1. Dr. S. K. Kudari,
Professor,
Department of Mechanical Engineering,
B V B College of Engg. & Tech.,
HUBLI

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.

4. Whirling (pusaran ) of shafts
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. Unbalance in rotor / disc For perfect balancing
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
rotor
Geometric centre
Mass centre e m
Top view of a rotor
Dr. S. K. Kudari, Professor, BVB College of Engg. & Tech., Hubli

6. RECAP Whirling of shafts Static dynamic Top view of the disc O P G d e
Gaya sentrifugal ini menyebabkan poros membengkok ( gaya ini bekerja secara radial keluar melalui G)
RECAP
Whirling of shafts
Top view of the disc O P G d e
Gaya sentrifugal, Fc=MA=M(d+e)
P- Geometric center
G- centre of gravity
O- center of rotation
Static
dynamic
Dr. S. K. Kudari, Professor, BVB College of Engg. & Tech., Hubli

7. 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. the disc at the mid-span has an unbalance
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. 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 : x y O P
G (xg,yg) d e   t
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 E=2x105 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 D 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 = 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
= rad/s
= 0.289
= 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 D 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
= Damping ratio
= 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)
= 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 E=1.96x1011 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 D 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. Numerical problems Problem-3 Amp. of steady state vibrn
Space for
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size Picture
Space for
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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. Numerical problems Problem-3 = 314.15 rad/s = 2.61 = 0.023 mm
Space for
2 inch x 2 inch
size Picture
Space for
2 inch x 2 inch
size Picture
Numerical problems
Problem-3
= rad/s
= 2.61
= mm
Dr. S. K. Kudari, Professor, BVB College of Engg. & Tech., Hubli

32. Numerical problems Problem-3
Space for
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Space for
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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. Numerical problems Problem-4
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Space for
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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. Numerical problems Problem-4 Given data D =15 mm D L =1 m m =15 kg
Space for
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Space for
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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. Numerical problems Problem-4 Steps Find K
Space for
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Space for
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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. Space for
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size Picture
Space for
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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. Space for 2 inch x 2 inch size Picture Space for 2 inch x 2 inch
Dr. S. K. Kudari, Professor, BVB College of Engg. & Tech., Hubli