Dr. S. K. Kudari,

1
Dr. S. K. Kudari, Professor, Department of
Mechanical Engineering, B V B College of Engg.
Tech., HUBLI email skkudari_at_bvb.edu
2
CHAPTER-5
Topics covered Whirling of shafts neglecting
damping Whirling of shafts with damping
Numerical Problems/Discussions
10/04/07
11/04/07
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
3
Getaran pada poros

• Suatu penomena yg dapat terjadi pada poros-poros
  yg berputar pada suatu kecepatan tertentu adalah
  getaran yg berlebihan, meskipun dapat terjadi
  bahwa poros tersebut berputar sangat halus pada
  kecepatan kecepatan lain. Bila getaran
  berlebihan, dapat terjadi hal-hal spt poros
  patah, bantalan rusak, bagian-bagian mesin tidak
  dapat bekerja baik spt. pada sudu-sudu turbin
  dimana clearance antara sudu dan rumah turbin
  sangat kecil. Untuk dapat terjadi getaran pada
  suatu sistem diperlukan minimum 2 hal yaitu
  massa dan elastisitas.

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RECAP
Whirling (pusaran ) of shafts
Shaft

• Problems in shaft and a rotor systems
• Unbalance in rotor/disc
• (ii) Improper assembly
• (iii) Weaker bearings

disc
bearings
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
5
RECAP
Whirling of shafts
Unbalance in rotor / disc

• For perfect balancing
• Mass centre (centre of gravity) has to co-inside
  with the geometric centre
• (ii) m.e unbalance 0

Top view of a rotor
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
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Gaya sentrifugal ini menyebabkan poros membengkok
( gaya ini bekerja secara radial keluar melalui G)
Whirling of shafts
RECAP
Top view of the disc
Gaya sentrifugal, FcMAM(de)
P- Geometric center G- centre of gravity O-
center of rotation
Static
dynamic
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
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RECAP
Whirling of shafts
Whirling (pusaran) is defined as the rotation of
plane made by the bent shaft and line of centers
of bearings as shown in Figure ( rotasi bidang yg
dibuat oleh poros melengkung
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
8
RECAP
Whirling of shafts

• Assumptions
• the disc at the mid-span has an unbalance
• (ii) the shaft inertia is negligible and the
  shaft stiffness is same in all directions

Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
9
r
Whirling of shafts neglecting damping
It is desired to run the shaft at speed much
higher than the natural frequency of the shaft
rotor system
Critical speed
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
10
Whirling of shafts with damping
The deflection of shaft is
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
11
Theory questions
What do you understand by critical speed of
shafts. Derive the necessary relations and thus,
explain what is happening in the system carrying
a shaft having an unbalanced disc as its centre
is operated above and below critical speed. (VTU
Exam Jan 2006 for 12 marks)
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
12
Theory questions
The speed of the shaft under the condition
when r 1, i.e ??n is referred as critical
speed of shaft.
Derive the relation
Critical speed
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
13
Numerical problems
Problem-1
A power transmission shaft has diameter of 30 mm
and 900mm long, and simply supported. The shaft
carries a rotor of 4 kg at its mid-span. The
rotor has an eccentricity of 0.5 mm. Calculate
the critical speed of shaft and deflection of the
shaft at the mid-span at 1000 rpm. Neglect mass
of the shaft, take E2x105 MPa (Ref Mechanical
Vibrations By Kelly and Kudari, Schamus outline
series)
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
14
Numerical problems
Problem-1
Given data
D 30 mm L 900 mm m 4 kg e 0.5 mm E 2x105
MPa 2x1011 Pa N 1000 rpm
D
L
Find Nc, the critical speed And deflection of
shaft d
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
15
Numerical problems
Problem-1
To Find Nc, the critical speed, it is required to
find natural frequency of the system
To Find stiffness, K of the shaft
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
16
Numerical problems
Problem-1
To Find stiffness, K of the shaft
Simply supported shaft
W
?
L
Deflection of the beam at mid span
Stiffness of beam
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
17
Numerical problems
Problem-1
The stiffness of the shaft
Substitute D in meters
39.76x10-9 m4
K 523598.75 N/m
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
18
Numerical problems
Problem-1
361.80 rad/s
The critical speed ?cr?n
Ncr 3455 rpm
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
19
Numerical problems
Problem-1
Deflection of the shaft at 1000 rpm
104.72 rad/s
0.289
0.0455 mm
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
20
Numerical problems
Problem-2
A disc of mass 4 kg is mounted mid-way between
bearings, which may be assumed to be simple
supports. The bearing span is 0.48 m. The steel
shaft which is horizontal is 0.09 m in diameter.
The centre of gravity of the disc is displaced 3
mm from the geometric centre. The equivalent
viscous damping at the centre of the disc-shaft
is 49 N.s/m. If the shaft rotates at 760 rpm,
find the maximum stress in the shaft and compare
it with the dead load stress in the shaft. Also
find the power required to drive the shaft at
this speed.
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
21
Numerical problems
Problem-2
Given data
D 0.09 m L 0.48 m m 4 kg e 3 mm c 49
N.s/m N 760 rpm E 2x1011 Pa
D
L
Find maxm stress in the shaft And power required
to drive the shaft
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
22
Numerical problems
Problem-2
79.58 rad/s Forcing frequency
The Mod. of elasticity of the material is not
given
Assume E 2x1011 MPa
The stiffness of the shaft
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
23
Numerical problems
Problem-2
K 27957.3 N/m
83.6 rad/s
0.073 Damping ratio
0.951 Frequency ratio
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
24
Numerical problems
Problem-2
Deflection of shaft
d 0.016 m
Dynamic load on the shaft (Restoring force)
452.03 N
Static load on the shaft (Self weight)
39.24 N
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
25
Numerical problems
Problem-2
Total force (maxm force)
491.27 N
Maxm stress in the shaft
Z section modulus
58.95 N/m
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
26
Numerical problems
Problem-2
8.23x108 N/m2
6.58x107 N/m2
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
27
Numerical problems
Problem-2
power required to drive the shaft
power required to overcome damping
Friction force c?d
Friction torque c?d2
Power 2?NT/60
90 Watts
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
28
Numerical problems
Problem-3
A rotor having mass of 5 kg is mounted mid-way on
1 cm diameter shaft supported at the ends by two
bearings. The bearing span is 40 cm. Because of
certain manufacturing inaccuracy, the CG of disc
is 0.02 mm away from the geometric centre of the
rotor. If the system rotates at 3000 rpm, find
the amplitude of steady state vibrations and
dynamic force transmitted to bearings. Neglect
damping and weight of shaft. Take E1.96x1011 MPa
(Ref VTU Exam Jan 2007 for 12 marks)
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
29
Numerical problems
Problem-3
Given data
D 1 cm L 40 cm m 5 kg e 0.02 mm E 2x1011
N/m2 N 3000 rpm Damping-neglected
D
L
Find d, Amp of steady state vibrn. Dynamic force
transmitted to bearings
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
30
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Numerical problems
Problem-3
Amp. of steady state vibrn Deflection of the
shaft
The stiffness of the shaft
K 72158 N/m
120.3 rad/s
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
31
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Numerical problems
Problem-3
314.15 rad/s
2.61
0.023 mm
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
32
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Numerical problems
Problem-3
Dynamic force transmitted to bearings Restoring
force due to spring
1.68 N
Load on each bearings
(1.68/2) N
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
33
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Numerical problems
Problem-4
A horizontal shaft 15 mm diameter and 1 m long is
held on simply supported bearings. The mass of
the disc at the mid span 15 kg and eccentricity
is 0.3 mm. The young's modulus of the shaft
material is 200 GPa. Find the critical speed of
shaft (Ref VTU Exam July 2006 for 10 marks)
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
34
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Numerical problems
Problem-4
Given data
D 15 mm L 1 m m 15 kg e 0.3 mm E 200 GPa E
2x1011 Pa
D
L
Find the critical speed of the shaft
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
35
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Numerical problems
Problem-4
Steps
Find K Find natural frequency of the
system critical speed of the shaft can be
obtained by equating forcing frequency to natural
frequency
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
36
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Summary
Due unbalance in a shaft-rotor system, rotating
shafts tend to bend out at certain speed and
whirl in an undesired manner
The speed of the shaft under the condition when
r 1, i.e ??n is referred as critical speed of
shaft.
The theory developed helps the design engineer to
select the speed of the shaft, which gives
minimum deflection
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
37
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Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli