BALANCING OF ROTATING MASSES

1 VTU EDUSATPROGRAMME-17 DYNAMICSOF MACHINESS ubjectCode-10ME54 BALANCING OFROTATINGMASSESINTRODUCTION:Whenmaninve ntedthewheel,heveryquicklylearntthatifit wasn tcompletelyroundandifitdidn trotateevenlyaboutit scentralaxis,thenhehadaproblem!Whatthepr oblemhehad?Thewheelwouldvibratecausingda magetoitselfandit ssupportmechanismandinseverecases, :Theconditionwhichexistsinarotorwhenvibr atoryforceormotionisimpartedtoitsbearing sasaresultofcentrifugalforcesiscalledunb alanceortheunevendistributionofmassabout arotor Compiled by:VIJAYAVITHALBONGALEASSOCIATE PROFESSOR DEPARTMENT OFMECHANICALENGINEERINGMALNADCOLLEGEOFEN GINEERING HASSAN-573 202. KARNATAKA EDUSATPROGRAMME-17 Rotatingcenterline:Therotatingcenterline beingdefinedastheaxisaboutwhichtherotorw ouldrotateifnot constrained byits bearings.

2 (Alsocalled thePrincipleInertiaAxis orPIA). the two centerlines are coincident, then the rotor will be in a state of balance. When theyareapart, therotorwill types of unbalance can be defined by the relationship between the two centerlines. Theseinclude:Static Unbalance where the PIA is displaced parallel to the geometric centerline. (Shown above)CoupleUnbalance wherethePIAintersectsthegeometriccenterl ineatthecenterof gravity.(CG)DynamicUnbalance wherethePIAandthegeometriccenterlinedono tcoincideor :Inthedesignofrotatingpartsofamachineeve rycareistakentoeliminateanyoutofbalanceo rcouple, variation in in in ofunbalancethat is acceptableat alowspeed is completelyunacceptableat a , ,theforcequadruples;ifthespeedistripledt heforceincreasesDYNAMICSOFMACHINESVIJAYA VITHALBONGALEDYNAMICSOFMACHINESVIJAYAVIT HALBONGALEVTU EDUSATPROGRAMME-17byafactorofnine!

3 Thecentreofgravityofthesystem remains stationeryduring acomplete revolution ofthecrank shaft thecouples involved in acceleration ofthedifferent movingparts balanceeach :a)StaticBalancing:i)Staticbalancingis abalanceofforces duetoaction )Abodyis said to bein staticbalancewhenits centreofgravityis in the axis )Dynamicbalancing:i)Dynamicbalanceis abalancedueto theaction )Abodyissaidtobeindynamicbalancewhenther esultantmomentsorcouples, )Theconditions ofdynamicbalancearemet,theconditions ofstatic balancearealso ,thedynamicforcesaresetupandforcesnotonl yincreaseloadsonbearingsandstressesinthe variouscomponents, , EDUSATPROGRAMME-17 Inarevolvingrotor, , ,enginecrankshafts,rotorsofcompressors, ,impartvibratorymotionandnoise,thereareh umandiscomfort, involves redistributing the mass which may be carried out by addition or removal ofmass from various :Asystemofrotatingmassesissaidtobeinstat icbalanceifthecombinedmasscentreof thesystem lies on theaxis ofrotationDYNAMICBALANCING.

4 Whenseveralmassesrotateindifferentplanes ,thecentrifugalforces,inadditiontobeingo utofbalance, 2eDYNAMICSOFMACHINESVIJAYAVITHALBONGALEG mVTU EDUSATPROGRAMME-17 MASSBY A OF A SINGLE ROTATING MASSROTATING INTHE SAME PLANEC onsideradisturbingmass m1 which is attached to ashaft rotatingat radiusofrotationofthemassm1 distancebetweentheaxisofrotationofthesha ftandthecentreofgravityofthemassm1 Thecentrifugal forceexerted bymass m1 on theshaft is given by,2c11 1F m r (1) , EDUSATPROGRAMME-17 Let,r2 radiusofrotationofthemassm2 distancebetweentheaxisofrotationofthesha ftandthecentreofgravityofthemassm2 Thereforethecentrifugal forcedueto mass m2 will be,2c222F m r (2)221122or m1r1 m2r2 (3)m r m rEquatingequations (1)and (2), wegetFc1 )thenetdynamicforceactingontheshaftmustb eequaltozero, )thenetcoupleduetothedynamicforcesacting ontheshaftmustbeequaltozero, )andii) EDUSATPROGRAMME-17 CASE2(I):THEPLANEOFTHEDISTURBINGMASSLIES INBETWEENTHEPLANES OFTHETWO BALANCING , r1 and r2 betheradii ofrotation ofthemasses in planes A, M and Nrespectively.

5 LetL1,L2 andLbethedistancebetween Aand M,Aand N, and Mand Nrespectively. Now,Thecentrifugal forceexerted bythemass m in planeAwill be,(1)2cF m r Similarly,Thecentrifugal forceexerted bythemass m1 in planeM will be,(2)2c111F m r DYNAMICSOFMACHINESVIJAYAVITHALBONGALEVTU EDUSATPROGRAMME-17 And thecentrifugal forceexerted bythemass m2 in planeNwill be,(3)2c222F m r 2112222or m r m r m rForthecondition ofstaticbalancing,Fc Fc1 Fc2LL222112 Therefore, (5)m1r1L mrL2orm1r1 mror m rxL m m1r1 m2r2 (4)Now,todeterminethemagnitudeofbalancin gforceintheplane M orthedynamicforceatthebearing O ofashaft,takemomentsabout P ,Fc1xL FcxL2 LDYNAMICSOFMACHINESVIJAYAVITHALBONGALELT herefore,12 222221 (6)or mr mrm2r2L mrL1orm r xL m rxLSimilarly,inordertofindthebalancingfo rceinplane N orthedynamicforceatthebearing P ofashaft,takemomentsabout O ,Fc2xL FcxL1 Fordynamicbalancingequations (5)or(6)mustbesatisfied alongwith equation (4).

6 VTU EDUSATPROGRAMME-17 CASE2(II):WHENTHEPLANEOF THEDISTURBINGMASSLIESONONEENDOFTHE TWO PLANESCONTAININGTHEBALANCING m r m r m rForstaticbalancing,Fc1 Fc m1r1 mr m2r2 (1)Fordynamicbalancethenetdynamicforceac tingontheshaftandthenetcoupledueto dynamicforces actingontheshaft is equal to M orthedynamicforceatthebearing O ofa shaft, takemoments about P . 122112 Therefore,or mr mr (2)m1r1L mrL2or m rxL m rxLVTU EDUSATPROGRAMME-17F xL FxLc1c2LL12 2or mr mr (3)m2r2L mrL1 Similarly, to find thebalancingforcein theplane N , takemoments about O , ,F xL FxLc2c1or m 2rxL m 2rxL221 Therefore,CASE3:BALANCINGOFSEVERALMASSES ROTATINGINTHESAMEPLANEC onsiderarigidrotorrevolvingwithaconstant angularvelocity , m r m r m r m r (1)VTU EDUSATPROGRAMME-17 Ifm1,m2,m3 andm4 arethemassesrevolvingatradiir1,r2,r3 andr4 respectivelyinthe forces exerted byeach ofthemasses areFc1, Fc2, Fc3 and Fc4 respectively.

7 LetFbethevectorsum oftheseforces. Fc1 Fc2 Fc3 , ,thenintroduceacounterweight(balanceweig ht)ofmass m atradius r tobalancetherotorsothat,2222211223344orm 1r1 m2r2 m3r3 m4r4 mr 0 (3)m r m r m r m r m r 0 (2)Themagnitudeofeither m or r maybeselectedand theothercan becalculated. Ingeneral, if miriis thevectorsum ofm1r1,m2r2,m3r3,m4r4etc, then, miri mr 0 (4)Theaboveequation canbesolved Analytical Method:Procedure:Step 1: Find out the centrifugal force or the product of mass and itsradius of rotationexerted by each of MASSES on the ROTATING shaft, since 2is same for each mass,therefore the magnitude of the centrifugal force for each mass is proportional to the product oftherespectivemass and its radius :Resolvetheseforcesintotheirhorizontalan dverticalcomponentsandfindtheir sums.

8 ,Sumofthehorizontalcomponentsn miricos i m1r1cos 1 m2r2cos 2 m3r3cos 3 i 1 Sumoftheverticalcomponentsn mirisin i m1r1sin 1 m2r2 sin 2 m3r3 sin 3 i 1 VTU EDUSATPROGRAMME-17 Step 3: Determinethemagnitudeoftheresultant centrifugal force2 DYNAMICSOFMACHINESVIJAYAVITHALBONGALE n2 nR miri cos i miri sin i i 1 i 1 Step 4:If is theangle, which resultant forcemakes with thehorizontal, thenn mirisin ii 1n miricos ii 1tan Step 5: Thebalancingforceis then equal to theresultant force, but in oppositedirection. Step 6: Nowfind out themagnitudeofthebalancingmass, such thatR mrWhere, m =balancingmass and r=its radius ofrotation2. Graphical Method:Step 1:Drawthespacediagramwith thepositions oftheseveral MASSES , as 2:Findoutthecentrifugalforcesorproductof themassandradiusofrotationexertedby each 3:Now draw the vector diagram with the obtained centrifugal forces or product of the MASSES and radii ofrotation.

9 To drawvectordiagram , bc, cd, derepresents theforcesFc1, Fc2, Fc3and Fc4 on ab paralleltoforceFc1 ofthespacediagram,at b drawalineparalleltoforce Fc2. Similarlydrawlinescd, deparallel to Fc3 and Fc4 4:As per polygon law of forces, the closing side ae represents the resultant force in magnitudeand directionas shown in 5:Thebalancing forceis then , equal and oppositetotheresultant 6:VTU EDUSATPROGRAMME-17 Determinethemagnitudeofthebalancingmass( m)atagivenradiusofrotation(r), such that,2cF m rormr resultantofm1r1,m2r2,m3r3andm4r4 CASE4:BALANCINGOFSEVERALMASSESROTATINGIN DIFFERENTPLANESW henseveralmassesrevolveindifferentplanes , , , , , about thereferenceplanemust , theresultant couplemust be ,ingeneral, EDUSATPROGRAMME-17 Example:Consider four MASSES m1, m2, m3 and m4 attached to the rotor at radii r1, r2, r3 and r4 respectively.

10 Themassesm1, m2, m3 and m4 rotatein planes 1, 2, 3 and 4 )PositionofplanesofmassesChooseareferenc eplaneat O sothatthedistanceoftheplanes1,2,3and4fro m O areL1,L2, L .Chooseanotherplane M M isatadistanceofLmfromthereferenceplane L .Thedistancesofalltheotherplanestothelef tof L maybetakenasnegative(-ve)andtotherightma ybetakenaspositive(+ve). EDUSATPROGRAMME-17 Step 2:Construct the couple polygon first. (The couple polygon can be drawn by taking a convenient scale)Addtheknownvectorsandconsideringea chvectorparalleltotheradiallineofthemass drawthecouplediagram. Then theclosingvectorwill be mM rM LM .The vectord o on the couple polygon represents the balanced couple. Since the balancedcoupleCM is proportional to mM rM LM , therefore,DYNAMICSOFMACHINESVIJAYAVITHAL BONGALEP lane 1 Mass (m) 2 Radius (r) 3 Centrifugalforce/ 2 (m r)4 Distancefrom Ref.