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
th
Performed on: Wednesday December 16 Submitted on: Wednesday
, 2006
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
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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 ω t). The equation of motion for the system is:
..
. d2 (M − m) x + m 2 (x + e sin ω t) = - K x - C x dt ..
(5.1)
rearranging, ..
..
.
M x + C x + K x = (m e ω2 ) sin ω t
(5.2)
which becomes the equation of forced vibrations with the steady state solution given by: X=
meω 2
(5.3)
(K − M ω2 ) 2 + (C ω) 2
and, tan φ =
Cω K − M ω2
(5.4)
The complete solution can be found as follows.
x(t) = X1 e
- ξ ωn t
sin( 1 - ξ ωn t + φ1 ) + 2
m e ω2 sin( ω t - φ ) (K - M ω2 ) 2 + (C ω ) 2
(5.5)
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
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the machines. The balancing is achieved by modifying the distribution of mass in a controlled fashion.
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 α0. Adding a trial mass M causes velocity of vibration of the magnitude V1with an angle of α1 . The effect of trial mass alone can be determined from the vector diagram as VT and αT. The magnitude of the vector VT, can also be obtained by analytical vector mechanics as follows. VT = V0 + V1 − 2V0 V1 Cos ( α 0 −α1 ) 2
2
2
(5.6)
Using Cosine Law for the triangles in Figure 3: 2
2
2
V1 = V0 + VT − 2VT V0 Cosα T V 2 + VT 2 − V12 α T = Cos −1 0 2V V T 0
(5.7) (5.8)
Where αT 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. M0 =
V0 * MT VT
(5.9)
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: αcomp = - αT + α0 + 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. 2
2
2
V2 = VT2 + V0 − 2VT2 V0 cosα
(5.11)
From the Figure (4): 2
2
2
(5.13)
2
2
2
(5.12)
V1 = VT1 + V0 + 2VT1 V0 cosα V1 = VT1 + V0 − 2VT1 V0 cosβ
since, α=π-β means cos α = -cos β, therefore also VT1 = VT2 =VT .Thus Equations (5.11) and (5.13) yield
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2 2 2 V1 2 − V2 2 V1 + V2 − 2V0 α T = π − Cos and VT = 2 4VT V0 −1
(5.14)
Where αT is the angle between VT (first possition of trial mass) and V0. Thus we can obtain the correction mass, MC as follows:
MC =
V0 * MT VT
(5.15)
This correction mass has to be fixed at an angle αT 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.
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(7)
Run the unit again to record velocities and phase angles for both the planes in table-1 as before.
(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:
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Q1 . (V11 - V10) + Q2 . (V21 - V10) = - V10
(5.16)
Q1 . (V21 - V20) + Q2 . (V22 - V20) = - V20
(5.17)
Equations (5.13) and (5.14) can be solved simultaneously to get vector equations:
Q1 =
− V10 − Q2.(V12 − V10) = a + jb (V11 − V10)
Q2 =
− V20(V11 - V10) − V10.(V21 − V20) = c + jd (V21 − V20)(V12 - V10) - (V22 - V20).(V11 - V10)
(5.18) (5.19)
where, Q1 = (a) 2 + (b) 2
Q2 = (c) 2 + (d) 2
b α1 = tan −1 a d α 2 = tan −1 c Therefore, the correction mass for both of the planes can be computed as: M C1 = M T1 . Q1 and M C2 = M T2 . Q2
with their respective positions of α1 and α2.
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Sample Calculations Calculations for plane 1: Graphical method VT = 2.29 mm/s αT = 58 deg Therefore, the correction mass is calculated as follows: V 4.072 mo = mT o = 0.4 × = 0.7113 gm VT 2.29 The error in correction mass is 0.8 − 0.7113 % Error = × 100 = 11.09 % 0.8 The position of the correction mass is calculated as follows: α comp = −α T + α 0 + 180 = −58 + 146 + 180 = 268 ° Method 1 Velocity due to the trail mass is calculated as follows: VT = V02 + V12 − 2V0V1 cos( α 0 − α 1 ) VT =
( 4.072 ) 2 + ( 4.73) 2 − 2 × 4.072 × 4.73 cos(146 − 117 )
= 2.294 mm / s
Phase angle due to trail mass is calculated as follows: V 2 + VT2 − V12 α T = cos −1 0 2VT V0 ( 4.072 ) 2 + ( 2.294 ) 2 − ( 4.73) 2 α T = cos 2 × 2.294 × 4.072 −1
= 91.62 °
Therefore, the correction mass is calculated as follows: V 4.072 mo = mT o = 0.4 × = 0.71 gm VT 2.294 The error in correction mass is 0.8 − 0.71 % Error = × 100 = 11.25 % 0.8
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The position of the correction mass is calculated as follows: α comp = −α T + α 0 + 180 = −91.62 + 146 + 180 = 234.38 ° Method 2 Velocity due to the trail mass is calculated as follows: V12 + V22 − 2V02 VT = 2 VT =
( 4.73) 2 + ( 3.913) 2 − 2 × ( 4.072 ) 2 2
= 1.504 mm / s
Phase angle due to trail mass is calculated as follows: 2 2 −1 V1 − V2 α T = π − cos 4 V V T 0 ( 4.73) 2 − ( 3.913) 2 α T = π − cos 4 × 1.504 × 4.072 −1
= 106.75 °
Therefore, the correction mass is calculated as follows: V 4.072 mo = mT o = 0.4 × = 1.08 gm VT 1.504 The error in correction mass is 0.8 − 1.08 % Error = × 100 = 35 % 0.8 The position of the correction mass is calculated as follows: α comp = −α T + α 0 + 180 = −106.75 + 146 + 180 = 219.25 °
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Observation Data Tables Plane-1: The vibration velocity of factory balanced disk, Vf1
=
7.934
µm/s
The initial unbalancing mass connected, M0
=
0.80
gm
Velocity of the initial unbalance, V0
=
4.072
mm/s
Phase angle, α0
=
146
Trial mass, MT
=
0.4
gm
Velocity of vibration of the magnitude V1
=
4.73
mm/s
Phase angle, α1
=
117
deg
deg
Velocity of vibration of the magnitude V2 with MT rotated 180°= 3.913 mm/s Phase angle, α2 Phase angle, αfactory
= =
174 217
deg deg
Plane-2: The vibrational velocity of factory balanced disk, Vf1
=
731.1
µm/s
The initial unbalancing mass connected, M0
=
0.40
gm
Velocity of the initial unbalance, V0
=
3.06
mm/s
Phase angle, α0
=
69
deg
Trial mass, MT
=
0.80
gm
Velocity of vibration of the magnitude V1
=
7.537
mm/s
Phase angle, α1
=
115
deg
Velocity of vibration of the magnitude V2 with MT rotated 180°= 3.852 mm/s Phase angle, α2 Phase angle, αfactory
= =
327 127
deg deg
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Temporary trial mass in Plane-1, MT1 =
0.80
gm
Temporary trial mass in Plane-2, MT2 =
0.80
gm
Table 1: Dynamic balancing of two plane demo machine
Plane-1 Velocity, A Phase, P
Run 1 2 3
No trial mass Trial mass on plane-1 Trial mass on plane-2
(mm/s) 2.854 4.301 2.490
Plane-2 Velocity, A Phase, P
(deg) 144 226 113
(mm/s) 6.198 5.996 8.093
(deg) 111 125 91
Table 2: Correction masses for static balancing in Plane-1 and Plane-2
Corrections Graphical Method 1 Method 2
Correction in plane-1 Mass, MC1 Angle (deg) (gm) 0.7113 268.00 0.71 234.38 1.08 219.25
Correction in plane-2 Mass, MC2 Angle (deg) (gm) 0.42 206.00 0.419 150.33 0.476 117.20
Table 3: Residual unbalancing in two plane demo machine for Static balancing
Residual unbalance
Plane-1 Velocity, A Phase, P (deg) (µm/s) 174.7 32
Plane-2 Velocity, A Phase, P (deg) (µm/s) 801.9 135
Table 4: Correction masses and residual unbalancing of two plane demo machine
Corrections
Correction in plane-1 Mass, MC1 Angle (deg)
Correction in plane-2 Mass, MC2 Angle (deg) 17
(gm) 0.473
-242.797
(gm) 1.425
-123.043
Table 5: Residual unbalancing in two plane demo machine
Residual unbalance
Plane-1 Velocity, A Phase, P (deg) (µm/s) 8.846 223
Plane-2 Velocity, A Phase, P (deg) (µm/s) 6.224 176
Discussion 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 18
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.
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.
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•
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.
Nomenclature e m
eccentricity out-of-balance mass
t
time
x(t)
total response
A
amplidue of velocity vector used in program WT8121 20
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
α1
Phase angle for V1
α2
Phase angle for V2
αcomp position of correction mass αT
phase angle for velocity due to trial mass alone
φ
phase lag for steady state amplitude
ω
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,
3.
Reading, Massachusetts, U. S. A., 1995 pp 565 - 575.
Broch J. T., Mechanical Vibrations & Shock Measurement, Bruel & Kjaer, Denmark, 1991.
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Appendix
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