NUS ME3112-2 lab

ME3112-2 DYNAMIC BALANCING & GYROSCOPIC EFFECTS

SEMESTER 5

SESSION 2014/2015

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

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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

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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

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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 vector ~ ri 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:

 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   m ~z i

i

i

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  mb rb 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)

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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 7

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

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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.

N i (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

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