UNIT 6 INTRODUCTION TO BLNCING Intoduction to Balancing Stuctue 6.1 Intoduction Objectives 6. Foce on Shaft and Beaing due to Single Revolving ass 6.3 Balancing of a Single Revolving ass 6.4 Pocedue fo Balancing 6. Extenal Balancing of Single Revolving ass 6.6 Static and Dynamic Balancing 6.7 Seveal asses Revolving in Same Tansvese Plane 6.8 Balancing of Seveal asses in Diffeent Tansvese Planes 6.9 Summay 6.10 nswes to SQs 6.1 INTRODUCTION In the system of otating masses, the otating masses have eccenticity due to limited accuacy in manufactuing, fitting toleances, etc. mass attached to a otating shaft will otate with the shaft and if the cente of gavity of the otating mass does not lie on the axis of the shaft then the mass will be effectively otating about an axis at cetain adius equal to the eccenticity. Since the mass has to emain at that adius, the shaft will be pulled in the diection of the mass by a foce equal to the centifugal foce due to inetia of the otating mass. The otating centifugal foce povides hamonic excitation to system which theeby causes foced vibation of the machines. e will discuss how such a foce can be balanced to emove the effect of unbalance. The unbalance is expessed as poduct of mass and eccenticity. Objectives fte studying this unit, you should be able to undestand what is unbalanced foce, and the effect of this, how is unbalanced foce due to single otating mass balanced, and how is unbalanced foce due to seveal otating masses in the same plane detemined? 6. FORCE ON SHFT ND BERING DUE TO SINGLE REVOLVING SS Figue 6.1 shows a evolving mass attached to a hoizontal shaft, which is suppoted by two beaings. The mass is at a adius fom the axis of the shaft. The mass is attached to the shaft at a distance a fom beaing on the left and at distance b fom ight hand beaing so that the span of the shaft between the beaings is a + b. The eccenticity is due to the toleances assigned, the limited accuacy of the manufactuing machines and nonhomogenity of the mateial. The shaft is otating with an angula velocity ad/s. dynamic foce F will pull the shaft towads the connected mass. The magnitude of F is given by F... (6.1) 1
Theoy of achines The foce F will be a bending foce on the shaft and will cause bending moment. dditional bending stess will be induced and eactions at beaings and B will occu. The eactions R and R B can be calculated by consideing the equilibium. They ae b R a b... (6.) a RB a b... (6.3) a F b B Figue 6.1 : Revolving ass attached to a Hoizontal Shaft These eactions on beaings will otate with the mass, hence will cause fatigue damage. The wea of beaing all ove the cicumfeence will also incease. The shaft will be subjected to bending moment whose maximum value will occu at the section whee the mass is connected to the shaft. The bending moment unde the evolving mass will be ab a b... (6.4) when the eactions at the suppots become moe than the toleable limits of beaings, the balancing is done. Example 6.1 shaft of cicula coss-section of diamete 0 mm is suppoted in two beaings at a distance of 1 m. mass of 0 kg is attached to the shaft such that its cente of gavity is mm fom the axis. The mass is placed at a distance of 400 mm fom left hand beaing. To avoid unequal weaing of beaings, the designe places the mass in the cente of the span. Calculate eactions at beaings, maximum bending moments and bending stesses if the shaft otates at 70 pm. Solution The foce caused on shaft due to otation = F N 70 78.4 ad/s 60 60 F g Use 3 0 kg, 78.4 ad/s, 10 m 16 3 F 0 (78.) 10 616.8 N... (i) Case I ass at a = 400 mm fom left hand (LH) beaing b = 1000 400 = 600 mm a + b = 1000 mm left, B ight hand beaing (Figue 6.1) R b F a b 600 616.8 1000
o R 370.1 N... (ii) and R F R 616.8 370.1... (iii) o B RB 46.74 N aximum B, R. a 370.1 400 o 3 B 148.04 10 Nmm... (iv) Bending stess is given by 3 B b d 3 Intoduction to Balancing whee d = 0 mm = shaft diemete o Case II 3 148.04 b 3 10 (0) 3 1.06 N/mm... (v) b ass at the cente of span, i.e. a = b = 00 mm R aximum B, F 616.8 RB 308.4 N F 3 B b d 3 a 308.4 00 14.13 10 Nmm... (vi) 3 14.13 3 10 (0) 3 3 o 17.3 N/mm... (vii) b 6.3 BLNCING OF SINGLE REVOLVING SS Balancing is a pocess of the edistibution of the mass in the system such that the eactions at the beaings ae within the toleable limits of the beaings. Thee ae two methods of achieving this. System ethod The effects of an off-axis o eccentic mass connected to a otating shaft, as bought out above, has to be nullified. One simple way by which this is achieved is by attaching anothe mass 1 at a adius 1, exactly opposite to as shown in Figue 6.. The shaft is otating at an angula speed of. B 1 1 1 1 Figue 6. : Balancing of a Single Revolving ass : System ethod 17
Theoy of achines The mass 1 and its adius ae so chosen that it is equal to the centifugal foce due to, i.e. 1 F g g which means that 1 1... (6.) 1 If Eq. (6.) is satisfied then the esultant foce on the shaft and hence on beaing will be zeo. Thus additional eaction o oveload on the beaings is zeo and B = 0, hence no additional stess in the shaft will be induced. The system is now called intenally balanced. Intenal balance is achieved by adding a balancing mass exactly opposite to evolving mass which causes unbalance. Thus the distubing and balancing masses ( and 1, espectively) ae in the same plane fo intenal balance and they satisfy the condition given by Eq. (6.). This method is used fo balancing auto wheels, etc. Second ethod In this method, instead of meutalising centifugal foce, the eccenticity o adius is educed. By doing this, we intend to educe the magnitude of the centifugal foce. This method is used fo thicke discs like flywheel whee it is possible to take out mass by shallow dilling. The side of the disc which consists of cente of gavity is called heavy side as shown in Figue 6.3(a). The opposite side to the heavy side is called light side. Since heavy side consists of moe mass, the mass 1 as given by the Eq. (6.) can be taken out by dilling a shallow hole of diamete d and depth b as given by the following elation 4 d b 1 whee b is less than thickness of the disc and is density of mateial. Heavy Side + CG 1 + CG + Light Side Shallow Hole Figue 6.3 : Balancing of a Single Revolving ass : Second ethod 6.4 PROCEDURE FOR BLNCING 18 thin disc like flywheel o a ca wheel may be mounted on an axle o a shaft. It can be otated by hand. The side which comes down can be maked. It is otated again. If the same side comes down, this is heavy side and opposite to this is light side. If thee is no foce measuing device and otating device, some mass can be mounted wheneve it is possible on light side. The cae should be taken that the mounting distance is as lage as possible. If maked heavy side again comes down, moe mass can be mounted on the light side. This pocess is epeated till any side comes down. Now it is faely balanced. If thee is a machine like wheel balancing machine, it indicates the magnitude of mass and the location whee balancing mass should be mounted. These methods ae tial and eo methods and ae time consuming methods. This cannot be used in industies whee time available pe piece is less. The industies have balancing machines which have
otating device and tansduces to povided magnitude of balancing mass and its location. In pactice, we neve aim the pefect balancing. The machine o the component is balanced till eactions ae within a toleable limit of the beaings. Intoduction to Balancing SQ 1 (a) (b) (c) hat do you mean by unbalance and why it is due to? hat do you mean by balancing? hy all the otating systems ae not balanced? Example 6. Solution In Example 6.1 find what weight of the balancing mass will achieve complete balance if the balancing mass has its cente of gavity at a distance of 7. mm fom the axis of otation. ill this be tue fo both positions of distubing mass in Example 6.1. Use (Eq. (18.) with kg, mm, 1 7. mm 1 1 7. 1 16.67 kg... (i) 7. Since the Eq. (6.) is independent of distance along the shaft the position of distubing mass will not affect the magnitude of balancing mass. So balancing mass is same as at Eq. (i) fo at a = 400 mm o in the cente of the span. 6. EXTERNL BLNCING OF SINGLE REVOLVING SS The single evolving mass connected to the shaft at adius causes the unbalance foce and eactions at the suppot. Howeve, if two masses 1 at adius 1 and at adius ae attached to the shaft, espectively in the same axial plane then also balancing of foce due to otation of can be achieved. The condition of balance in case as shown in Figue 6.3 will be g g g 1 1 The bending moments due to the foces due to otating masses will be balanced if g g g 1 a 1 a1 a g g g 1 b 1 b1 b a, b, a 1, b 1, a and b ae shown in Figue 6.4. 19
Theoy of achines The equations ae witten again by canceling out g fom both sides. 1 1... (6.6) a 1 1 a1 a... (6.7) b 1 1 b1 b... (6.8) Thus the distubing foce o unbalanced foce on the shaft is emoved. Thee is no excess eaction at any of beaings and B. This is known as extenal balancing. L l m a b a 1 b 1 a b 1 1 1 Figue 6.4 : Extenal Balancing of Single Revolving ass The extenal balancing with two otating masses (Figue 6.4) is esoted to when it is not possible to intoduce the balancing mass exactly opposite to distubing mass in the same adial plane. It may be wothwhile to note that a single mass placed in the same axial plane but in a diffeent adial plane may satisfy the condition that 1 1 ( 0) but and 1 will togethe cause a couple to act upon the shaft. This moment of the couple will tend to ock the shaft in the beaings. The balancing masses in the same axial plane but in two diffeent adial planes can satisfy the conditions of zeo foce tansvese to beam and zeo moment. The Eqs. (6.6), (6.7) and (6.8) ae such conditions. If we define thee adial planes fo thee masses, 1 and as, L and, espectively and call distance between and L as l and that between and as m then fom Figue 6.3 it is seen that a = a 1 + l and b = b + m. Then eplacing a by (a 1 + l) and b by (b + m) in Eqs. (6.7) and (6.8), espectively following ae obtained. o l ( l m) l l m and m 1 1 l m... (6.9)... (6.10) Note that same esults may be obtained if we take moments about sections L and of the shaft. lso note that Eq. (6.6) implies that eactions at suppots ae zeo. Eqs. (6.9) and (6.10) ae moe convenient to use along with Eq. (6.6) fo solving a poblem on extenal balancing. gain note that (l + m) is the distance between two adial planes in which balancing masses ae placed. e undestand that the Eqs. (6.9) and (6.10) ae applicable to a situation as shown in Figue 6.3 but if both L and ae on one side of plane then also these equations ae tue but one of 1 and will be on the same side of the shaft as. (l + m) can be denoted by d, so that l... (6.11) d 160 m and 1 1... (6.1) d
Example 6.3 mass of 100 kg is fixed to a otating shaft so that distance of its mass cente fom the axis of otation is 8 mm. Find balancing masses in following two conditions : (a) (b) Two masses one on left of distubing mass at a distance of 100 mm and adius of 400 mm, and othe on ight at a distance of 00 mm and adius of 10 mm. Two masses placed on ight of the distubing mass espectively at distances of 100 and 00 mm and adii of 400 and 00 mm The masses ae placed in the same axial plane. Intoduction to Balancing Solution Fo Case (a) see Figue 6.. L 100 = 8 00 = 10 1 = 400 Figue 6. : Figue fo Example 6.3 = 8 mm, l = 100 mm, m = 00 mm, d = l + m = 100 + 00 = 300 mm, 1 = 400 mm, = 10 mm, 100 g kg, 1 =?, =? Fom Eq. (6.6) Fom Eq. (6.11) 1 1 100 8 400 10 (g cancels out)... (i) 1 100 10 100 8 300 0.67 kg... (ii) Fom Eq. (6.1) 100 8 00 1 400 300 o 1 38 kg... (iii) Check with Eq. (i) 1 800 = 100 + 7600 161
Theoy of achines Fo Case (b) see Figue 6.6 00 100 100 Fom Eq. (6.11) Figue 6.6 : Figue fo Example 6.3 100 8 100 ( d 100) 10 100 o 1 kg... (iii) Fom Eq. (6.1) 100 8 00 1 400 100 o 1 114 kg... (iv) Fom Eq. (18.6) 100 8 114 400 10 800 4600 o 10 1 kg... (v) The negative sign indicates is on the othe side of 1. 6.6 STTIC ND DYNIC BLNCING If the cente of gavity of all otating masses is made to coincide with the axis of otation, a state is achieved when beaings will cay no additional eaction. Howeve, the masses may still cause some net bending moment on the shaft. Such bending moment will keep changing its plane and thus cause shaft to vibate. Connecting the masses in such a way that bending moment is made to vanish will esult in situation when shaft will not vibate. The balancing when only centes of gavity of attached mass system lies on axis of otation is known as static balancing. The balancing with centes of attached mass system made to coincide with axis of otation and no net bending moment acting on shaft is called dynamic balancing. In dynamic balancing foces and moments both ae to be balanced. 6.7 SEVERL SSES REVOLVING IN SE TRNSVERSE PLNE 1 16 numbe of masses weighing 1,, etc. may be connected to the shaft such that thei espective centes of gavity ae at distances of 1,, etc. Each of these masses may be placed at its own angula position in the tansvese o adial planes as depicted in Figue 6.7(a). Each will exet centifugal foce which will be popotional to the poduct
of mass and adius (i.e. ). If we wish to ascetain if the net effect will be an unbalanced foce, then we daw a polygon of foces. If the foce polygon does not close then the esultant unbalance is equal to the closing side of the polygon. In this case the closing side is, O. Thus a foce equivalent to, O in the diection to O will close the polygon. Hence balancing mass may be connected paallel to line joining and O and the length of this side will be the poduct of mass and adius. Intoduction to Balancing 3 4 4 4 3 4 3 4 1 1 0 a 1 1 1 3 3 Example 6.4 Solution (a) Figue 6.7 : Seveal asses Revolving in same Tansvese Plane Fou masses 1,, 3 and 4 at adii of mm, 17 mm, 0 mm and 300 mm ae connected at angles of zeo, 4 o, 7 o and 10 o fom hoizontal line as shown. If the shaft otates at 00 pm, find what unbalanced foce acts upon the shaft and at what angle fom mass 1. If a mass to balance the system can be placed at a adius of 00 mm, find the weight of the mass. Take 1 = 1000 N, = 100 N, 3 = 100 N and 4 = 800 N 1 1. 10.6 10 3 3 3 10 4 4.4 10 Figue 6.7 shows the oientation of distubing foces with magnitudes (popotional to ). Foce polygon is shown in Figue 6.8(b). Fom the polygon the unbalanced = 7.7 10 is at an angle of 07. o. (b) 4 4 10 o.4 3 3 3.6 7 o 10 7 4. 07. o 1 1 4 o (a) (b) Figue 6.8 : Figue fo Example 6.4 163
Theoy of achines The unbalanced foce g 7.7 10 00 10 9.81 60.166 10 N 3 If adius at which balancing mass is placed is 00 mm. Then 7.7 10 7.7 10 00 387 N ltenative method is to esolve along hoizontal and vetical diection and find thei esultant. ( H ) (..6 cos 4 3 cos 7.4 cos 10) 10 (. 1.86 0.776 1.) 10 = 3.676 10 ( V ) (.6 sin 4 3 sin 7.4 sin 10) 10 (1.86.9.078) 10 = 6.83 10 10 [ ( H)] [ ( V )] 10 [3.676] [6.83] = 7.76 10 ( V ) 6.83 tan tan tan 1.883 61.7 ( H) 3.67 1 1 1 o Note that is the esultant unbalance foce which will act upwad. The balancing mass will be placed opposite to it, i.e. downwad as shown by boken line in Figue 18.7(a). 07. o is the measued angle. The calculated value of the angle is 360 90 61.7 = 08.3 o. Compae the Values Fom polygon constuction Fom calculation o 7.7, 07. o 7.76, 08.3. 6.8 BLNCING OF SEVERL SSES IN DIFFERENT TRNSVERSE PLNES 164 To begin with we will name the planes in which masses evolve as, B, C and D. The masses otating in these planes ae espectively a, b, c and d. The adii at which centes of gavity lie fom axis of otation in these planes ae espectively a, b, c and d.
The angula sepaation between masses stating fom and B ae, and. The balancing masses will be placed in two planes L and which ae between and B and between C and D, espectively. The Figue 6.9 depicts the system. Distance between planes L and any of planes, B, C and D is denoted by l with appopiate suffix. The same distances fo plane is denoted by m. Intoduction to Balancing c m L B C D b l a l b l c l d a m d m a L m c m d d Figue 6.9 : Balancing of Seveal asses in Diffeent Tansvese Plane The method appaently is same as used in Section 6.4 in which balance masses in two planes L and wee found. So we need to epeat the pocedue fo mass in plane and balancing mass in planes L and. Thus we poceed in steps of planes, B, C and D. table of the kind shown below as Table 6.1 will be helpful. Rows will be dedicated to planes in which masses evolve which could be known o unknown. Fo a mass of weight, evolving at adius, the foce is popotional to as is a g constant (Eq. (6.1)). Two planes L and ae chosen which ae espectively at distances of l and m fom the plane of evolving mass and balancing masses ae placed in planes L and as given by Eqs. (6.11) and (6.1). So the columns of the table will descibe plane (, B, C, etc.) weight ; adius ; foce ; distances l and m; balancing foces in L and. Plane Table 6.1 : Calculation of Balancing asses in Two Planes eight Radius Foce g Distance Fom Plane L l Plane m a a a a l a m a B b b b b l b m b Balancing Foce Plane L a a m d m b b d a b g Plane a l a a d b l b b d The sign of the foces in last two columns will be decided by obsevation. Yet as a ule if a foce is in the same diection as the distubing foce, then it will be positive and if in the opposite diection it will be negative. fte the balancing foces have been calculated in planes L and which will be paallel to foces in planes and B, etc. they ae combined to give a single esultant. Thei inclination to foce in plane can be detemined. The above pocedue will be followed in solving the example. Example 6. In Figue 6.9 fou masses a = 1000 N, b = 100 N, c = 100 N and d = 1300 N evolve espectively at adii of a = mm, b = 17 mm, c = 0 mm and d = 300 mm in planes, B, C and D. Two planes L and ae 16
Theoy of achines Solution selected to place balancing masses l and m at a adius of 600 mm. The masses b, c and d ae espectively at angles of 4 o, 7 o and 13 o fom a and distances between planes ae : l a = 300 mm, l b = 37 mm, l c = 70 mm, l d = 100 mm, m a = 1800 mm, m b = 87 mm, m c = 00 mm, m d = 0 mm. Find the balancing masses and oientation of thei adii fom adius of mass a. See Figue 6.9. Poceed as pe Table 6.1. The distance between planes L and, d = l d m d = 100 87 = 8 mm. Plane eight N Radius mm Foce g Distance of Plane Fom Plane, L l (mm) Plane, m (mm) Balancing Foce Plane L m/d Nmm g Plane l/d Nmm 10 3. 10 300 1800 6.48 10 1.08 10 1.7 B 1. 10 3 17.6 10 37 87 3.67 10 10 C 1. 10 3 0 3.00 10 70 00.4 10 3.6 10 D 1.3 10 3 300 3.90 10 100 0 1.6 10 9.36 10 The last two columns show balancing foces fo those in planes and B, etc. Hence, these foces will act in opposite diection to a, b, etc. fou foces in planes L and will be equal to one foce by a single otating mass. Thus, two balancing masses one each in planes L and will be obtained. Balancing foces in L plane ae shown in Figue 6.10, along with distubing foces a, b, c and d. To find esultant of all balancing foces we go fo thei components along hoizontal and vetical diections. The elevant angles ae shown in figue. c d b 6.48 10 1.6 10 13 4 o 4 7 4 o 9 o 7 o 11 o a 1.364 10 3.67 10.4 10 Figue 6.10 : Figue fo Example 6. ( H) 6.48 10 3.67 10 cos 4.4 10 cos 7 1.6 10 cos 4 (6.48.6 0.61 1.1) 10 10.8 10 ( V) 3.67 10 sin 4..4 10 sin 7 1.6 10 sin 4 (.6.3 1.1) 10 166 6.0 10
(Resultant) [ ( H)] [ ( V )] Intoduction to Balancing ith The angle with a 116.64 36.4 10 1.364 10 Nmm 1.364 10 600 mm, 060.7 N 600 ( V ) 6.0 tan tan tan 0.74 9 ( H) 10.8 1 1 1 o Balancing foces in plane ae shown in Figue 6.11 along with distubing foces a, b, c and d. The esultant is found as above. ( H) (1.08 1.7 cos 13 3.6 cos 10 9.36 cos 4) 10 (1.08 1.11 0.93 6.6) 10.66 10 ( V) (1.7 sin 13 3.6 sin 10 9.36 sin 4) 10 (1.11 3.48 6.6) 10 11. 10 ( R).66 11. 10 1. 10 t = 600 mm, the balancing mass in plane will be 1. 10 091 N 600 This will be placed at angle with diection of a 11. tan tan 1.98 63..66 1 1 o c d b 1.08 10 a 4 o 1.7 10 10 o 13 o 9.36 10 3.6 10 1. 10 Figue 6.11 : Figue fo Example 6. 167
Theoy of achines SQ Thus we see that fo balancing the evolving masses in planes, B, C and D we have to connect two masses of 060.7 N and 091 N, espectively which ae between and B and between C and D. (a) (b) (c) (d) (e) hat do you undestand by balancing of evolving masses? If not balanced what effects ae induced on shaft beaing system due to unbalanced otating masses. How do you achieve balance of otating masses which lie in paallel tansvese planes of a shaft? Five masses, B, C, D and E evolve in the same plane at equal adii., B and C ae espectively 10, and 8 kg in mass. The angula diection fom ae 60 o, 13 o, 10 o and 70 o. Find the masses D and E fo complete balance. shaft caies thee pulleys, B and C at distance apat of 600 mm and 100 mm. The pulleys ae out of balance to the extent of, 0 and 30 N at a adius of mm. The angula position of out of balance masses in pulleys B and C with espect to that in pulley ae 90 o and 10 o espectively. It is equied that the pulleys be completely balanced by poviding balancing masses evolving about axis of the shaft at adius of 1 mm. The two masses ae to be placed in two tansvese planes midway between the pulleys. 6.9 SURY The masses that ae connected to shaft and whose centes of gavity do not lie on axis of the otation, evolve about the axis at constant adius. oving in cicula path they ae subjected to centifugal foce which may cause bending stess in the shaft and otating eactions in the beaing. To nullify thei effects, evolving masses can be povided in the plane of distubing mass o in some othe paallel plane. If balancing masses ae placed in the plane of unbalance fo single mass, only one balancing plane is equied othewise two balancing planes shall be equied. The balancing pocess equies that both the bending foces and moments on the shaft be made to vanish. 6.10 NSERS TO SQs 168 SQ (d) Since the adii ae equal, the foces ae popotional to masses. The Figue 6.11 shows the foces and thei oientation. Since the system is balance the foce polygon must close wheeby we can find the unknown foces and coesponding masses. e plot only masses hence sides of polygon will diectly give the masses. In the foce (popotional foce) polygon known values ae witten inside the polygon and measued values ae witten outside. ass D = 8 kg ass E = 6 kg
8 C B Intoduction to Balancing.6 D 60 o 13 o 10 10 o 70 o E Oientation of Foces c d 8 8 b 6 10 e a SQ Popotional Foce Polygon Figue 6.11 L B C b N mm 0 N mm mm 30 N 10 o 90 o a 600 100 mm l a 300 l b 300 m b 100 l c m c c m a 300 300 900 D = 100 300 = 100 mm Plane (N) (mm) (Nmm) Distance Fom L (l) (m) Balancing Foce L m/d l/d 6 300 900 468.7 16. B 0 00 300 300 1.00 1.00 C 30 70 100 300 + 187.0 37.00 169
Theoy of achines L Plane ( H) 468.7 187. cos 30 631.13 ( V) 1 187. sin 30 18.7 468.7 30 o 187. 1.00 ( R) 631.13 18.7 668 Nmm 1 mm 668.34 N 1 Plane 18.7 tan tan 0.347 19.14 631.13 1 1 o ( H) 16. 37 cos 30 481 ( V) 37 sin 30 16. 31. ( R) 481 31. 48 37.00 30 o 16. 1 mm 1 48 3.86 N 1 31. tan 0.06 481 o 3.7 170