jQ

j j j j

j

j j jQ

j

j

− − + + − − +=

− + + − + += −

− − − − − +=

+= +




A5/ The correction masses are calculated as follows:

tc

tc

mQm

mQm

×=

×=

22

11

However, the trial mass tm is linked to the Maximum residual mass MMR as follows:

MMRmMMR t ×≤≤× 105

Preferably:

MMRmt ×=10

The maximum residual mass MMR is calculated as follows:

t

rotor

R

MPRUMMR

×=

With;

PRU: Permissible Residual Unbalance, determined as depicted by Figure 7.4:

MMR=280gmm/kg

Rt: Arbitrary radial position of the trial mass tm on balancing planes. We choose Rt=150 mm.

kgm gMMR t 840084010840150

450280 =×==×=

( ) gmQm

gmQm

tc

tc

8319840097.02.0

7701840071.058.0

22

212

22

11

=×−+=×=

=×+=×=

The correction phases are calculated as follows:

( )

( ) °−=

−==

°=

==

35.782.0

97.0

75.5058.0

71.0

22

11

ArctgQArg

ArctgQArg

c

c

ϕ

ϕ

It should be noted that these correction masses will be placed on the same radial position

used previously for the trial mass.




Figure 6.4. Determination of permissible residual unbalance.

280

Balancing of rigid rotors _Integrated course - ISET Gafsa

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