www.getyun.co YIS OF HIES IG OF ROTTIG SSES
www.getyun.co Rotatng centelne: The otatng centelne beng defned as the axs about whch the oto would otate f not constaned by ts beangs. (lso called the Pncple Ineta xs o PI). Geoetc centelne: The geoetc centelne beng the physcal centelne of the oto. When the two centelnes ae concdent, then the oto wll be n a state of balance. When they ae apat, the oto wll be unbalanced. ffeent types of unbalance can be defned by the elatonshp between the two centelnes. These nclude: Statc Unbalance whee the PI s dsplaced paallel to the geoetc centelne. (Shown above) ouple Unbalance whee the PI ntesects the geoetc centelne at the cente of gavty. (G) ynac Unbalance whee the PI and the geoetc centelne do not concde o touch. The ost coon of these s dynac unbalance. auses of Unbalance: In the desgn of otatng pats of a achne evey cae s taken to elnate any out of balance o couple, but thee wll be always soe esdual unbalance left n the fnshed pat because of. slght vaaton n the densty of the ateal o. naccuaces n the castng o. naccuaces n achnng of the pats. Why balancng s so potant?. level of unbalance that s acceptable at a low speed s copletely unacceptable at a hghe speed.. s achnes get bgge and go faste, the effect of the unbalance s uch oe sevee.. The foce caused by unbalance nceases by the squae of the speed. 4. If the speed s doubled, the foce quaduples; f the speed s tpled the foce nceases
www.getyun.co by a facto of nne! Identfyng and coectng the ass dstbuton and thus nzng the foce and esultant vbaton s vey vey potant IG: alancng s the technque of coectng o elnatng unwanted neta foces o oents n otatng o ecpocatng asses and s acheved by changng the locaton of the ass centes. The objectves of balancng an engne ae to ensue:. That the cente of gavty of the syste eans statoney dung a coplete evoluton of the cank shaft and. That the couples nvolved n acceleaton of the dffeent ovng pats balance each othe. Types of balancng: a) Statc alancng: ) Statc balancng s a balance of foces due to acton of gavty. ) body s sad to be n statc balance when ts cente of gavty s n the axs of otaton. b) ynac balancng: ) ynac balance s a balance due to the acton of neta foces. ) body s sad to be n dynac balance when the esultant oents o couples, whch nvolved n the acceleaton of dffeent ovng pats s equal to zeo. ) The condtons of dynac balance ae et, the condtons of statc balance ae also et. In oto o ecpocatng achnes any a tes unbalance of foces s poduced due to neta foces assocated wth the ovng asses. If these pats ae not popely balanced, the dynac foces ae set up and foces not only ncease loads on beangs and stesses n the vaous coponents, but also unpleasant and dangeous vbatons. alancng s a pocess of desgnng o odfyng achney so that the unbalance s educed to an acceptable level and f possble elnated entely. IG OF ROTTIG SSES When a ass oves along a ccula path, t expeences a centpetal acceleaton and a foce s equed to poduce t. n equal and opposte foce called centfugal foce acts adally outwads and s a dstubng foce on the axs of otaton. The agntude of ths eans constant but the decton changes wth the otaton of the ass.
www.getyun.co In a evolvng oto, the centfugal foce eans balanced as long as the cente of the ass of oto les on the axs of otaton of the shaft. When ths does not happen, thee s an eccentcty and an unbalance foce s poduced. Ths type of unbalance s coon n stea tubne otos, engne cankshafts, otos of copessos, centfugal pups etc. eω G e The unbalance foces exeted on achne ebes ae te vayng, pat vbatoy oton and nose, thee ae huan dscofot, pefoance of the achne deteoate and detental effect on the stuctual ntegty of the achne foundaton. alancng nvolves edstbutng the ass whch ay be caed out by addton o eoval of ass fo vaous achne ebes alancng of otatng asses can be of. alancng of a sngle otatng ass by a sngle ass otatng n the sae plane.. alancng of a sngle otatng ass by two asses otatng n dffeent planes.. alancng of seveal asses otatng n the sae plane 4. alancng of seveal asses otatng n dffeent planes STTI IG: syste of otatng asses s sad to be n statc balance f the cobned ass cente of the syste les on the axs of otaton YI IG; When seveal asses otate n dffeent planes, the centfugal foces, n addton to beng out of balance, also fo couples. syste of otatng asses s n dynac balance when thee does not exst any esultant centfugal foce as well as esultant couple.
www.getyun.co SE. IG OF SIGE ROTTIG SS Y SIGE SS ROTTIG I THE SE PE onsde a dstubng ass whch s attached to a shaft otatng at ω ad/s. et adus of otaton of the ass dstance between the axs of otaton of the shaft and the cente of gavty of the ass The centfugal foce exeted by ass on the shaft s gven by, F c ω ( ) Ths foce acts adally outwads and poduces bendng oent on the shaft. In ode to counteact the effect of ths foce F c, a balancng ass ay be attached n the sae plane of otaton of the dstubng ass such that the centfugal foces due to the two asses ae equal and opposte.
www.getyun.co et, adus of otaton of theass dstance between the axs of otaton of the shaft and the cente of gavty of the ass Theefoe the centfugal foce due to ass wll be, Fc ω () Equatng equatons () and (), we get F c F ω ω c o () The poduct can be splt up n any convenent way. s fo as possble the adus of otaton of ass that s s geneally ade lage n ode to educe the balancng ass. SE : IG OF SIGE ROTTIG SS Y TWO SSES ROTTIG I IFFERET PES. Thee ae two possbltes whle attachng two balancng asses:. The plane of the dstubng ass ay be n between the planes of the two balancng asses.. The plane of the dstubng ass ay be on the left o ght sde of two planes contanng the balancng asses. In ode to balance a sngle otatng ass by two asses otatng n dffeent planes whch ae paallel to the plane of otaton of the dstubng ass ) the net dynac foce actng on the shaft ust be equal to zeo,.e. the cente of the asses of the syste ust le on the axs of otaton and ths s the condton fo statc balancng ) the net couple due to the dynac foces actng on the shaft ust be equal to zeo,.e. the algebac su of the oents about any pont n the plane ust be zeo. The condtons ) and ) togethe gve dynac balancng.
www.getyun.co SE (I): THE PE OF THE ISTURIG SS IES I ETWEE THE PES OF THE TWO IG SSES. onsde the dstubng ass lyng n a plane whch s to be balanced by two otatng asses and lyng n two dffeent planes and whch ae paallel to the plane as shown. et, and be the ad of otaton of the asses n planes, and espectvely. et, and be the dstance between and, and, and and espectvely. ow, The centfugal foce exeted by the ass n plane wll be, F ω c () Slaly, The centfugal foce exeted by the ass n plane wll be, Fc ω ()
www.getyun.co nd the centfugal foce exeted by the ass n plane wll be, Fc ω Fo the condton of statc balancng, F F c o.e. c + F ω c () ω + + ω (4) ow, to detene the agntude of balancng foce n the plane o the dynac foce at the beang O of a shaft, take oents about P whch s the pont of ntesecton of the plane and the axs of otaton. Theefoe, F c o x F ω x Theefoe, c x ω x o (5) Slaly, n ode to fnd the balancng foce n plane o the dynac foce at the beang P of a shaft, take oents about O whch s the pont of ntesecton of the plane and the axs of otaton. Theefoe, F c o x F ω x Theefoe, c x ω o x (6) Fo dynac balancng equatons (5) o (6) ust be satsfed along wth equaton (4).
www.getyun.co SE (II): WHE THE PE OF THE ISTURIG SS IES O OE E OF THE TWO PES OTIIG THE IG SSES. Fo statc balancng, F c o.e. F + F c ω c ω + ω + () Fo dynac balance the net dynac foce actng on the shaft and the net couple due to dynac foces actng on the shaft s equal to zeo. To fnd the balancng foce n the plane o the dynac foce at the beang O of a shaft, take oents about P..e.
www.getyun.co F c o x F ω x Theefoe, c x ω x o () Slaly, to fnd the balancng foce n the plane, take oents about O,.e., F c o x F ω x Theefoe, c x ω x o () SE : IG OF SEVER SSES ROTTIG I THE SE PE onsde a gd oto evolvng wth a constant angula velocty ω ad/s. nube of asses say, fou ae depcted by pont asses at dffeent ad n the sae tansvese plane.
www.getyun.co If,, and 4 ae the asses evolvng at ad,, and 4 espectvely n the sae plane. The centfugal foces exeted by each of the asses ae F c, F c, F c and F c4 espectvely. et F be the vecto su of these foces..e. F F c + F ω c + F + c + F ω c4 + ω + 4 ω 4 () The oto s sad to be statcally balanced f the vecto su F s zeo. If the vecto su F s not zeo,.e. the oto s unbalanced, then ntoduce a counteweght ( balance weght) of ass at adus to balance the oto so that, ω + ω + ω + ω + ω () + + + 4 o + 4 4 4 () The agntude of ethe o ay be selected and the othe can be calculated. In geneal, f s the vecto su of,,, 4 4 etc, then, + (4) The above equaton can be solved ethe analytcally o gaphcally.. nalytcal ethod: Pocedue: Step : Fnd out the centfugal foce o the poduct of ass and ts adus of otaton exeted by each of asses on the otatng shaft, snce ω s sae fo each ass, theefoe the agntude of the centfugal foce fo each ass s popotonal to the poduct of the espectve ass and ts adus of otaton. Step : Resolve these foces nto the hozontal and vetcal coponents and fnd the sus..e., Su of n cos θ Suof the n the hozontal vetcal cos θ sn θ sn θ coponents + coponents cos θ + sn θ + + sn θ cos θ + +
www.getyun.co Step : etene the agntude of the esultant centfugal foce R n n cos θ + sn θ Step 4: If θ s the angle, whch esultant foce akes wth the hozontal, then tan θ n n sn θ cos θ Step 5: The balancng foce s then equal to the esultant foce, but n opposte decton. Step 6: ow fnd out the agntude of the balancng ass, such that R Whee, balancng ass and ts adus of otaton. Gaphcal ethod: Step : aw the space daga wth the postons of the seveal asses, as shown. Step : Fnd out the centfugal foces o poduct of the ass and adus of otaton exeted by each ass. Step : ow daw the vecto daga wth the obtaned centfugal foces o poduct of the asses and ad of otaton. To daw vecto daga take a sutable scale. et ab, bc, cd, de epesents the foces F c, F c, F c and F c4 on the vecto daga. aw ab paallel to foce F c of the space daga, at b daw a lne paallel to foce F c. Slaly daw lnes cd, de paallel to F c and F c4 espectvely. Step 4: s pe polygon law of foces, the closng sde ae epesents the esultant foce n agntude and decton as shown n vecto daga. Step 5: The balancng foce s then, equal and opposte to the esultant foce. Step 6:
www.getyun.co etene the agntude of the balancng ass ( ) at a gven adus of otaton ( ), such that, F ω c o esultantof,, and 4 4 SE 4: IG OF SEVER SSES ROTTIG I IFFERET PES When seveal asses evolve n dffeent planes, they ay be tansfeed to a efeence plane and ths efeence plane s a plane passng though a pont on the axs of otaton and pependcula to t. When a evolvng ass n one plane s tansfeed to a efeence plane, ts effect s to cause a foce of sae agntude to the centfugal foce of the evolvng ass to act n the efeence plane along wth a couple of agntude equal to the poduct of the foce and the dstance between the two planes. In ode to have a coplete balance of the seveal evolvng asses n dffeent planes,. the foces n the efeence plane ust balance,.e., the esultant foce ust be zeo and. the couples about the efeence plane ust balance.e., the esultant couple ust be zeo. ass placed n the efeence plane ay satsfy the fst condton but the couple balance s satsfed only by two foces of equal agntude n dffeent planes. Thus, n geneal, two planes ae needed to balance a syste of otatng asses.
www.getyun.co Exaple: onsde fou asses,, and 4 attached to the oto at ad,, and 4 espectvely. The asses,, and 4 otate n planes,, and 4 espectvely. a) Poston of planes of asses hoose a efeence plane at O so that the dstance of the planes,, and 4 fo O ae,, and 4 espectvely. The efeence plane chosen s plane. hoose anothe plane between plane and 4 as shown. Plane s at a dstance of fo the efeence plane. The dstances of all the othe planes to the left of ay be taken as negatve( -ve) and to the ght ay be taken as postve (+ve). The agntude of the balancng asses and n planes and ay be obtaned by followng the steps gven below. Step : Tabulate the gven data as shown afte dawng the sketches of poston of planes of asses and angula poston of asses. The planes ae tabulated n the sae ode n whch they occu fo left to ght.
www.getyun.co Plane ass () Radus () entfugal foce/ω ( ) 4 stance fo Ref. plane () 5 ouple/ ω ( ) 6 - - 4 4 4 4 4 4 4 4 4 Step : onstuct the couple polygon fst. (The couple polygon can be dawn by takng a convenent scale) dd the known vectos and consdeng each vecto paallel to the adal lne of the ass daw the couple daga. Then the closng vecto wll be. The vecto d o on the couple polygon epesents the balanced couple. Snce the balanced couple s popotonal to, theefoe,
www.getyun.co o vecto vecto ' ' d o ' ' d o Fo ths the value of n the plane can be detened and the angle of nclnaton φ of ths ass ay be easued fo fgue (b). Step : ow daw the foce polygon (The foce polygon can be dawn by takng a convenent scale) by addng the known vectos along wth. The closng vecto wll be. Ths epesents the balanced foce. Snce the balanced foce s popotonal to, vecto eo o vecto eo Fo ths the balancng ass can be obtaned n plane and the angle of nclnaton of ths ass wth the hozontal ay be easued fo fgue (b). Pobles and solutons Poble. Fou asses,, and ae attached to a shaft and evolve n the sae plane. The asses ae kg, kg, 8 kg and 5 kg espectvely and the ad of otatons ae 4, 5, 6 and. The angula poston of the asses, and ae 6, 5 and 7 fo ass. Fnd the agntude and poston of the balancng ass at a adus of. Soluton: Gven: ass() kg kg (efeence ass) Radus().4 entfugal foce/ω ( ) kg-.48 kg- kg.5.5 kg- 8 kg.6.8 kg- 5 kg..45 kg- ngle(θ ) θ θ 6 θ 5 θ 7 To detene the balancng ass at a adus of.. The poble can be solved by ethe analytcal o gaphcal ethod.
www.getyun.co nalytcal ethod: Step : aw the space daga o angula poston of the asses. Snce all the angula poston of the asses ae gven wth espect to ass, take the angula poston of ass asθ. Tabulate the gven data as shown. Snce the agntude of the centfugal foces ae popotonal to the poduct of the ass and ts adus, the poduct can be calculated and tabulated. Step : Resolve the centfugal foces hozontally and vetcally and fnd the su. Resolvng,, and hozontally and takng the su gves, n cos θ.48 x cos + +.5 x cos 6 + +.8 x cos5.48+.5+ (.764) +.4 kg + +.45 x cos 7 () Resolvng,, and vetcally and takng the su gves,
www.getyun.co n sn θ.48 x sn + + +.5 x sn 6 + +.8 x sn5 +.45 x sn7 +.4 +.764 + (.45).747 kg () Step : etene the agntude of the esultant centfugal foce n n R cos θ sn θ + (.4) + (.747).748kg Step 4: The balancng foce s then equal to the esultant foce, but n opposte decton. ow fnd out the agntude of the balancng ass, such that R.748kg Theefoe, R.748 7.48 kg ns. Whee, balancng ass and ts adus of otaton Step 5: etene the poston of the balancng ass. If θ s the angle, whch esultant foce akes wth the hozontal, then tanθ n n cos θ and θ 87.4 o 9.6.747.4.97 Reebe STUETS TKE OPY.e. n fst quadant all angles (, and tanθ) ae postve, n second quadant only sn θ s postve, n thd quadant only tan θ s postve and n fouth quadant only cos θ s postve. Snce nueato s postve and denonato s negatve, the esultant foce akes wth the hozontal, an angle (easued n the counte clockwse decton) θ 9.6
www.getyun.co The balancng foce s then equal to the esultant foce, but n opposte decton. The balancng ass les opposte to the adal decton of the esultant foce and the angle of nclnaton wth the hozontal s, θ 87.4 angle easued n the clockwse decton. Gaphcal ethod: Step : Tabulate the gven data as shown. Snce the agntude of the centfugal foces ae popotonal to the poduct of the ass and ts adus, the poduct can be calculated and tabulated. aw the space daga o angula poston of the asses takng the actual angles( Snce all angula poston of the asses ae gven wth espect to ass, take the angula poston of ass as θ ).
www.getyun.co Step : ow daw the foce polygon (The foce polygon can be dawn by takng a convenent scale) by addng the known vectos as follows. aw a lne ab paallel to foce F (o the poduct to a pope scale) of the space daga. t b daw a lne bc paallel to F (o the poduct ). Slaly daw lnes cd, de paallel to F (o the poduct ) and F (o the poduct ) espectvely. The closng sde ae epesents the esultant foce R n agntude and decton as shown on the vecto daga. Step : The balancng foce s then equal to the esultant foce, but n opposte decton. R Theefoe, R 7.48 kg ns The balancng ass les opposte to the adal decton of the esultant foce and the angle of nclnaton wth the hozontal s, θ 87.4 angle easued n the clockwse decton.
www.getyun.co Poble : The fou asses,, and ae kg, 5 kg, kg and kg attached to a shaft and evolve n the sae plane. The coespondng ad of otatons ae.5 c, 7.5 c, 5 c and c and the angles easued fo ae 45, and 55. Fnd the poston and agntude of the balancng ass, f the adus of otaton s 6 c. Soluton: nalytcal ethod: Gven: ass() kg kg (efeence ass) Radus().5 entfugal foce/ω ( ) kg-.5 kg- 5 kg.75 6.5 kg- kg.5 kg- kg. 9 kg- ngle(θ ) θ θ 45 θ θ 55 θ?.6? Step : aw the space daga o angula poston of the asses. Snce all the angula poston of the asses ae gven wth espect to ass, take the angula poston of ass asθ. Tabulate the gven data as shown. Snce the agntude of the centfugal foces ae popotonal to the poduct of the ass and ts adus, the poduct can be calculated and tabulated.
www.getyun.co Step : Resolve the centfugal foces hozontally and vetcally and fnd the su. Resolvng,, and hozontally and takng the su gves, n cos θ.5 x cos + + + 6.5 x cos 45 +.5 + 8.56 + ( 5) + (.) 5.97 kg () + x cos + 9 x cos 55 Resolvng,, and vetcally and takng the su gves, n sn θ.5 x sn + + 6.5 x sn + 45 + + 8.56 + 5.98 + ( 7.67) 6.87 kg () + x sn + 9 x sn 55 Step : etene the agntude of the esultant centfugal foce n n R + ( 5.97) + ( 6.87) 7.9 kg Step 4: The balancng foce s then equal to the esultant foce, but n opposte decton. ow fnd out the agntude of the balancng ass, such that R 7.9 kg Theefoe, R 7.9 8.98 kg.6 ns Whee, balancng ass and ts adus of otaton Step 5: etene the poston of the balancng ass. If θ s the angle, whch esultant foce akes wth the hozontal, then
www.getyun.co tanθ n n and θ.8 6.87 5.97.4 The balancng ass les opposte to the adal decton of the esultant foce and the angle of nclnaton wth the hozontal s, θ.8 angle easued n the counte clockwse decton. Gaphcal ethod: Step : Tabulate the gven data as shown. Snce the agntude of the centfugal foces ae popotonal to the poduct of the ass and ts adus, the poduct can be calculated and tabulated.
www.getyun.co Step : aw the space daga o angula poston of the asses takng the actual angles (Snce all angula poston of the asses ae gven wth espect to ass, take the angula poston of ass asθ ). Step : ow daw the foce polygon (The foce polygon can be dawn by takng a convenent scale) by addng the known vectos as follows. aw a lne ab paallel to foce F (o the poduct to a pope scale) of the space daga. t b daw a lne bc paallel to F (o the poduct ). Slaly daw lnes cd, de paallel to F (o the poduct ) and F (o the poduct ) espectvely. The closng sde ae epesents the esultant foce R n agntude and decton as shown on the vecto daga. Step 4: The balancng foce s then equal to the esultant foce, but n opposte decton. R Theefoe, R 9 kg ns The balancng ass les opposte to the adal decton of the esultant foce and the angle of nclnaton wth the hozontal s, θ angle easued n the counte clockwse decton.
www.getyun.co Poble : oto has the followng popetes. Plane ass agntude Radus ngle 9 kg 7 kg 8 kg 4 4 6 kg xal dstance fo fst ass θ - θ 6 6 θ 5 θ 7 56 If the shaft s balanced by two counte asses located at ad and evolvng n planes dway of planes and, and dway of and 4, detene the agntude of the asses and the espectve angula postons. Soluton: nalytcal ethod: ass () kg Radus () entfugal foce/ω ( ) kg- 4 stance fo Ref. plane 5 ouple/ ω ( ) kg- 6 ngle θ 9...9 -.8 -.7?.. θ? 7...84.8.67 6 8..4..4.688 5?...6 l.6 θ? 4 6.. 4 4.7.48.456 7 Fo dynac balancng the condtons equed ae, 7 + + ----------(I) fo foce balance l+ l --------------(II) fo couple balance
www.getyun.co Step : Resolve the couples nto the hozontal and vetcal coponents and fnd the sus. Su of the hozontal coponents gves, On substtuton we get.7 cos +.456cos7.e. l +.6 +.67 cos 6 Su of the vetcal coponents gves,.e. l sn θ +.7 sn.6 +.456 sn7 sn θ l +.6 On substtuton we get Squang and addng () and (), we get l +.688cos5.85 () +.67 sn 6 +.6.97 () +.688sn5
www.getyun.co l.e.,.6 Theefoe, (.85) + (.97).484.484 6.9kg.6 ns vdng () by (), we get.97 tan θ.85 and θ.7 Step : Resolve the foces nto the hozontal and vetcal coponents and fnd the sus. Su of the hozontal coponents gves, On substtuton we get.9 cos +.e. + +.84 cos 6 Su of the vetcal coponents gves, On substtuton we get.9 sn +.e. sn θ + + +. cos5 +.x6.9xcos.7 +.84 sn 6.69 () sn θ + +.x6.9xsn.7 Squang and addng () and (4), we get.698 (4) +. sn5 +.7 cos7 +.7sn7.e.,. Theefoe, (.69) + (.698).58.58 5.8 kg. ns vdng (4) by (), we get.698 tanθ and θ.6.69 ns
www.getyun.co Gaphcal Soluton:
www.getyun.co Poble 4: The syste has the followng data.. kg. 8 kg. 4 kg. 5 @. 4. 8 @ 48. 8. 4 @ 5. 4 The dstances of planes n etes fo plane ae: l.854, l.7,l.96,l.97 Fnd the ass-adus poducts and the angula locatons needed to dynacally balance the syste usng the coecton planes and. Soluton: nalytcal ethod Plane ass () kg Radus () entfugal foce/ω ( ) kg- 4 stance fo Ref. plane 5 ouple/ ω ( ) kg- 6 ngle θ? θ?..5.6.854.648.4.8.8.4796.7.568 48.8.4.4.496.96 5.984 5.4?.97.97 θ? 7
www.getyun.co Step : Resolve the couples nto the hozontal and vetcal coponents and fnd the sus. Su of the hozontal coponents gves, On substtuton we get.648 cos.4 +.97.e. l + Su of the vetcal coponents gves,.648 sn.4 +.97.e. l + l On substtuton we get +.568 cos 48.8.766 ().97 l Squang and addng () and (), we get vdng () by (), we get +.568 sn 48.8.769 ().97.766.769 +.97.97.97 kg + 5.984 cos5.4 + 5.984 sn5.4 tanθ.769.766 and θ 75.7 ns Step : Resolve the foces nto the hozontal and vetcal coponents and fnd the sus. Su of the hozontal coponents gves,
www.getyun.co On substtuton we get.6 cos.4 + Theefoe + Su of the vetcal coponents gves, On substtuton we get.6 sn.4 + Theefoe + + +.4796 cos 48.8 +.97 cos75.7.66 () + +.97 sn75.7 Squang and addng () and (4), we get.876 (4) +.4796 sn 48.8 (.66) + (.876).887 kg vdng (4) by (), we get +.496 cos5.4 +.496 sn 5.4 tanθ.876.66 and θ 8.5 ns Poble 5: shaft caes fou asses,, and of agntude kg, kg, 4 kg and kg espectvely and evolvng at ad 8, 7, 6 and 8 n planes easued fo at, 4 and 7. The angles between the canks easued antclockwse ae to 45, to 7 and to. The balancng asses ae to be placed n planes X and Y. The dstance between the planes and X s, between X and Y s 4 and between Y and s. If the balancng asses evolve at a adus of, fnd the agntudes and angula postons.
www.getyun.co Gaphcal soluton: et, X be the balancng ass placed n plane X and Y be the balancng ass placed n plane Y whch ae to be detened. Step : aw the poston of the planes as shown n fgue (a). et X be the efeence plane (R.P.). The dstances of the planes to the ght of the plane X ae taken as postve (+ve) and the dstances of planes to the left of X plane ae taken as negatve(-ve). The data ay be tabulated as shown Snce the agntude of the centfugal foces ae popotonal to the poduct of the ass and ts adus, the poduct can be calculated and tabulated. Slaly the agntude of the couples ae popotonal to the poduct of the ass, ts adus and the axal dstance fo the efeence plane, the poduct l can be calculated and tabulated as shown.
www.getyun.co Plane ass () kg Radus () entfugal foce/ω ( ) kg- 4 stance fo Ref. plane X 5 ouple/ ω ( ) kg- 6 ngle θ.8 6 -. -.6 - X X?. X X. X θ X?.7. 4. to 45 4.6 4. 7. to 7 Y Y?. Y Y. Y.4 Y Y l Y.4 Y θ?.8 6.6 9.6 to Step : ssung the ass as hozontal daw the sketch of angula poston of asses as shown n fgue (b). Step : aw the couple polygon to soe sutable scale by takng the values of l (colun no. 6) of the table as shown n fgue (c). 7 Y aw lne o a paallel to the adal lne of ass. t a daw lne a b paallel to adal lne of ass. Slaly, daw lnes b c, c d paallel to adal lnes of asses and espectvely. ow, jon d to o whch gves the balanced couple.
www.getyun.co We get,.4 o Y Y vecto d'o' 7.kg 8.5 kg ns Step 4: To fnd the angula poston of the ass Y daw a lne o Y n fgue (b) paallel to d o of the couple polygon. y easueent we get θ Y n the clockwse decton fo. Step 5: ow daw the foce polygon by consdeng the values of (colun no. 4) of the table as shown n fgue (d). Follow the sla pocedue of step. The closng sde of the foce polygon.e. e o epesents the balanced foce. o X X vectoeo 5.5 kg X 55kg ns Step 6: The angula poston of X s detened by dawng a lne o X paallel to the lne e o of the foce polygon n fgue ( b). Fo fgue (b) we get, θ X 45, easued clockwse fo. ns Poble 6:,, and ae fou asses caed by a otatng shaft at ad, 5, and 5 espectvely. The planes n whch the asses evolve ae spaced 6 apat and the ass of, and ae kg, 5 kg and 4 kg espectvely. Fnd the equed ass and elatve angula settngs of the fou asses so that the shaft shall be n coplete balance. Soluton: Gaphcal ethod: Step : et, be the balancng ass placed n plane whch s to be detened along wth the elatve angula settngs of the fou asses. et be the efeence plane (R.P.). ssue the ass as hozontal aw the sketch of angula poston of ass (lne o ) as shown n fgue (b). The data ay be tabulated as shown.
www.getyun.co VTU EUST PROGRE - 7 Plane (R.P.) ass () kg Radus () entfugal foce/ω ( ) kg- 4 stance fo Ref. plane 5 ouple/ ω ( ) kg- 6 ngle θ?.. θ?.5.5.6.75 θ 5.... θ? 4.5.6.8.8 θ? 7 Step : To detene the angula settngs of ass and the couple polygon s to be dawn fst as shown n fg (c). Take a convenent scale aw a lne o b equal to.75 kg- paallel to the lne o. t pont o and b daw vectos o c and b c equal to. kg- and.8 kg- espectvely. These vectos ntesect at pont c. Fo the constucton of foce polygon thee ae fou optons. ny one opton can be used and elatve to that the angula settngs of ass and ae detened.
www.getyun.co Step : ow n fgue (b), daw lnes o and o paallel to o c and b c espectvely. Fo easueent we get, θ and θ 4 ns Step 4: In ode to fnd and ts angula settng daw the foce polygon as shown n fgue (d). losng sde of the foce polygon od epesents the poduct..e.
www.getyun.co.7kg- Theefoe,.7 7kg ns Step 5: ow daw lne o paallel to od of the foce polygon. y easueent, we get, θ 55 ns Poble 7: shaft caes thee asses, and. Planes and ae 6 c and c fo., and ae 5 kg, 4 kg and 6 kg espectvely at a adus of.5 c. The angula poston of ass and ass wth ae 9 and espectvely. Fnd the unbalanced foce and couple. lso fnd the poston and agntude of balancng ass equed at c adus n planes and dway between and, and and. Soluton: ase (): Plane (R.P.) ass () kg Radus () entfugal foce/ω ( ) kg- 4 stance fo Ref. plane 5 ouple/ ω ( ) kg- 6 5.5.5 4.5..6.6 6.5.5..8 ngle θ 7 θ θ 9 θ nalytcal ethod Step : etenaton of unbalanced couple Resolve the couples nto the hozontal and vetcal coponents and fnd the sus. Su of the hozontal coponents gves, l.6 cos 9 +.8cos Su of the vetcal coponents gves, l.6 sn9 +.8 sn.559 (). ()
www.getyun.co Squang and addng () and (), we get unbalanced (-.559) + (-.).588 kg Step : etenaton of unbalanced foce Resolve the foces nto the hozontal and vetcal coponents and fnd the sus. Su of the hozontal coponents gves,.5 cos +. cos 9.5+ + (.99).49 () Su of the vetcal coponents gves,.5 sn +. sn 9 +.5 cos +.5 sn +. + (.75).5 (4) Squang and addng () and (4), we get F unbalanced Gaphcal soluton: (-.49) + (.5).548 kg
www.getyun.co ` b b.8.6 o o Unbalanced foce c.5. c Unbalanced couple o.5 a ouple polygon Foce polygon ase (): To detene the agntude and dectons of asses and. et, be the balancng ass placed n plane and be the balancng ass placed n plane whch ae to be detened. The data ay be tabulated as shown.
www.getyun.co Plane ass () kg Radus () entfugal foce/ω ( ) kg- 4 stance fo Ref. plane 5 ouple/ ω ( ) kg- 6 5.5.5 -. -.75 (R.P.) ngle θ 7 θ?.. θ? 4.5... θ 9?...6.6 θ? 6.5.5.9.5 θ nalytcal ethod: Step : Resolve the couples nto the hozontal and vetcal coponents and fnd the sus. Su of the hozontal coponents gves, l + On substtuton we get.e. -.75 cos -.75 + +.6.6 l +. cos 9.544.544.6 Su of the vetcal coponents gves, l + On substtuton we get -.75 sn.e. +.+.6.6 l.75.75.6 +. sn 9 Squang and addng () and (), we get +.6 + (.69) 5.74 () 6.5 () +.6 + (.675) +.5 cos +.5 sn
www.getyun.co ( ) + ( ) (5.74) + (6.5) 7.6.e. 7.6 and 6.5 kg ns vdng () by (), we get 6.5 tanθ and θ.65 5.74 ns Step : Resolve the foces nto the hozontal and vetcal coponents and fnd the sus. Su of the hozontal coponents gves, + On substtuton we get.5 cos.5+. + +.574+ (.99) Theefoe. and +. +. cos 9 +.55 + +.649 cos.65.55 5.5 (). +.5cos Su of the vetcal coponents gves, + On substtuton we get.5 sn +. Theefoe. and +. +.875 + +. sn 9 + +.65+ (.75) +.649 sn.65.875 8.75 (4). +.5sn Squang and addng () and (4), we get ( ) + ( ) (-5.5) + (-8.75) 74.9.e. 74.9 and 6.7 kg ns vdng (4) by (), we get
www.getyun.co 8.75 tanθ and θ 9. -5.5 ns The balancng ass s at an angle 9. + 8 99. easued n counte clockwse decton. Gaphcal ethod:..5.75.6 OUPE POYGO..5...5 FORE POYGO
www.getyun.co Poble 8: Fou asses,, and ae copletely balanced. asses and ake angles of 9 and espectvely wth n the sae sense. The planes contanng and ae apat. asses,, and can be assued to be concentated at ad of 6, 48, 4 and espectvely. The asses, and ae 5 kg, 5 kg and kg espectvely. etene ) ass and ts angula poston ) poston of planes and. Soluton: nalytcal ethod Plane (R.P.) Step : aw the space daga o angula poston of the asses. Snce the angula poston of the asses and ae gven wth espect to ass, take the angula poston of ass asθ. Tabulate the gven data as shown. ass () kg Radus () entfugal foce/ω ( ) kg- 4 stance fo Ref. plane 5 ouple/ ω ( ) kg- 6 ngle θ?.6.6 θ? 5.48 7. l? 7. l 5.4 6. l? 6. l. 6. l? 6. l θ 7 θ 9 θ
www.getyun.co Step : ass be the balancng ass placed n plane whch s to be detened along wth ts angula poston. Refe colun 4 of the table. Snce s to be detened ( whch s the only unknown),esolve the foces nto the hozontal and vetcal coponents and fnd the sus. Su of the hozontal coponents gves, On substtuton we get.6 Theefoe.6 + 7. cos + + 6. cos 9 -.4 () Su of the vetcal coponents gves, + + 6. cos + sn θ On substtuton we get.6 Theefoe.6 + 7. sn + + 6. sn9 -. () Squang and addng () and (), we get.6 ( (.4).6. kg ns.6 vdng () by (), we get ) + (.) + + + 6. sn.6. tanθ and Resutltant akes an angle 56.6 -.4 The balancng ass akes an angle of θ 6.6 ns
www.getyun.co Step : Resolve the couples nto the hozontal and vetcal coponents and fnd the sus. Su of the hozontal coponents gves, l On substtuton we get l + 7.l 7.l cos + + 6.l 5.96l l cos9 - + + 6.l - - - - - - - - l cos - - () + l Su of the vetcal coponents gves, l On substtuton we get l + 7.l + + 6.l sn + l + 6.l l - sn 9 + + 6.l - - - - - - - - - l - (4) sn + l ut fo fgue we have, On substtutng ths n equaton (4), we get l 6.( l.e. 6.l l +.) l l +..8 - - - - - - - - - -(5) Thus we have two equatons ( ) and (5), and l.5 and 7.l 6.l l 5.96 l l On solvng the equatons, we get.976 - - - - - - - - - - - ().8 two unknowns l - - - - - - - - - -(5) s pe the poston of planes of asses assued the dstances shown ae postve (+ ve ) fo the efeence plane. ut the calculated values of dstances l and l ae negatve. The coected postons of planes of asses s shown below., l
www.getyun.co Refeences:. Theoy of achnes by S.S.Rattan, Thd Edton, Tata cgaw Hll Educaton Pvate ted.. Kneatcs and ynacs of achney by R.. oton, Fst Edton n SI unts, Tata cgaw Hll Educaton Pvate ted.. Pe on ynac alancng auses, oectons and onsequences y J yons Intenatonal Sales anage IR alancng v. EntekIR Intenatonal