ME3112-2 DYNAMIC BALANCING & GYROSCOPIC EFFECTS
SEMESTER 5
SESSION 2009/2010
Department of Mechanical Engineering
National University of Singapore
CONTENTS
LIST OF ILLUSTRATIONS
LIST OF TERMINOLOGY & SYMBOLS
INTRODUCTION
PART A:
PART B:
UNBALANCED DYNAMIC FORCES
1.
Description of Equipment & Instrumentation
2.
Principle and Theory of Operation
3.
Experimental Procedure
4.
Discussion
5.
Tabular Working Sheet
GYROSCOPIC EFFECTS
1.
Description of Equipment & Instrumentation
2.
Principle and Theory of Operation
3.
Experimental Procedure
4.
Discussion
5.
Tabular Working Sheet
REFERENCES
i
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
ii
LIST OF TERMINOLOGY AND SYMBOLS USED
PART A:
UNBALANCED DYNAMIC FORCES
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)
PART B:
GYROSCOPIC EFFECTS

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
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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.
PART A:
1.
UNBALANCED DYNAMIC FORCES
Description of Equipment & Instrumentation
Fig. A1: High Precision Balancing Machine - IRD B01/S
1
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 twoplane 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
2
Fig. A3: Rotor Specimen for Balancing Experiment
2.
Principle and Theory of Operation
2
Me
M
e

O
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
Me2. 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).
Added mass, m
Rotating Disc
Center of rotation
( new center of
gravity)
Original center
of gravity
Center of gravity
(a)
(a) 'Natural' Position of the non-rotating
unbalanced disc.
(b)
(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
3
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.
 m ~r  0
(A1)
i i
where i = 1, 2, ... which denotes the total number of masses and m denotes the mass. The
ri represents the eccentricity e of the mass mi from the axis of rotation.
vector ~
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:
 m ~z  ~r  0
i i
(A2)
i
where ~z i 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 ~z i . Mathematically, if the distance between these
two plane is ~z o , then
 m (~z  ~z )  ~r  0
i
i
o
(A3)
i
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.
3.
Experimental Procedure
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.
4
(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

c
d
b
c2 = a2 + b2 - 2ab cos 
d2 = (2a)2 + b2 - 2(2a)b cos 
2a2 = d2 - 2c2 + b2
solving
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:
 m~z  ~r
 m ~z  ~r
o
a
i i
i

a
b
(A4)
It is also clear that if the counter-balance mass mb is to be placed on the unbalanced
disc, then:
 m ~z  ~r
i
i
i

 m ~z
b
o
~
rb
(A5)
5
where ~
rb is the vector representing the position of the counter-balanced mass mb.
Therefore, combining Eqs. (A4) and (A5) gives:
mra a

mrb b
(A6)
If the counter-balance mass is to be placed at the same radius as the trial mass used in
the experiment, then:
mb 
b
m
a
(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.
5.
Tabular Working Sheet
Table A1:
Unbalanced Dynamic Forces --- Two-Plane Balancing
Initial unbalanced mass introduced:
weight: _______________
Trial Mass Added
M
(g)
angular position: ____________________
Deflections of the spring
b
(mm)
c
(mm)
D
(mm)
6
Resultants
A
(mm)

Counter balance
mass required
Mb
(g)
PART B:
1.
GYROSCOPIC EFFECTS
Description of Equipment & Instrumentation
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.
2.
Principle and Theory of Operation
T
P
O

H'
B
H
H
A

S
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 velocityr .
For the given direction
of spin, the angular moment H ( = I) is represented by vector OA at
r
one instant and OB at some future time. As the angular momentum is a vector quantity, the
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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
 ( I )
t
(B1)


But  ( I )  H  AB  OA   where  is the angle through which the axis of spin rotates
in time t
 d
 T  OA
 I  
dt
Where  
(B2)
d
= precession rate in radian per second.
dt

The vector AB lies in the plane SOT and in the limit when  is very small, its direction is

perpendicular to OA 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 AB , i.e. direction OT.
Following guideline can be used in determining the sense of gyroscopic torque.
S (spin)
CW
CW
ACW
P (precession)
CW
ACW
ACW
T (torque)
CW
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:
 

T    ( I )
(cross product)
(B3)
P (precession axis)
T (torque axis)
E
F
A

O
H
H
l
D
H'
C
B
S (spin axis)
F
Fig. B3: Torque-Precession Relations for a Spinning Disc Undergoing Gyroscopic Motion
8
TABLE B3
Knob Position
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
T1
7.0
7.5
8.0
T2
7.0
8.0
8.0
X
5.0
5.0
5.0
B = 15
3.
Rotor SpinSpeed(rpm)
C = 28
P.C.D = 35.5
Experimental Procedure
(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, T  M rotor gl , 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).
9
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.
5.
Tabular Working Sheet
Table B1: Gyroscopic Effects
Spin Rate
S/No.
Ni (rpm)
i
rad/sec
Precession Rate
 1
  =  sec/rad
 
t i sec
 (=
2
) rad/sec
ti
1
2
3
4
5
Results
Texpl (Nm)
(T / I) expt (= a)
T / Itheo
10
Difference (T / I) expt -
T / Itheo
Table B2: Regression analysis for results of gyroscopic effects experiment
 i'
S/No.
i ' i
 i' 2
i
1
2
3
4
5
n=

 ' 
i
  '
2
i
i
  '  

  '
2
i
i

Equations for Least Square Method:
T  1 T 
       '  a 'b
 I   I 
a
n  1 i '  i   i '   i
n  1  i ' 2    i '2
  '     '  ' 
b
n  1  '    '
2
i
i
i
i
i
i
2
2
i
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
11
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