CHAPTER 2 TORSIONAL VIBRATIONS

Transcription

1 Dr Tiwari, Associae Professor, De. of Mechaical Egg., T Guwahai, (riwari@iig.ere.i) CHAPTE TOSONAL VBATONS Torsioal vibraios is redomia wheever here is large discs o relaively hi shafs (e.g. flywheel of a uch ress). Torsioal vibraios may origial from he followig forcigs (i) ieria forces of recirocaig mechaisms (such as isos i C egies) (ii) imulsive loads occurrig durig a ormal machie cycle (e.g. durig oeraios of a uch ress) (iii) shock loads alied o elecrical machiery (such as a geeraor lie faul followed by faul removal ad auomaic closure) (iv) orques relaed o gear mesh frequecies, urbie blade assig frequecies, ec. For machies havig massive roors ad flexible shafs (where he sysem aural frequecies of orsioal vibraios may be close o, or wihi, he source frequecy rage durig ormal oeraio) orsioal vibraios cosiue a oeial desig roblem area. such cases desigers should esure he accurae redicio of machie orsioal frequecies ad frequecies of ay orsioal load flucuaios should o coicide wih he orsioal aural frequecies. Hece, he deermiaio of orsioal aural frequecies of he sysem is very imora.. Simle Sysem wih Sigle oor Mass Cosider a roor sysem as show Figure.(a). The shaf is cosidered as massless ad i rovides orsioal siffess oly. The disc is cosidered as rigid ad has o flexibiliy. f a iiial disurbace is give o he disc i he orsioal mode ad allow i o oscillae is ow, i will execue he free vibraios as show i Figure.. shows ha roor is siig wih a omial seed of ad excuig orsioal vibraios, θ(), due o his i has acual seed of ( + θ()). should be oed ha he siig seed remais same however agular velociy due o orsio have varyig direcio over a eriod. The oscillaio will be simle harmoic moio wih a uique frequecy, which is called he orsioal aural frequecy of he roor sysem. K θ K l θ, θ θ, θ Fixed ed Figure.a A sigle-mass cailever roor sysem Figure.(b) Free body diagram of disc

2 Dr Tiwari, Associae Professor, De. of Mechaical Egg., T Guwahai, (riwari@iig.ere.i) θ θ θ θ Figure. Torsioal vibraios of a roor From he heory of orsio of shaf, we have K T GJ = = () θ l where, K is he orsioal siffess of shaf, is he roor olar mome of ieria, kg-m, J is he shaf olar secod mome of area, l is he legh of he shaf ad θ is he agular dislaceme of he roor. From he free body diagram of he disc as show i Figure.(b) Exeral orque of disc = θ K θ = θ () Equaio () is he equaio of moio of he disc due o free orsioal vibraios. The free (or aural) vibraio has he simle harmoic moio (SHM). For SHM of he disc, we have θ( ) = θˆ si f so ha θ = θˆsi = θ (3, 4) f f f where ˆ θ is he amliude of he orsioal vibraio ad f is he orsioal aural frequecy. O subsiuig Eqs. (3) ad (4) io Eq. (), we ge ( θ) Kθ = or = K / (5) f f. A Two-Disc Torsioal Sysem θ K θ Fricioless bearigs Figure.3 A wo-disc orsioal sysem 7

3 Dr Tiwari, Associae Professor, De. of Mechaical Egg., T Guwahai, (riwari@iig.ere.i) A wo-disc orsioal sysem is show i Figure.3. his case whole of he roor is free o roae as he shaf beig moued o fricioless bearigs. θ,θ (θ - θ )K (θ - θ )K θ,θ (a) Disc (b) Disc Figure.4 Free body diagram of discs From he free body diagram i Figure.4(a) Exeral orque = θ ad Exeral orque= θ or ( ) ad ( ) θ θ K = θ θ θ K = θ θ + Kθ Kθ = or θ + Kθ Kθ = ad () For free vibraio, we have SHM, so he soluio will ake he form θ = θ ad f θ = θ (3) f Subsiuig equaio (3) io equaios () & (), i gives θ + Kθ Kθ = ad θ + Kθ Kθ = f which ca be assembled i a marix form as K f K θ = K K θ f or [ K]{ θ } = { } (4, 5) The o-rial soluio of equaio (5) is obaied by akig deermia of he marix [K] as which gives K = 73

4 Dr Tiwari, Associae Professor, De. of Mechaical Egg., T Guwahai, (riwari@iig.ere.i) ( K )( K ) K = or ( ) The roos of equaio (6) are give as + K = (6) 4 f f ( ) K ( ) f = ad f = +.5 (7) From equaio (4) corresodig o firs aural frequecy for f =, we ge θ = θ (8) θ θ Figure.5 Firs mode shae From Eq. (8) i ca be cocluded ha, he firs roo of equaio (6) rereses he case whe boh discs simly rolls ogeher i hase wih each oher as show i Figure.5. is he rigid body mode, which is of a lile racical sigificace. This mode i geerally occurs wheever he sysem has free-free ed codiios (for examle aerolae durig flyig). From equaio (4), for f = f, we ge ( ) K ˆ θ K ˆ θ = f ( ) K ˆ ˆ + K θ Kθ = or ( ) which gives relaive amliudes of wo discs as ˆ θ ˆ θ = (9) The secod mode shae (Eq. 9) rereses he case whe boh masses vibrae i ai-hase wih oe aoher. Figure.6 shows mode shae of wo-roor sysem, showig wo discs vibraig i oosie direcios. 74

5 Dr Tiwari, Associae Professor, De. of Mechaical Egg., T Guwahai, (riwari@iig.ere.i) B Elasic lie Node θ C l l θ Figure.6 Secod mode shae From mode shaes, we have θ θ θ l = = () l l θ l Sice boh he masses are always vibraig i oosie direcio, here mus be a oi o he shaf where orsioal vibraio is o akig lace i.e. a orsioal ode. The locaio of he ode may be esablished by reaig each ed of he real sysem as a searae sigle-disc cailever sysem as show i Figure.6. The ode beig reaed as he oi where he shaf is rigidly fixed. Sice value of aural frequecy is kow (he frequecy of oscillaio of each of he sigle-disc sysem mus be same), hece we wrie = K = K () f where f is defied by equaio (7), K ad K are orsioal siffess of wo (equivale) sigleroor sysem, which ca be obaied from equaio (), as K = ad K = f f The legh l ad l he ca be obaied by (from equaio ) l = GJ K ad l = GJ K wih l+ l = l () 75

6 Dr Tiwari, Associae Professor, De. of Mechaical Egg., T Guwahai, (riwari@iig.ere.i).3 Sysem wih a Seed Shaf l l l 3 a b d d d 3 (a) l e l e l e3 a b (b) Figure.7(a) Two discs wih seed shaf (b) Equivale uiform shaf Figure.7(a) shows a wo-disc seed shaf. such cases he acual shaf should be relaced by a useed equivale shaf for he urose of he aalysis as show i Fig..7(b). The equivale shaf diameer may be same as he smalles diameer of he real shaf. The equivale shaf mus have he same orsioal siffess as he real shaf, sice he orsioal srigs are coeced i series. The equivale orsioal srig ca be wrie as = + + K K K K e 3 Nohig equaio (), we have l J = l J + l J + l J e e 3 3 which gives l = l J J + l J J + l J J = l + l + l e e e 3 e 3 e e e3 (3) le = l J /, /, / e J le = l J e J le = l J e J wih

7 Dr Tiwari, Associae Professor, De. of Mechaical Egg., T Guwahai, (riwari@iig.ere.i) where l, l, l are equivale leghs of shaf segmes havig equivale shaf diameer d 3 ad l e is e e e 3 he oal equivale legh of useed shaf havig diameer d 3 as show i Figure.7(b). From Figure.7(b) ad oig equaios () ad (), i equivale shaf he ode locaio ca be obaied as ( ) l + a= GJ e e f le + b= GJ ( ) 3 e ad (4) where ( + ) / K = + + e ad K = e l GJ l GJ l3 GJ3 From above equaios he ode osiio a & b ca be obaied i he equivale shaf legh. Now he ode locaio i real shaf sysem ca be obaied as follows: From equaio (3), we have J π π l = l, J = d, J = d e e 4 4 e 3 J 64 4 Sice above equaio is for shaf segme i which ode is assumed o be rese, we ca wrie a= a J J ad b= b J J e e above equaios ca be combied as a a = (5) b b So oce a & b are obaied from equaio (4) he locaio of ode i acual shaf ca be obaied equaio (5) i.e. he fial locaio of ode o he shaf i real sysem is give i he same roorio alog he legh of shaf i equivale sysem i which he ode occurs..4 MODF Sysems Whe here are several umber of discs i he roor sysem i becomes is muli-dof sysem. Whe he mass of he shaf iself may be sigifica he he aalysis described i revious secios (i.e. sigle or wo-discs roor sysems) is iadequae o model such sysem, however, hey could be exeded o allow for more umber of lumed masses (i.e. rigid discs) bu resulig mahemaics 77

8 T k Dr Tiwari, Associae Professor, De. of Mechaical Egg., T Guwahai, (riwari@iig.ere.i) becomes cumbersome. Aleraive mehods are: (i) rasfer marix mehods (ii) mehods of mechaical imedace ad (iii) fiie eleme mehods..4. Trasfer marix mehod: A muli-disc roor sysem, suored o fricioless suors, is show i Fig. 7. Fig. 8 shows he free diagram of a shaf ad a disc, searaely. A aricular saio i he sysem, we have wo sae variables: he agular wis θ ad Torque T. Now i subseque secios we will develo relaioshi of hese sae variables bewee wo eighbourig saios ad which ca be used o obai goverig equaios of moio of he whole sysem..poi marix: 3 k k k 3 4 θ θ θ 3 Figure.8 A muli-disc roor sysem LT T LT Fig..9(a) Free body diameer of shaf secio (b) Free body diagram of roor secio θ The equaio of moio for he disc is give by (see Figure.9(b)) T T = θ (6) L For free vibraios, agular oscillaios of he disc is give by θ = ˆsi θ so ha θ = ˆsi θ = θ (7) f f Subsiuig back io equaio (6), we ge T T = θ (8) L f Agular dislacemes o he eiher side of he roor are equal, hece 78

9 Dr Tiwari, Associae Professor, De. of Mechaical Egg., T Guwahai, (riwari@iig.ere.i) θ = θ (9) L Equaios (8) ad (9) ca be combied as θ T = θ f L T or { S} [ P] { S} = (, ) L where {S} is he sae vecor a saio ad [P] is he oi marix for saio.. Field marix: For shaf eleme as show i Figure.9(a), he agle of wis is relaed o is orsioal siffess ad o he orque, which is rasmied hrough i, as T θ θ= () K Sice he orque rasmied is same a eiher ed of he shaf, hece T = T (3) L Combiig () ad (3), we ge L θ k θ = T T (4) which ca be wrie as L { S} [ F] { S} = (5) where [F] is he field marix for he shaf eleme. Now we have { } = [ ] { } = [ ][ ] { } == [ ] { } S P S P F S U S where [U] is he rasfer marix, which relaes he sae vecor a righ of saio o he sae vecor a righ of saio. O he same lies, we ca wrie { S} = [ U] { S} { S} = [ U] { S} = [ U] [ U] { S} { S} = [ U] { S} = [ U] [ U] [ U] { S} { S} = [ U] { S} = [ U] [ U] [ U] { S} = [ T]{ S} (6) 79

10 Dr Tiwari, Associae Professor, De. of Mechaical Egg., T Guwahai, (riwari@iig.ere.i) where [T] is he overall sysem rasfer marix. The overall rasfermaio ca be wrie as θ T = θ T (7) For free-free boudary codiios, he each ed of he machie orque rasmied hrough he shaf is zero, hece T = T = (8) O usig equaio (8) io equaio(7), he secod se of equaio gives θ = which gives ( f) = sice θ (9) which is saisfied for some values of f, which are sysem aural frequecies. These roos may be foud by ay roo-searchig echique. Agular wiss ca be deermied for each value of f from firs se of equaio of equaio (7), as θ = T O akig θ =, we ge ( ) θ we ge θ f = = (3) Eq. (3), coais so for each value of differe value of θ 4 is obaied ad usig Eq. (7) relaive dislacemes of all oher saios ca be obaied, by which mode shaes ca be loed. Examle.. Obai he orsioal aural frequecy of he sysem show i Figure. usig he rasfer marix mehod. Check resuls wih closed form soluio available. Take G =.8 N/m..6m. m.6 kgm 5.66 kgm Figure. Examle. Soluio: We have followig roeries of he roor 8

11 Dr Tiwari, Associae Professor, De. of Mechaical Egg., T Guwahai, (riwari@iig.ere.i) π 4 G=.8 N/m -6 ; l =.6 m; J = (.) = 9.8 m 3 The orsioal siffess is give as 4 k GJ = = = l Nm/rad Aalyical mehod: The aural frequecies i he closed form are give as ( ) ( + ) k = ; ad rad/sec = = = Mode shaes are give as For ad = { θ} = { θ} = rad/s { θ} = { θ} = 4.{ θ} Trasfer marix mehod: Sae vecors ca be relaed bewee saios & ad &, as { S} = [ P] { S} { S} = [ P] [ F] { S} = [ P] [ F] [ P] { S} The overall rasformaio of sae vecors bewee & is give as θ θ k θ T T T k = = ( ) k k k θ = ( ) ( ) T k k O subsiuig values of various roor arameers, i gives θ T ( ) ( ) θ = T (A) Sice eds of he roor are free, he followig boudary codiios will aly 8

12 Dr Tiwari, Associae Professor, De. of Mechaical Egg., T Guwahai, (riwari@iig.ere.i) T = T = O alicaio of boudary codiios, we ge he followig codiio = [ ]{ θ} = 5 4 Sice { θ}, we have [ ] = 5 which gives he aural frequecy as = ad = rad/sec which are exacly he same as obaied by he closed form soluio. Mode shaes ca be obaied by subsiuig hese aural frequecies oe a a ime io equaio (A), as For ad = { θ} = { θ} rigid body mode = rad/s { θ} = 4.{ θ} ai-hase mode which are also exacly he same as obaied by closed form soluios. Examle.. Fid orsioal aural frequecies ad mode shaes of he roor sysem show i Figure. B is a fixed ed ad D ad D are rigid discs. The shaf is made of seel wih modulus of rigidiy G =.8 () N/m ad uiform diameer d = mm. The various shaf leghs are as follows: BD = 5 mm, ad D D = 75 mm. The olar mass mome of ieria of discs are: =.8 kg-m ad =. kg-m. Cosider he shaf as massless ad use (i) he aalyical mehod ad (ii) he rasfer marix mehod. Figure. Examle. B D D Soluio: Aalyical mehod: From free body diagrams of discs as show i Figure., equaios of moio ca be wrie as 8

13 Dr Tiwari, Associae Professor, De. of Mechaical Egg., T Guwahai, (riwari@iig.ere.i) θ + kθ + k ( θ - θ ) = θ + k ( θ - θ ) = The above equaios for free vibraios ad hey are homogeeous secod order differeial equaios. free vibraios discs will execue simle harmoic moios. k θ k ( θ -θ ) k ( θ -θ ) θ θ (a) D (b) D Figure. Free body diagram of discs For he simle harmoic moio θ = θ, hece equaios of moio ake he form k+ k - k θ = k k θ O akig deermia of he above marix, i gives he frequecy equaio as ( k + k + k ) + k k = 4 which ca be solved for, as ( ) k + k + k ± k + k + k 4k k = For he rese roblem he followig roeries are gives GJ GJ k = = 878 N/m ad k = = N/m l l =.8 kgm ad =. kgm Naural frequecies are obaied as 83

14 Dr Tiwari, Associae Professor, De. of Mechaical Egg., T Guwahai, (riwari@iig.ere.i) = rad/s ad = 75. rad/s The relaive amliude raio ca be obaied as (Figure.3) k - θ θ = k =.336 for ad -.7 for (a) For Figure.3 Mode shaes (b) For Trasfer marix mehod k k Figure.4 Two-discs roor sysem wih saio umbers For Figure.4 sae vecors ca be relaed as { θ} = [ P] [ F] [ P] [ F] { θ} The above sae vecor a various saios ca be relaed as / k / k θ θ θ θ = ad = T - - T T T + + k k which ca be combied o give 84

15 Dr Tiwari, Associae Professor, De. of Mechaical Egg., T Guwahai, (riwari@iig.ere.i) θ T - + k k k k θ = T + k k (A) wih = + k Boudary codiios are give as A saio θ = ad T = (assumed) ad a righ of saio T = O alicaio of boudary codiios he secod equaio of equaio (A), we ge k = + + k T sice T ad o subsiuig for, we ge = k k k which ca be solved o give k k k k k k = + ± should be oed ha i is same as obaied by he aalyical mehod. 85

16 Dr Tiwari, Associae Professor, De. of Mechaical Egg., T Guwahai, (riwari@iig.ere.i) Exercise.. Obai he orsioal criical seed of a roor sysem as show i Figure E.. Take he olar mass mome of ieria, =.4 kg-m. Take shaf legh a =.3 m ad b =.7 m; modulus of rigidiy G =.8 N/m. The diameer of he shaf is mm. Bearig A is flexible ad rovides a orsioal srig of siffess equal o 5 erce of he siffess of he shaf segme havig legh a ad bearig B is a fixed bearig. Use eiher he fiie eleme mehod or he rasfer marix mehod. A B a b Figure E. A overhag roor sysem Exercise.. Fid he orsioal criical seeds ad he mode shaes of he roor sysem show i Figure E. by rasfer marix mehod. B ad B are fricioless bearigs ad D ad D are rigid discs. The shaf is made of seel wih modulus of rigidiy G =.8 () N/m ad uiform diameer d = mm. The various shaf leghs are as follows: B D = 5 mm, D D = 75 mm, ad D B = 5 mm. The olar mass mome of ieria of discs are: J d =.8 kg-m ad J d =. kg-m. Cosider shaf as massless. Figure E. B B D D Exercise.3. Obai he orsioal criical seed of a overhag roor sysem as show i Figure E.3. The ed B of he shaf is havig fixed ed codiios. The disc is hi ad has. kg-m of olar mass mome of ieria. Neglec he mass of he shaf. Use (i) he fiie eleme ad (ii) he rasfer marix mehod. B D Figure E.3 Exercise.4 Fid he orsioal aural frequecies ad he mode shaes of he roor sysem a show i Figure E.4 by ONLY rasfer marix mehod. B ad B are fixed suors ad D ad D are rigid discs. The shaf is made of seel wih modulus of rigidiy G =.8 () N/m ad uiform diameer d 86

17 Dr Tiwari, Associae Professor, De. of Mechaical Egg., T Guwahai, (riwari@iig.ere.i) = mm. The various shaf leghs are as follows: B D = 5 mm, D D = 75 mm, ad D B = 5 mm. The olar mass mome of ieria of discs are: J d =.8 kg-m ad J d =. kg-m. Cosider shaf as massless. D B D B Figure E.4 Exercise.5 Fid all he orsioal aural frequecies ad draw corresodig mode shaes of he roor sysem show i Figure E.5. B ad D rerese bearig ad disc resecively. B is fixed suor (wih zero agular dislaceme abou shaf axis) ad B ad B 3 are simly suored (wih o-zero agular dislaceme abou shaf axis). The shaf is made of seel wih modulus of rigidiy G =.8 () N/m ad uiform diameer d = mm. The various shaf leghs are as follows: B D = 5 mm, D B = 5 mm, B D = 5 mm, D B 3 = 5 mm, ad B 3 D 3 = 3 mm. The olar mass mome of ieria of he discs are: = kg-m, = kg-m, ad 3 =.8 kg-m. Use boh he rasfer marix mehod ad he fiie eleme mehod so as o verify your resuls. Give all he deailed ses i obaiig he fial sysem equaios ad alicaio of boudary codiios. Cosider he shaf as massless ad discs as lumed masses. Figure E.5 B B B 3 D D D 3 Exercise.6 Obai he orsioal criical seed of urbie-coulig-geeraor roor as show i Figure E.6 by he rasfer marix ad fiie eleme mehods. The roor is assumed o be suored o fricioless bearigs. The olar mass mome of ierias are T = 5 kg-m, C = 5 kg-m ad G = 5 kg-m. Take modulus of rigidiy G =.8 N/m. Assume he shaf diameer hroughou is. m ad leghs of shaf bewee bearig-urbie-coulig-geeraor-bearig are m each so ha he oal sa is 5 m. Cosider shaf as massless. 87

18 Dr Tiwari, Associae Professor, De. of Mechaical Egg., T Guwahai, (riwari@iig.ere.i) Bearig Turbie Coulig Geeraor Bearig Figure E.6 A urbie-geeraor se Exercise.7 a laboraory exerime oe small elecric moor drives aoher hrough a log coil srig ( urs, wire diameer d, coil diameer D). The wo moor roors have ierias ad ad are disace l aar, (a) Calculae he lowes orsioal aural frequecy of he se-u (b) Assumig he eds of he srig o be buil-i o he shafs, calculae roaioal seed (assume exciaio frequecy will be a he roaioal frequecy of he shaf) of he assembly a which he coil srig bows ou a is ceer, due o whirlig. 88