nus me3112-2 lab

ME3112-2 DYNAMIC BALANCING & GYROSCOPIC EFFECTS

SEMESTER 5

SESSION 2014/2015

Department of Mechanical Engineering

National University of Singapore

i

CONTENTS

LIST OF ILLUSTRATIONS

LIST OF TERMINOLOGY & SYMBOLS

INTRODUCTION

PART A: UNBALANCED DYNAMIC FORCES

1. Description of Equipment & Instrumentation

2. Principle and Theory of Operation

3. Experimental Procedure

4. Discussion

5. Tabular Working Sheet

PART B: GYROSCOPIC EFFECTS




4. Discussion


REFERENCES

ii

LIST OF ILLUSTRATIONS

Fig. A1: High Precision Balancing Machine - IRD B01/S

Fig. A2: Balancing Machine for Experiment

Fig. A3: Rotor Specimen for Balancing Experiment

Fig. A4: Unbalanced Disc.

Fig. A5: Illustration of the 'Simplest' Method of Static Balancing

Fig. A6: Moment Diagram of Unbalanced Forces

Fig. B2 Experimental Setup for Study of Gyroscopic Effects

Fig. B2: Disc Spinning about OS Undergoing Precession About OP

Fig. B3: Torque-Precession relations for a Spinning Disc Undergoing

GyroscopicMotion

Table A1: Unbalanced Dynamic Forces --- Two-Plane Balancing

Table B1: Gyroscopic Effects

Table B2: Regression Analysis for Results of Gyroscopic Effects' Experiment

iii

LIST OF TERMINOLOGY AND SYMBOLS USED


M mass of the unbalanced rotor g

e eccentricity about the rotating axis mm

angular speed of the rotor rad/s

Fc centrifugal force (N or kN)


angular velocity or 'spin' of the disc rad/s

angular velocity or 'precession' of the disc rad/s

I mass moment of inertial of the revolving disc Nm2

T gyroscopic torque or couple Nm

H angular momentum of the disc gm2/s

1

INTRODUCTION

Both unbalanced dynamic forces and gyroscopic effects have profound influence on the

working and of rotating machinery like turbines, compressors, pumps, motors etc. Since they

have different effects on the behavior and performance of the rotating system, they are being

considered separately and described under Part A and Part B.

Part A refers to the experiment connected with unbalanced dynamic forces. These forces act

directly on the bearings supporting the rotor and thus increase the loads and accelerate the

fatigue failure. These unbalanced forces induce further mechanical vibrations in the

machinery and connected parts thereby creating environmental noise problem through

radiation of sound. Hence it is desirable to balance all such uncompensated masses and thus

reduce the effect of unbalance forces in a dynamics balancing machine.

Part B refers to the experiments connected with gyroscopic effects. This effect is felt in all

rotating machinery whenever the axis of rotation(spin) undergoes a change of direction (for

example as in an aircraft, ship, automobile etc.). Such a system experiences an additional

moment or torque resulting in higher stresses on the bearings. However gyroscopic effects

have certain beneficial effects as compared to the unbalance dynamic forces in applications

like inertial guidance, gyrostabilizers, navigation etc. It is therefore necessary to account for

gyroscopic effects while designing rotating machinery and system.



Fig. A1: High Precision Balancing Machine - IRD B01/S

2

Figure A1 shows a high precision balancing machine used in production balancing. It is

capable of balancing rotor of mass up to 5 kg. It provides solution for single-plane and two-

plane problem for between bearing and overhung rotors. It can achieve balance accuracy up to

2 micron eccentricity. The equipment is capable of storing up to 50 different rotor

configurations. The rotor is driven by a variable speed motor via a flat belt and the rotor

rotational speed is captured through a digital tachometer. The vibration due to the unbalanced

rotor is measured using the linear variable differential transducer (LVDT). All these captured

information are fed directly into the on-board processor of the balancing machine. Hence it is

a complete automatic operation. With standard operation procedure, the unbalance in the

rotor is detected and the compensations for the unbalance are automatically computed. After

applying the compensations to the unbalanced rotor (either by removing or introducing

counter-balance masses), the balancing exercise is repeated to determine the reduction in the

rotor unbalance. Further balancing exercise is conducted until the desired balanced rotor

condition is achieved.

The high precision balancing machine is only for demonstration purpose. A simplified

version of this equipment is used for conducting the experiment instead. Figure A2 illustrates

the experimental setup which is designed for two-plane balancing. Figure A3 shows the rotor

used in the experiment. It consists of a shaft and two attached end discs. The discs are located

exactly at the position of the balancing plane and they have provision to attach masses in

order to introduce unbalance. The driving and vibration monitoring instrumentation are

similar to the high precision balancing machine. With the rotor rotational speed captured

using a hand-held tachometer and the amplitude of vibration measured using a digital

oscilloscope, the unbalance in the rotor can be determined and the compensation for the

unbalance can be calculated. The following section presents the theory for the dynamic

balancing exercise.

Fig. A2: Balancing Machine for Experiment

3

Fig. A3: Rotor Specimen for Balancing Experiment


O

e

M

Me

2

Fig. A4: Unbalanced Disc

Consider a body of mass M rotating with a uniform angular velocity about O with

eccentricity e, as shown in Fig. A4. The centrifugal force Fc acting on the axis of rotation is

Me2. This force is therefore very sensitive to speed and hence there is a need to reduce this

force either by operating at lower speeds or by decreasing the eccentricity (as shown in Fig.

A5).

Center of rotation

( new center of

gravity)

Rotating Disc Added mass, m

Center of gravityOriginal center

of gravity

(a) (b)

(a) 'Natural' Position of the non-rotating

unbalanced disc.

(b) Sketch illustrating the result of static balancing of the disc. (The

center of gravity of the disc now coincides with the center of

rotation).

Fig. A5: Illustration of The 'Simplest' Method of Static Balancing

4

In general, it is desirable to make e as low as possible and this is done in a balancing machine.

If the rotor thickness to diameter ratio is less than 0.5 and speed are below 1000 rpm, the

rotor can be single plane balanced - otherwise two-plane balancing is necessary. Further if the

operating speed of a rotor is less than 30% of its critical speed, it is considered as a rigid rotor

for balancing purposes. Most of the flexible rotors operate at a speed of at least 70% higher

than of the critical speed for safe operation. In the single plane balancing technique, with a

rotor of mass, mi and radius ri from the axis of rotation and lying in the same plane, the

condition for static balance is used.

0~ii rm (A1)

where i = 1, 2, ... which denotes the total number of masses and m denotes the mass. The

vector ir~ represents the eccentricity e of the mass mi from the axis of rotation.

When the bodies are rotating in several planes, the condition for dynamic balance has to be

satisfied in addition to that for static balance. This is given by:

0~~ iii rzm (A2)

where iz~ is the axial coordinate vector of the mass mi measured for a chosen datum. In the

two-plane balancing technique, instead of satisfying Eqs. (A1) and (A2) explicitly, Eq. (A2) is

used with two different datum planes for iz~ . Mathematically, if the distance between these

two plane is oz~ , then

0~)~~( ioii rzzm (A3)

It is therefore, clear that Eqs. (A2) and (A3) imply the satisfaction of Eq. (A1). Conceptually,

it means that if a system of bodies rotating in several plane is in dynamic balance with respect

to two different datum planes, then the system is also in static balance. This is the principle of

the two-plane balancing technique.


The procedure described below is for two-plane balancing exercise. It should be conducted

for one plane at a time. In the experiment, balancing exercise is conducted for only one of the

two plane.

(a). As the existing discs of the rotor are initially balanced, it is necessary to introduce some

imbalance into the rotor before performing the balancing exercise. This can be done by

randomly adding masses to the one side of the disc.

(b). Place the rotor on the balancing machine, with the unbalanced disc on the free swinging

pivot support. Fit the flat belt over the rotor shaft and the driving pulley on the variable

speed DC motor.

5

(c). Switch on the LVDT and observe the output signal from the digital oscilloscope. Adjust

the zero level to an appropriate reference value before starting the experiment.

(d). Start the driving motor and observe the vibration caused by the unbalanced rotor over

the digital oscilloscope (refer to the calibration data attached to each balancing machine

for the actual displacement measured).

(e). Adjust the motor speed such that amplitude of vibration is about 0.1 to 0.2 mm. Record

the rotor speed, (rpm) using a hand-held digital tachometer and the amplitude of

vibration, b.

(f). Add a trail mass m to the unbalanced disc at any location with a radial distance ra, Bring

the rotor to speed the initial speed and note the new amplitude of vibration, c.

(g). Increase the trial mass to 2m at the same location and repeat step f. Record the new

amplitude of vibration, d.

(h). Let a, be the amplitude of vibration due to the trial mass m alone. The following

relations are obtained between the various variables by using Eq. (A2) in the form of a

moment diagram as shown in Fig. A6. From which the amplitude of vibration, a and the

phase angle, can be determined.

2a

a

b

cd

c2 = a2 + b2 - 2ab cos

d2 = (2a)2 + b2 - 2(2a)b cos

solving 2a2 = d2 - 2c2 + b2

Fig. A6: Moment Diagram of Unbalanced Forces

(i). From step h, determine the amount of counter-balance mass, mb required to balance the

disc. This can be done by first noting that the magnitude of a and b in the moment

diagram are proportional to the moment of the unbalance forces due to the trial mass m

and the initial unbalanced masses, respectively. Thus:

b

a

rzm

rzm

iii

ao

~~

~~

(A4)

It is also clear that if the counter-balance mass mb is to be placed on the unbalanced

disc, then:

bobiii rzmrzm~~~~ (A5)

6

where br~ is the vector representing the position of the counter-balanced mass mb.

Therefore, combining Eqs. (A4) and (A5) gives:

b

a

rm

mr

bb

a (A6)

If the counter-balance mass is to be placed at the same radius as the trial mass used in

the experiment, then:

mb

amb (A7)

Finally, in order to specify the correct angular location of the counter-balance mass, the

value of obtained from step h must be used.

4. Discussion

Explain why it is necessary to have 2 trail masses added to determine the counter-balance

mass and the location.


Table A1: Unbalanced Dynamic Forces --- Two-Plane Balancing

Initial unbalanced mass introduced:

weight: _______________ angular position: ____________________

Trial Mass Added Deflections of the spring Resultants Counter balance

mass required

m

(g)

b

(mm)

c

(mm)

d

(mm)

a

(mm) Mb

(g)

7



Fig B1: Experimental Setup for the Study of Gyroscopic Effects

Figure B1 shows a view of the gyroscopic model on which tests are to be conducted for

determining the rate of precession and its sense for a given spin magnitude and direction, and

an applied torque. This consists of a rotor-disc system supported on bearing pedestals and

driven by a variable speed D.C. motor through a pinion-gear mechanism. The motor can be

withdrawn through swinging it freely about a vertical axis with the help of a revolving

platform. The torque is applied to the disk by quickly removing one of the supports. The

precession rate is possible to give only one sense of direction for spin as well as torque.

Hence direction of precession is always the same.


O

A

B

H

H'

S

TP

H

Fig. B2: Disk Spinning About OS Undergoing Precession About OP

Refer to Fig. B2. It shows a disc spinning in a vertical plane with angular velocity and the

axis of spin is simultaneously rotating in a horizontal plane SOT with an angular velocity .

For the given direction of spin, the angular moment H ( = I) is represented by vector OAr

at

one instant and OBr

at some future time. As the angular momentum is a vector quantity, the

8

resultant moment is found by applying the right hand screw rule (RHSR). The change in H

(i.e. H) is only produced by the action of a couple or torque on the disc. Hence by Newton's

Law

t

IT

)( (B1)

But AOBAHI

)( where is the angle through which the axis of spin rotates

in time t

Idt

dAOT

(B2)

Where d

dt

= precession rate in radian per second.

The vector BA

lies in the plane SOT and in the limit when is very small, its direction is

perpendicular to AO

and therefore to the plane SOP. The gyroscopic couple thus acts in the

plane SOP, and its sense must be clockwise when viewed the direction BA

, i.e. direction OT.

Following guideline can be used in determining the sense of gyroscopic torque.

S (spin) P (precession) T (torque)

CW CW CW

CW ACW ACW

ACW ACW CW (CW - Clockwise; ACW - Anti-Clockwise)

To overcome this gyroscopic effect, a couple is applied on the bearings in a rotor-bearing

system (Fig. B3) in the opposite sense, which act as an additional load. Vectorially

represented the sense of torque may be determined by applying the RHSR to the following

relation:

)(

IT (cross product) (B3)

H

A

BC

D

E

P (precession axis)

T (torque axis)

S (spin axis)

O

lF

F

H

H'

Fig. B3: Torque-Precession Relations for a Spinning Disc Undergoing Gyroscopic Motion

9

TABLE B3

Knob Position Rotor SpinSpeed(rpm) Set-up Dimensions (mm)

Set-up G4 G5 G6 Dimension G4 G5 G6

3 2793 3120 2598 L1 101.0 101.0 100.0

4 3858 4538 3803 L2 88.0 87.0 89.0

5 4919 6073 5077 L3 50.0 51.0 46.0

6 5976 7566 6306 D1 9.5 9.5 9.5

7 7057 8989 7528 D2 8.0 8.0 8.0

Density of Steel : 7800 kg/m3 D3 69.0 70.0 70.0

Density of Gear : 1190 kg/m3 D4 20.0 20.0 20.0

C1 21.0 20.0 20.0

Dimensions of Gear (mm) C2 10.0 11.0 12.5

A = 10 C = 28 T1 7.0 7.5 8.0

B = 15 P.C.D = 35.5 T2 7.0 8.0 8.0

X 5.0 5.0 5.0


(a). Rotate the disc at some speed by adjusting the knob position and note the spin rate,

from Table B3.

(b). Apply the torque instantly by quickly withdrawing the removable support and record the

processional rate with a stop-watch.

(c). Repeat the above procedure for different spin rates by varying the knob position and

note the corresponding processional rates.

(d). Using the equation of moment, calculate the gyroscopic torque, glMT rotor , where

Mrotor = total mass of spinning assembly

g = acceleration due to gravity

l = distance between center of gravity of spinning assembly to

the center of fixed support

Calculate the moment inertial, I from the given dimensions in Fig. B4 and Table B3 and

thus obtain ( / )T I theo.

(e). Using the Least Square Method, plot against ' (=1/) determine the experimental value of ( / )T I expt from the slope of the line. (Equations apply in Least Square Method

can be found with Table B2).

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Fig. B4

4. Discussion

Compare the two values of ( / )T I and comment on the errors involved in the experiments as

well as the model setup.


Table B1: Gyroscopic Effects

Spin Rate Precession Rate

S/No. Ni (rpm) i rad/sec

1=

sec/rad

t i sec (=2

t )

i

rad/sec

1

2

3

4

5

Results

Texpl (Nm) (T / I)expt (= a) theoI/T Difference (T / I)expt - theoI/T

11

Table B2: Regression analysis for results of gyroscopic effects experiment

S/No. i' i i'2 i i'

1

2

3

4

5

n = 'i i 2'i i i'

2

'i

Equations for Least Square Method:

baI

T

I

T

''

1

22 ''1''1

ii

iiii

n

na

22

2

''1

'''

ii

iiiii

nb

Note: Theoretically 'a' should correspond to (T/I)expl and 'b' should be zero.

REFERENCES

(1) J.E. Shigley, 'Theory of Machines', McGraw-Hill

(2) Ham, Crane, Rogers, 'Mechanics of Machinery', McGraw-Hill

(3) Hannah & Rogers. 'Mechanics of Machines', Edward Arnold Press

nus me3112-2 lab

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