DYNAMICS OF MACHINES By Dr.K.SRINIVASAN, Professor, AU-FRG Inst. for CAD/CAM, Anna University

1. DYNAMICS OF MACHINES By Dr.K.SRINIVASAN, Professor, AU-FRG Inst. for CAD/CAM, Anna University Topic : Balancing of Rotating masses

2. 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

3. Balancing • "is the process of attempting to improve themass distribution • of a body so that it rotates in its bearings withoutunbalanced centrifugal forces”

4. 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

5. Rotating components for balancing

6. Shaft with rotors Bearing 1 Bearing 2 Unbalanced force on the bearing –rotor system

7. Cut away section of centrifugal compressor

11. Unbalance is caused by the displacement of the mass centerline from the axis of rotation. • Centrifugal forceof "heavy" point of a rotor exceeds the centrifugal force exerted by the light side of the rotor and pulls the entire rotorin the direction of the heavy point. • Balancingis the correction of this phenomena by the removal or addition of mass

12. 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.

13. 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.

14. 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.

15. ● 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

16. Unbalance due to unequal distribution of masses • Unbalance due to • unequal distance of masses

17. . Types of UnbalanceStaticUnbalanceDynamicUnbalance

18. STATIC BALANCING (SINGLE PLANE BALANCING)

19. Single plane balancing Adequate for rotors which are short in length, such as pulleys and fans

20. F = m r 2 m  Magnitude of unbalance r O2 Vibration occurs Elasticity of the bearing

21. 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

22. 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

23. 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.

24. Dynamic or "Dual-Plane" balancing Dynamic balancing is required for components such as shafts and multi-rotor assemblies.

25. 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 2l)/ L

26. On an arbitrary plane C

27. Several masses revolving in different planes Apply dynamic couple on the rotating shaft  Dynamic unbalance

28. Balancing of several masses rotating in different planes B C D A F c End view F b  F d F a L M

30. 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

31. Fc Fm =? Cc Fd Fb Fc Fa Cb Fm Ca Cd Fb Fl=Ml rl Fa Cm=Mmrmd 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

32. 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 oand 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

33. 18 kg B C A D  =100o  =190 o 10 cm 20 cm 12.5 kg l d End view M a

34. B C D 18 kg A  =100o  =190 o 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

35. 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

36. Shaft with rotors Bearing 1 Bearing 2 Unbalanced force on the bearing –rotor system

37. Cut away section of centrifugal compressor

41. Balancing machines: • static balancing machines • dynamic balancing machines Measurement static unbalance: Hard knife edge rails Thin disc Chalk mark

42. Static balancing machines: Universal level used in static balancing machine

43. A helicopter- rotor assembly balancer

44. Field balancing of thin rotors : Thin disc Signal from Sine wave generator

45. 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 83ofrom the reference signal. Determine the magnitude and position of the required balancing mass

46. Field balancing with 3-mass locations (without sine wave generator) A3 2A1 A2 Required balancing mass , mb = mt [ad/dc]

47. Dynamic balancing machines Vibration amplitude versus rotating unbalance Mu r M [ {1- (/n)2}2 +{2(/n)}2] ½ X= 180o X (Mur)  90o 2 3 1 0 /n 2 3 1 /n Phase angle versus rotating speed Amplitude versus rotating speed

48. 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.

49. CRADLE TYPE BALANCING MACHINE

50. rotor Pivoted-cradle balancing machine with specimen mounted