Balancing of Rotating Disks

Kuwait University

College of Engineering and Petroleum

Mechanical Engineering Department

Fall 2009/2010

ME 475

Dynamics of Machines and Mechanical Vibration Laboratory

Experiment No.4

Balancing of Rotating Disks

Omar Saleem – ID No. 205111466

Performed on: Wednesday December 16th, 2006

Submitted on: Wednesday December 30th, 2006

Instructors:

Prof. Mohammad Al-ansary

Eng. Tallaa Kamel

Table of Contents

List of Tables ……………………………………………………………………….3

Introduction ………………………………………………………………………...4

Objectives …………………………………………………………………………..5

Safety Precautions ……………………………………………………………….....5

Theory …………………………………………………………………………...…6

Experimental Setup ………………………………………………………………...8

Experimental Procedure ……………………………………………………………9

Sample Calculations ………………………………………………………………14

Observation Data Tables..…………………………………………………………16

Analysis of Results and Discussion ………………………………………………19

Conclusions …………………………………………….….……………………...20

Nomenclature ……………………………………….…….………………………21

References …………………………………………….….……………………….22

Appendix ……………………………………………….…….…………………...23

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List of Tables

Table 1: Dynamic balancing of two plane demo machine ……………………..……..17

Table 2: Correction masses for static balancing in Plane-1 and Plane-2 ………..……17

Table 3: Residual unbalancing in two plane demo machine for Static balancing …....17

Table 4: Correction masses and residual unbalancing of two plane demo machine.…18

Table 5: Residual unbalancing in two plane demo machine ……………………....…18

3

Introduction

The balancing is an important factor in many designing and running of many machines

and systems. It is known that even small imbalance in machinery or components of the

system can cause the whole system to fail. So, checking for unbalance and fixing it is an

important task and is vital for proper functioning of systems.

In our experiment imbalance was introduced in a rotating disk and the effect of this

imbalance on the disk is recorded as the velocity due the imbalance. Two types of

balancing operations are carried out which are dynamic and static balancing. Static

balancing is carried out resolving system into one plane and balancing by adding mass in

that plane only. Whereas , dynamic balancing involves multiple planes. A trail mass is

introduced at a known position in the system which will change the velocity value . This

is done to find a position and magnitude of counter mass which will balance the system.

In static balancing three methods are used. The first method is the graphical method

where the velocity is obtained graphically using vector representation. The other two

methods use analytical techniques to obtain the velocity. In dynamic balancing the results

were interpreted using a computer program(MATLAB). This procedure is adapted due to

the complexity in solving the equations in two planes.

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Objectives

1. To learn how to perform static (single plane) balancing of a rotating disk.

2. To learn how to perform dynamic (two planes) balancing.

3. To compare results obtained by methods of graphical and analytical vector mechanics

for the balancing with the known system of out-of-balance masses.

Safety Precautions

(i) Follow all electrical and electronic apparatus safety rules.

(ii) Check that the electrical equipment are set to match the available mains voltage and

correct fuse.

(iii) Do not perform any internal adjustments, maintenance or repair.

(iv) Follow moving object safety rules.

(v) All the connections of accelerometer to analyzer should carefully be made straight

using adhesive tape and multichannel selector.

(vi) To avoid undesirable excitation of the bearing of the motor be careful in increasing

the motor speed.

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Theory

Unbalance in rotating machine element is the result of uneven distribution of the

rotor's mass, and causes vibrations to be transmitted to the bearings. The principal

component of each of these vibrations has the same frequency as the frequency of

rotation. This can theoretically be proven by considering a single out-of-balance mass on

a rigid disk as follows.

As shown in figure (1), the non rotating mass is equal to (M- m) and the displacement of

mass, m is given by the expression (x + e sin w t). The equation of motion for the system

is:

rearranging,

which becomes the equation of forced vibrations with the steady state solution given by:

and,

The complete solution can be found as follows.

Thus to eliminate the unwanted vibrations, balancing is required. The balancing of rotors

reduces both the dynamic load on the bearings and the vibration isolation requirements of

the machines. The balancing is achieved by modifying the distribution of mass in a

controlled fashion.

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There are two kinds of the balancing as follows.

(1) STATIC BALANCING: The process where primary forces caused by unbalanced

mass components in a rotating object may be resolved into one plane and balanced by

adding a mass in that plane only. As the object would now be completely balanced in the

static condition, this is known as static balancing.

(2) DYNAMIC BALANCING or MULTI-PLANE BALANCING: This describes the

process where the primary forces and secondary forces caused by unbalanced mass

components in a rotating object may be resolved into two or more planes and balanced by

adding mass increments in those planes. This balancing process is known as dynamic

balancing. After being balanced dynamically the object would be completely balanced in

both static and dynamic conditions.

The balancing methods are described under Experimental Procedure.

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Experimental Setup

The experimental setup shown in figure (2) consists of the following:

(i) The vibration table (concrete beds)

(ii) Two accelerometers with their accessories

(iii) Multichannel selector

(vi) Vibration analyzer (Type 2515) or a multichannel analyzer 3550

(vii) Balancing demonstration unit (Type WA-155)

(viii) Electronic Balance

(ix) Set of known trial masses.

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Experimental Procedure

Case 1: Static Balancing

Method 1: Refer to figure (3) which is a graphical representation of the

velocity vectors. In the vector diagram the initial unbalance is indicated by V0 with an

angle a0. Adding a trial mass M causes velocity of vibration of the magnitude V1with an

angle of a1 . The effect of trial mass alone can be determined from the vector diagram as

VT and aT. The magnitude of the vector VT, can also be obtained by analytical vector

mechanics as follows.

Using Cosine Law for the triangles in Figure 3:

Where aT is the angle between VT and V0, and can be found as

Thus the position of the correction mass can be given by the angle referred to the position

of the trial mass as positive in the direction of rotation.

The initial unbalance and consequently the correction mass is given by:

and the position of the correction mass can be given by the angle referred to the datum as:

acomp = - aT + a0 + 1800 (5.10)

Method 2: Follow the steps given below making a total number of four runs:

(1) Weigh a trial mass (e.g. MT = 4.7 g)

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(2) Connect Demo machine to vibration analyzer via a multi channel selector to be

able to make observations for plane-2 as well.

(3) Since the demo unit is initially well balanced, connect an external known

mass to make it unbalanced.

(4) Record the vibration velocity while running the unit (say V0 mm/s).

(5) Fix a trial mass MT at a known position and run the unit with the same speed to

record the vibration velocity V1 due to combined unbalance (i.e. initial + trial

mass at 1).

(6) Move the same trial mass MT to position 2 which is 1800 apart at the same radial

position and run the unit again as before to record V2.

(7) Draw the vector diagram for the known vector magnitudes and with unknown

angles as follows:

(a) Draw two concentric circles of radii V1 and V2 which are proportional to

the velocities.

(b) Draw two radii in respective circles at arbitrary angles.

(c) Join points A and B.

From the Figure (4):

since, a=- means cos a = -cos , therefore

also VT1 = VT2 =VT .Thus Equations (5.11) and (5.13) yield

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Where aT is the angle between VT (first possition of trial mass) and V0. Thus we can

obtain the correction mass, MC as follows:

This correction mass has to be fixed at an angle aT referenced to the trial mass.

Case 2: Dynamic Balancing

Method : The dynamic balancing is more complex due to vector forces in more than

one plane. A computer program can be written to combat this difficulty. A balancing

program WT 8121 of Bruel & Kjaer can be employed to estimate the correction masses

and their positions required for dynamic balancing. The step by step procedure is given as

follows:

(1) Prepare the experimental setup as shown in Figure (2).

(2) Record the initial unbalanced velocities (do not forget to make the demo

machine unbalanced by connecting known masses at known positions) and phase

angles without any trial masses for Plane-1 and Plane-2 in table-1.

(3) Connect a known trial mass in Plane-1 and mark its position.

(4) Run the machine and record the velocities and phase angles in table-1 for both

the planes.

(5) Remove the trial mass from Plane-1.

(6) Connect another trial mass in Plane-2 and mark its position.

(7) Run the unit again to record velocities and phase angles for both the planes in

table-1 as before.

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(8) Feed the data recorded in Table-1 to run your program or use Pocket Computer

(Organizer II, Software WT8121) to find out the correction masses and angles.

(9) Remove the trial mass and connect the estimated correction masses at their

respective positions.

(10)Run the demo machine again to check the perfection of balancing and note down

the residual unbalance, if any.

Hints for developing the 2-Plane Balancing PROGRAM:

Let us assume the vibration velocity vectors as follows:

V10 velocity in Plane-1 with no trial mass (initial unbalance),

V20 velocity in Plane-2 with no trial mass (initial unbalance),

V11 velocity in Plane-1 with trial mass in Plane-1,

V21 velocity in Plane-2 with trial mass in Plane-1,

V12 velocity in Plane-1 with trial mass in Plane-2, and

V22 velocity in Plane-2 with trial mass in Plane-2.

Now,

V11 - V10 effect in Plane-1 of a trial mass in Plane-1,

V12 - V10 effect in Plane-1 of a trial mass in Plane-2,

V21 - V20 effect in Plane-2 of a trial mass in Plane-1, and

V22 - V20 effect in Plane-2 of a trial mass in Plane-2.

To balance the rotor completely, correction masses must be connected in both the

balancing planes in such a manner that their combined velocity vectors in each plane

cancel out the initial (original unbalancing) vectors V10 and V20. Let us assume that Q1

and Q2 are the changes in the two planes relative to the applied trial masses.

Now the balancing equations can be written as:

Q1 . (V11 - V10) + Q2 . (V21 - V10) = - V10 (5.16)

Q1 . (V21 - V20) + Q2 . (V22 - V20) = - V20 (5.17)

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Equations (5.13) and (5.14) can be solved simultaneously to get vector equations:

where,

Therefore, the correction mass for both of the planes can be computed as:

and

with their respective positions of a1 and a2.

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Sample Calculations

Calculations for plane 1:

Graphical methodVT = 2.29 mm/sαT = 58 deg

Therefore, the correction mass is calculated as follows:

The error in correction mass is

The position of the correction mass is calculated as follows:

Method 1 Velocity due to the trail mass is calculated as follows:

Phase angle due to trail mass is calculated as follows:




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Method 2

Velocity due to the trail mass is calculated as follows:

Phase angle due to trail mass is calculated as follows:




Observation Data Tables

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Plane-1:

The vibration velocity of factory balanced disk, Vf1 = 7.934 m/s

The initial unbalancing mass connected, M0 = 0.80 gm

Velocity of the initial unbalance, V0 = 4.072 mm/s

Phase angle, a0 = 146 deg

Trial mass, MT = 0.4 gm

Velocity of vibration of the magnitude V1 = 4.73 mm/s


Velocity of vibration of the magnitude V2 with MT rotated 180°= 3.913 mm/s


Phase angle, afactory = 217 deg

Plane-2:

The vibrational velocity of factory balanced disk, Vf1 = 731.1 m/s

The initial unbalancing mass connected, M0 = 0.40 gm

Velocity of the initial unbalance, V0 = 3.06 mm/s


Trial mass, MT = 0.80 gm

Velocity of vibration of the magnitude V1 = 7.537 mm/s


Velocity of vibration of the magnitude V2 with MT rotated 180°= 3.852 mm/s


Phase angle, afactory = 127 deg

Temporary trial mass in Plane-1, MT1 = 0.80 gm

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Temporary trial mass in Plane-2, MT2 = 0.80 gm

Table 1: Dynamic balancing of two plane demo machine

Run

Plane-1 Plane-2

Velocity, A

(mm/s)

Phase, P

(deg)

Velocity, A

(mm/s)

Phase, P

(deg)

1 No trial mass 2.854 144 6.198 111

2 Trial mass on plane-1 4.301 226 5.996 125

3 Trial mass on plane-2 2.490 113 8.093 91

Table 2: Correction masses for static balancing in Plane-1 and Plane-2

Corrections

Correction in plane-1 Correction in plane-2

Mass, MC1

(gm)Angle (deg)

Mass, MC2

(gm)Angle (deg)

Graphical 0.7113 268.00 0.42 206.00

Method 1 0.71 234.38 0.419 150.33

Method 2 1.08 219.25 0.476 117.20

Table 3: Residual unbalancing in two plane demo machine for Static balancing

Residual

unbalance

Plane-1 Plane-2

Velocity, A

(µm/s)Phase, P (deg)

Velocity, A


174.7 32 801.9 135

Table 4: Correction masses and residual unbalancing of two plane demo machine

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Corrections

Correction in plane-1 Correction in plane-2

Mass, MC1

(gm)Angle (deg)

Mass, MC2

(gm)Angle (deg)

0.473 -242.797 1.425 -123.043

Table 5: Residual unbalancing in two plane demo machine

Residual

unbalance

Plane-1 Plane-2

Velocity, A


Velocity, A


8.846 223 6.224 176

Discussion

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In this experiment the balancing operation is carried out on a system of rotating disk

which is imbalanced due to an excessive load. The magnitude of velocity of vibration act

as indicator of imbalance. If the magnitude of velocity of vibration is high this indicates

the presence of imbalance in the system, while a low magnitude will indicate that the

system has no imbalance and does not require balancing. Initially during experiment a

high velocity magnitude is recorded which shows imbalance so the system requires

balancing. Two types of balancing operations are applied which are static and dynamic

balancing.

In static balancing firstly, the graphical approach was applied where the Velocity of

vibration vectors were drawn for both initial imbalance and with trial mass added. A

resultant vector (VT ,αT) is found and is used to find the position and magnitude of counter

mass that is to be added to the system. Secondly, the analytical method 1 and 2 are used

to find the position and magnitude of the counter mass. In dynamic balancing a computer

program is used to compute position and magnitude of the counter mass due to increased

complexity of the problem.

To check the validity and accuracy of the results the mass of the unbalancing mass was

recorded using a mass balance before adding to the system. This value of the unbalance

mass was compared to the mass of imbalance found using different methods. Also, after

balancing the system residual unbalance was recorded. It can be seen that the magnitude

of velocity in the case of residual imbalance is very small ,which show that the imbalance

in the system has been successfully corrected.

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Conclusion

The following can be concluded from the experiment:

Static balancing can be used in a single phase system to correct imbalance.

Dynamic balancing can be used in multi-phase systems to correct imbalance.

Three different methods used for static balancing all produce results close to each

other so all of these methods can be successfully applied to balance a system.

20

Nomenclature

e eccentricitym out-of-balance mass

t time

x(t) total response

A amplidue of velocity vector used in program WT8121

C damping coefficient

K spring stiffness

M total rotating mass

M0 initial unbalance mass

MC correction mass

MC1 correction mass for plane-1

MC2 correction mass for plane-2

MT trial mass

MT1 trial mass in plane-1

MT2 trial mass in plane2

P phase angle of velocity vector in degree, used in program WT8121

V0 initial velocity

V1 velocity with trial mass

V2 velocity with trial mass rotated to 180 deg

X steady state amplitude

a1 Phase angle for V1

a2 Phase angle for V2

acomp position of correction mass

aT phase angle for velocity due to trial mass alone

f phase lag for steady state amplitude

w circular frequency in rad/sec

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References

1. Thomson W. T., Theory of Vibration with Applications, 3rd edition, Prentice

Hall, Unwin Hyman Ltd., London, 1988.

2. Rao S. S., Mechanical Vibrations, 3rd edition, Addison-Wesley Publishing

Company, Reading, Massachusetts, U. S. A., 1995 pp 565 - 575.

3. Broch J. T., Mechanical Vibrations & Shock Measurement, Bruel & Kjaer,

Denmark, 1991.

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Appendix

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Balancing of Rotating Disks

Documents

Transcript of Balancing of Rotating Disks