Dr. S. K. Kudari, Professor, Department of Mechanical Engineering, B V B College of Engg. & Tech., HUBLI email: skkudari@bvb.edu

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Numerical problems Problem-2 =8.23x108 N/m2 =6.58x107 N/m2 Dr. S. K. Kudari, Professor, BVB College of Engg. & Tech., Hubli

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

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

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

Numerical problems Problem-3 Amp. of steady state vibrn Space for 2 inch x 2 inch size Picture Space for 2 inch x 2 inch size Picture 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

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 = 314.15 rad/s = 2.61 = 0.023 mm Dr. S. K. Kudari, Professor, BVB College of Engg. & Tech., Hubli

Numerical problems Problem-3 Space for 2 inch x 2 inch size Picture Space for 2 inch x 2 inch size Picture 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

Numerical problems Problem-4 Space for 2 inch x 2 inch size Picture Space for 2 inch x 2 inch size Picture 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

Numerical problems Problem-4 Given data D =15 mm D L =1 m m =15 kg Space for 2 inch x 2 inch size Picture Space for 2 inch x 2 inch size Picture 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

Numerical problems Problem-4 Steps Find K Space for 2 inch x 2 inch size Picture Space for 2 inch x 2 inch size Picture 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

Space for 2 inch x 2 inch size Picture Space for 2 inch x 2 inch size Picture 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

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