DYNAMICS OF MACHINES

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DYNAMICS OF MACHINES By Dr.K.SRINIVASAN, Professo
r, AU-FRG Inst. for CAD/CAM, Anna
University Topic Balancing of Rotating masses
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What is balancing of rotating members?

• Balancing means a process of restoring a rotor
  which has unbalance to a balanced state by
  adjusting the mass distribution of the rotor
  about its axis of rotation

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Balancing "is the process of attempting to
improve the mass distribution of a body so that
it rotates in its bearings without unbalanced
centrifugal forces
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• Mass balancing is routine for rotating
  machines,some reciprocating machines,
• and vehicles
• Mass balancing is necessary for quiet operation,
  high speeds , long bearing life, operator
  comfort, controls free of malfunctioning, or a
  "quality" feel

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Rotating components for balancing
Pulley gear shaft assemblies Starter armatures Airspace components
High speed machine tool spindles flywheels Impellers
Centrifuge rotors Electric motor rotors Fan and blowers
Compressor rotors Turbochargers Precision shafts
 crank shafts  Grinding wheels Steam GasTurbine rotors
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Shaft with rotors
Bearing 1
Bearing 2
Unbalanced force on the bearing rotor system
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Cut away section of centrifugal compressor
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• Unbalance is caused by the displacement of the
  mass centerline from the axis of rotation.
• Centrifugal force of "heavy" point of a rotor
  exceeds the centrifugal force exerted by the
  light side of the rotor and pulls the entire
  rotor in the direction of the heavy point.
• Balancing is the correction of this phenomena by
  the removal or addition of mass

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Benefits of balancing

• Increase quality of operation.
• Minimize vibration.
• Minimize audible and signal noises.
• Minimize structural fatigue stresses.
• Minimize operator annoyance and fatigue.
• Increase bearing life.
• Minimize power loss.

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NEED FOR BALANCING

• Rotating a rotor which has unbalance
• causes the following problems.
• The whole machine vibrates.
• Noise occurs due to vibration of
• the whole machine.
• Abrasion of bearings may shorten
• the life of the machine.

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• Rotating Unbalance occurs due to the
  following reasons.
• ? The shape of the rotor is unsymmetrical. ?
  Un symmetrical exists due to a
• machining error. ? The material is not
  uniform, especially in
• Castings.
• ? A deformation exists due to a distortion.

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? An eccentricity exists due to a gap of
fitting. ? An eccentricity exists in the inner
ring of rolling bearing. ?
Non-uniformity exists in either keys or key
seats. ? Non-uniformity exists in the mass
of flange
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• Unbalance due to
• unequal distribution
• of masses
• Unbalance due to
• unequal distance of masses

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. Types of Unbalance
Static Unbalance
Dynamic Unbalance
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STATIC BALANCING (SINGLE PLANE BALANCING)
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Single plane balancing
Adequate for rotors which are short in length,
such as pulleys and fans
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F m r ?2
m
?
Magnitude of unbalance
r
O2
Vibration occurs
Elasticity of the bearing
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Balancing of several masses revolving in the
same plane using a Single balancing mass
m3r3 ?2
m2r2 ?2
?2
?1
?3
m1r1 ?2
bearing
m b
m4r4 ?2
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Graphical method of determination magnitude and
Angular position of the balancing mass
m4r4 ?2
m3r3 ?2
?b
m b r b ?2
m2r2 ?2
O
m1r1 ?2
Force vector polygon
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Determination of magnitude and Angular position
of the balancing mass
m1r1 ?2 cos ?1 m2r2 ?2 cos ? 2 m3r3
?2cos ? 3 m4r4 ?2 cos ? 4
mb cos ?b m1r1
?2 sin ?1 m2r2 ?2 sin ? 2 m3r3 ?2sin
? 3 m4r4 ?2 sin ? 4
mb sin ?b magnitude
m b and position ?b can be determined by
solving the above two equations.
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Dynamic or "Dual-Plane" balancing
Dynamic balancing is required for components
such as shafts and multi-rotor assemblies.
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Dynamic or "Dual-Plane" balancing
Statically balanced but dynamically unbalanced
m r ?2
r
r
Brg B
Brg A
l
m r ?2
Load on each support Brg due to
unbalance (m r ?2 l)/ L
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On an arbitrary plane C
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Several masses revolving in different
planes Apply dynamic couple on the rotating shaft
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Dynamic unbalance
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Balancing of several masses rotating in
different planes
B
C
D
A
F c
End view
F b
?
F d
F a
L
M
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Plane Mass M ( kg) Radius r (cm) Force / ?2, M r F , (kg. cm) Dist. From ref plane l , (cm) Couple / ?2 M r l C (kg cm 2)
A Ma ra Mara -la -Mara la
L (Ref.plane) Ml rl Ml rl 0 0
B Mb rb Mbrb lb Mbrb lb
C Mc rc Mcrc lc Mcrc lc
M Mm rm Mmrm d Mmrmd
D Md rd Mdrd ld Mdrdld
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A
C
B
D
Fm
la
Fc
lb
lc
Fb
ld
Fa
d
F l
Fd
M
L, Ref plane
End view
side view of the planes
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Fc
Fm ?
Cc
Fd
Fb
Fc
Fa
Cb
Fm
Ca
Cd
Fb
FlMl rl
Fa
CmMmrmd
F l ?
Fd
force polygon
Couple polygon
From couple polygon, by measurement, Cm Mm
X r m X d From force polygon, by
measurement, Fl Ml X rl
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Example
A shaft carries four masses in parallel planes
A,B,C,D in this order. The masses at B C are
18 kg 12.5 kg respectively and each has an
eccentricity of 6 cm. The masses at A D have an
eccentricity of 8 cm. The angle between the
masses at B C is 100 o and that between B A
is 190o both angles measured in the same sense.
The axial dist. between planes A B is 10cm and
that between B C is 20 cm. If the shaft is
complete dynamic balance, Determine, 1 masses
at A D 2. Distance between plane C D 3.
The angular position of the mass at D
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18 kg
B
C
A
D
? 190 o
? 100o
10 cm
20 cm
12.5 kg
l d
End view
M a
Plane Mass M kg Radius r cm Force / ?2, M r , kg. cm Dist. From ref plane l , cm Couple / ?2 M r l kg cm 2
A Ma? 8 8 Ma 0 0
B 18 6 108 10 1080
C 12.5 6 75 30 2250
D Md? 8 8 Md ld? 8 Md ld
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B
C
D
A
18 kg
? 190 o
? 100o
10 cm
20 cm
12.5 kg
M d
l d
M a
force polygon
Couple polygon
75
2,250
8 Md 63.5 kg. cm
108
1080
?d 203o
8 Ma 78 kg .cm
8 Md ld 2312 kg cm 2
O
O
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From the couple polygon, By
measurement, 8 Md ld 2,312 kg cm 2

? Md ld 2312 / 8 289 kg
cm
?d 203o From force polygon,
By measurement, 8 Md 63.5 kg cm
8 Ma 78.0
kg cm
Md 7.94 kg
Ma 9.75 kg
ld 289 /7.94 36.4 cm

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Shaft with rotors
Bearing 1
Bearing 2
Unbalanced force on the bearing rotor system
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Cut away section of centrifugal compressor
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• Balancing machines
• static balancing machines
• dynamic balancing machines

Measurement static unbalance
Hard knife edge rails
Thin disc
Chalk mark
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Static balancing machines
Universal level used in static balancing machine
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A helicopter- rotor assembly balancer
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Field balancing of thin rotors
Thin disc
Signal from Sine wave generator
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Oscilloscope signal
Required balancing mass , mb mt
oa/ab angular position of the mass
? During field balancing of thin disc using sine
wave generator, the measured amplitude of
vibration without trial mass is 0.6 mm and its
phase angle is 30o from the reference signal.
With trail mass attached, the amplitude is 1.0mm
and its phase angle is 83o from the reference
signal. Determine the magnitude and position of
the required balancing mass
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Field balancing with 3-mass locations (without
sine wave generator)
A3
2A1
A2
Required balancing mass , mb mt ad/dc
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Dynamic balancing machines Vibration amplitude
versus rotating unbalance
Mu r
M 1- (?/?n)22 2?(?/?n)2 ½
X
180o
X (Mur)
?
90o
2
3
1
0
2
3
?/?n
1
?/?n
Phase angle versus rotating speed
Amplitude versus rotating speed
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Pivoted carriage balancing machine

• Plane 1 coincides with pivot plane.
• Vibration levels as functions of angular position
  plotted.
• Minimum vibration level angle noted.
• Magnitude of trial mass varied by trial and error
  to reduce vibration.
• Repeated with plane 2 coinciding with the pivot
  plane.

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CRADLE TYPE BALANCING MACHINE
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rotor
Pivoted-cradle balancing machine with specimen
mounted
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Gisholt type balancing machine
stroboscope
Stroboscope flashes light
Angle marked end plate attached to the rotor
voltmeter

• Rotor mounted in spring supported half bearings.
• Vibration of bearing in particular direction used
  as direct measure of amount of unbalance in the
  rotor.
• Effect of unbalances in two planes separated by
  two electrical circuits one for each reference
  plane

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View of Precision Balance Machine