VIBRATION OF ELASTIC RINGS EXCITED BY PERIODICALLY-SPACED MOVING SPRINGS Etcheverry Hall, Berkeley, CA 94720

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1 ICSV Cairs Australia 9- July, 7 VIBRATION OF ELASTIC RINGS EXCITED BY PERIODICALLY-SPACED OVING SPRINGS Sripathi Vagipuram Cachi ad Robert G. Parker Departmet of echaical Egieerig, Uiversity of Califoria Berkeley 6 Etcheverry Hall, Berkeley, CA 97 Departmet of echaical Egieerig, Ohio State Uiversity W. 9 th Aveue, Columbus, OH parker.@osu.edu Abstract This work ivestigates parametric istabilities of i-plae bedig vibratios of a thi elastic rig subect to forces from discrete rotatig rigs of arbitrary umber, acig, ad orietatio. Several cofiguratios are examied, icludig systems with symmetric ad asymmetric circumferetial rig acig. The method of multiple scales is applied to aalytically idetify istability boudaries as closed-form expressios, ad two differet umerical approaches are used to verify these results. The effects of differet system parameters o the istability boudaries are studied aalytically: the bedig stiffess of the rig, the umber of rigs, ad their stiffess, locatio, orietatio ad rotatio eed. For several cases, well-defied properties for the occurrece or suppressio of istabilities are obtaied as simple relatios i the system parameters. INTRODUCTION The preset work addresses parametric excitatio of bedig vibratios i a statioary, thi rig subected to forces from rotatig rigs. The motivatio is from plaetary gears, which are commoly used i automotive trasmissios, helicopters, aircrafts ad wid turbies. Plaetary gear dyamics have historically bee aalyzed usig lumped-parameter models that take the rig, plaets, carrier ad the su as rigid bodies ad the gear tooth meshes as rigs. Recet studies, however, idicate that the deformable ature of the gear bodies, eecially a thi rig gear, must be icorporated to accurately model the mechaics. Bedig vibratios of the rig are parametrically excited by forces from the movig rigs (rig-plaet meshes), causig istability uder certai coditios. The bedig vibratios of rigs have bee extesively studied aalytically ad through experimets [-9]. However, little work o the vibratio of rigs subect to movig loads is

2 foud. Huag ad Soedel [, ] preseted closed-form solutios for the forced vibratio of rotatig rigs subected to harmoic ad periodic poit forces ad atially distributed forces. They compared those results with the iverted problem of a statioary rig with a movig poit force. etrikie ad Tochili [] studied the vibratios of a elastic rig with a time-varyig, movig poit force to model trai wheels. There seems to be o prior work o the vibratio of rigs subected to forces from movig rigs. A rig subected to movig rigs maifests as a parametrically excited system because the stiffess operator of the goverig equatio chages as the rig locatios chage. I this work, assumig that the movig rig stiffesses are small compared to the bedig stiffess of the rig, perturbatio methods are employed to aalytically idetify parametric istability boudaries as closed-form expressios. PROBLE FORULATION Figure a shows a statioary, thi rig of uiform cross-sectio with mea radius r subect to forces at its cetroidal surface from multiple rig-sets,!,,...,. Each rig-set cosists of two rigs of costat stiffess k ad k orieted i mutually perpedicular directios. The orietatio agle radial directio. The rig-sets are arbitrarily aced so that " ( $ " % # ) is the agle betwee the rig ad the k & ( $ & % # ) is the agular th coordiate of the rig-set measured from fixed E at iitial time t!. The above system describes the most geeral case of discrete rig forces o a statioary rig. The rig-sets rotate aroud the rig with a costat agular eed '. As they rotate, the orietatio agles " ad the relative agular acig (& & ( ( ) betwee ay two adacet rig-sets do ot chage. Referrig to Figure b, ) is the agular coordiate of ay poit o the rig i the iertial referece frame OEE, ad * is the agular coordiate of the same poit i the rotatig rigfixed referece frame Oe e. The agular coordiates of a material poit o the rig are related by )! * + '. t Oly i-plae bedig vibratios of the rig are cosidered i this work. Usig iextesibility of the rig cetroidal axis ad applyig Hamilto s priciple, the equatio for the tagetial dilacemet û is obtaied i odimesioal form as [] (a) E (b) e E w,z k u k e r φ β Ω β - k(-) ϕ θ Ω t φ - k (-) φ O E O E Figure (a) Rotatig rig o multiple rotatig rig-sets. (b) Defiitio of referece frames.

3 VI IV, u ( u - (, u + u + u- + # 6;.( c u + c u) () ( v (& )/ 7! ( ( + ) ( ( ( ) 9! ;. c u c u ) v & / )! : u EI ' k u!! t! v!! k EI b! r ˆ, <, <,,, r = Ar < # kb k! max( k, k ),!,,..., ; c! ( k cos " + k si " ), k c! c! ( k ( k )cos " si ", c! ( k si " + k cos " ) k k, () () Here, E is the Youg s modulus, A is the cross-sectioal area ad = is the desity of the rig. The importat odimesioal parameters are, which represets the ratio of the stiffess of the rig-sets to the bedig stiffess of the rig, ad the odimesioal rig rotatio k b eed v. The time-varyig rig forces parametrically excite the system as the agular locatios of the rig-sets chages periodically. Parametric istabilities occur for particular values of the magitude ( ) ad frequecy ( v ) of the time-varyig excitatio. Uder the assumptio that is a small quatity (stiffess of all the rigs are of the same order ad small compared to the bedig stiffess of the rig), perturbatio methods are used to obtai closedform approximatios for the regios of parametric istability i the v - plae. PARAETRIC INSTABILITY ANALYSIS To capture pricipal ad combiatio istabilities, a two-term Galerki discretizatio is applied to () usig the expasio (, ) ( ) i ) im) u )! > e + > m( ) e + cc, where cc represets the complex cougate of all precedig terms with m,? to elimiate rigid body motio. Substitutig ito (), ad formig the ier product of the resultig equatio with each of the basis fuctios yields the coupled equatios d > d d > d m ( i v i( m( ) v ( i( m+ ) v m m m + p > + q { G > + H e > + G e > + H e > }! ( i mv ( i( m( ) v ( i( m+ ) v m m m mm m mm m m m + p > + q { G > + H e > + G e > + H e > }! i( m( ) & i( m( ) & m! ; m + A m :! ; 7 ( + ) + ( ( ) :!! G i e c mc i mc c e ( i( m+ ) & ( i( m+ ) & m! ; 7 B m + C m :! ; 7 ( ( ) + (( ( ) :!! H i e c mc i mc c e p! ( ( ) /( + ), q! /( + ) m () ()

4 where G ad H deped o the rig-set parameters, ad p represets the odimesioal m m i atural frequecy for the bedig vibratios of a free rig i the odal diameter mode ( e D ) ). Although the oly excitatio frequecy i () is v, Galerki discretizatio to the modal coordiates i () results i the two excitatio frequecies ( m + ) v ad ( m ( ) v. The O( ) terms i () arise from the proectio oto the m odal diameter mode of the force the rotatig rigs exert whe the rig deflects i the odal diameter mode. Such a proectio results i forces varyig with the modulated frequecies ( m + ) v ad ( m ( ) v. Therefore, the system () may be viewed as havig the two differet excitatio frequecies ( m + ) v ad ( m ( ) v. Parametric istabilities arise whe the odimesioal rig rotatio eed v is close to particular combiatios of the free rig atural frequecies p m ad p. Applicatio of the method of multiple scales [] shows that terms leadig to resoat reose (secular terms) may arise whe ( m D ) v + p, which are coditios for summatio type combiatio istabilities of the first (plus sig: m + ) ad secod (mius sig: m ( ) kid, or whe ( m D ) v ( p, which are differece type combiatio istabilities of the first ad secod th kid. Pricipal istability correodig to the mode is obtaied with m!, givig v E p. (I arrivig at these expressios, m? ad v? are assumed without loss of geerality.) Cosiderig the parametric istability whe ( m + ) v + p, let ( m + ) v! p m + p + Fˆ where ˆ F is the detuig parameter. Elimiatio of terms leadig to ubouded, aperiodic reose i () yields the expressio for summatio combiatio istability boudaries of the first kid as v ( p + p ) + Fˆ q q q q Hˆ m m m, ˆ! F! ;@ + ;@ mm D ( m + ) p! pm! p pm ˆ ˆ ˆ ˆ ˆ ˆ ˆ ; ˆ ˆ ˆ ;!! H! I + R, I! ( C cosg ( B si G ) R! ( B cosg + C si G ) G! ( m + ) & () ad the th mode pricipal istability boudaries are give by v! ( p + Fˆ ) /. Cosiderig the parametric istability whe ( m ( ) v + p, let ( m ( ) v! p m + p + F!, ad the summatio combiatio istability boudaries of the secod kid are similarly obtaied as v ( p + p ) + F! q q q q H! m m m!,! F! ;@ + ;@ mm D ( m ( ) p! pm! p pm I R I ;!! R ;!!! m!! H!!! +!,!! (@ cosg + A si G )!! (@ cosg ( A si G ) G! ( ( )& (6)

5 It ca be show mathematically that differece type istabilities whe ( m D ) v ( p caot occur because such a coditio yields complex solutios for the detuig parameter. Numerical verificatio of the aalytical solutio is performed usig two approaches. I i the first method, Galerki discretizatio of () with basis fuctios as e D ) yields a time-varyig state matrix form x" ( )! P( ) x( ) with period T! # / v. Floquet s theorem is applied to obtai the regios of istability i the v - parameter plae. This method is computatioally itesive because of the umerical time itegratio, eecially for small v ad so log period T. Alteratively, the system may be aalyzed i a rig-fixed referece frame which allows a computatioally efficiet evaluatio of system stability. I this referece frame, the goverig equatio attais a time-ivariat state matrix form x" ( )! Qx( ) ad the system stability is dictated by the real part of the eigevalues of Q. By computig Q ad its eigevalues for rages of ad v (or other parameters), the regios of istability are obtaied []. RESULTS AND DISCUSSION The aalytical ad umerical results for the stability boudaries are plotted i the v - plae. Figure a shows these results for the case of oe radial rotatig rig. The aalytical stability boudaries are obtaied usig () ad (6) cosiderig the first three bedig modes, amely, modes with, ad odal diameters. The umerical istability regios are also computed takig the first three modes (startig with! ) to discretize the tagetial dilacemet. The agreemet betwee the aalytical ad umerical results is evidet, eve for relatively large values of the rig stiffess to bedig stiffess ratio. Both umerical methods yield the same istability regios.. Effect of umber of rig-sets, symmetry ad asymmetry The width of the parametric istability regios are govered by Ĥ ad H! defied i () ad (6), ad istabilities appear oly if Ĥ or H! are o-zero. Whe all the rig-sets are idetical with the same idividual rig stiffesses ad orietatio agles, ad whe the rig-sets are equally aced, the Ĥ is o-zero oly whe the odal diameters m ad are related to the total umber of rig-sets by m +! s, ad H! is o-zero oly whe m (! s where s!,,,... []. Hece, For the case of idetical ad equally aced rig-sets (referred to as the symmetric case), symmetry of the system suppresses may pricipal ad combiatio istabilities. A example with three radial rigs is preseted cosiderig the, ad odal diameter modes. The oly istabilities that appear i the symmetric case ( Figure b) are pricipal istability due to the odal diameter mode, ad combiatio istability of the first kid from iteractio of the ad odal diameter modes. I the asymmetric cases of idetical, uequally aced rig-sets (Figure c), or o-idetical, equally aced rig-sets (Figure d), all possible istabilities occur.. Parametric study The aalytical solutio shows how parametric istability regios chage due to a variatio i the system parameters. As examples, the effects of orietatio agle ( " ), stiffess agle (I ) ad

6 modulatio agle ( J ) are cosidered. For the case of oe rotatig rig, the orietatio agle is varied from "! o (radial) to "! 9 o (tagetial), ad the result is show i Figure a. Larger istability regios appear whe the same rig is orieted i the radial directio versus the tagetial directio, ad this result is easily verified aalytically []. (a) (b) (p +p )/(-) (p )/(+) v p /(+) 9 (p )/(-) v... ε p /(+) (p )/(+) (p )/(+) p /(+) (p +p )/(+) p /(+) ε (c) (d) (p +p )/(-) (p +p )/(-) 9 (p )/(-) 9 (p )/(-) v v p /(+) p /(+) (p )/(+) (p )/(+) (p )/(+) (p )/(+) p /(+) p /(+) (p +p )/(+) (p +p )/(+) p /(+) p /(+)... ε... ε Figure Parametric istability regios (a) Oe radial rotatig rig with k!, k! k, "! o (b) Symmetric case of three idetical ad equally aced rotatig radial rigs with k!, k! k, "! o. Asymmetric cases: (c) Three idetical but uequally aced rotatig radial rigs with k!, k! k, "! o, &! o, &! o, &! o. (d) Three o-idetical but equally aced rotatig radial rigs with k k! k! k, "! o., pricipal ad combiatio istabilities of first kid; --,! combiatio istability of secod kid; ***, umerical solutio. 6

7 (a) (b) (c) p /(+) (p )/(+) v p /(+) ustable regios p /(+) ϒ (degrees) Figure Parametric istability regios (a) Effect of rig orietatio agle: oe rotatig rig with k!, k! k. (b) Effect of stiffess agle: oe rotatig rig-set with k! k, "! o. (c) Effect of modulatio agle: two pairs of diametrically opposed radial rigs with! k / # k b!, k!,! k, "! o, &! o, &! 9 o + J o, &! o, &! 7 o + J o. k ( The stiffess agle is defied as I ta, k / k - k! k cosi, k! k sii where 7 o o! ( $ $ 9 ), so that k! k + k. Cosiderig a sigle rotatig rig-set defied by k, I, Figure b shows the istability zoes for differet values of I with "! o. Iterestigly, for "! o, the differet combiatio istabilities of the first kid, icludig pricipal istabilities, vaish for particular values of I, as idicated i Figure b by closig of the istability regios i the v - plae. Aalytical ivestigatio shows that combiatio istabilities of the first kid vaish for the ad m odal diameter modes if tai! / m []. Cosequetly, pricipal istabilities correodig to ad odal diameter modes vaish whe I!. o ad I! 6. o, reectively, ad the combiatio istability due to their iteractio vaishes whe I! 9.6 o ( Figure b). These results hold for arbitrary acig of rig-sets with differet k, so log as "! o, ad the stiffess agles for all the rig-sets are the same. Combiatio istabilities of the secod kid do ot exhibit similar behavior because there is o o o value of I ( $ I $ 9 ) for which H!!. Diametrically opposed rig-set pair cofiguratio is of practical importace i plaetary gear systems where equal plaet acig is ot possible due to assembly requiremets. The effect of agular acig betwee the diameters o the parametric istabilities is show i Figure c for two pairs of diametrically opposed radial rigs (! ) located at &!, &! 9 o + J o, &! o ad &! 7 o + J o, where J is the modulatio agle. The width of the istability regios vary with the modulatio agle ad are plotted for the rage J! o to J! 9 o. It may be easily show that for idetical ad diametrically opposed rig-sets, parametric istabilities caot occur if m D is odd []. This is cofirmed from Figure c. If m D is eve, however, Ĥ or H! may become zero depedig o the values of m,, & ad. For example, i Figure c, the pricipal istability from! vaishes whe J! o. I o

8 CONCLUSIONS I-plae bedig vibratios of a statioary rig are parametrically excited whe subect to multiple, rotatig rig-sets of arbitrary stiffess ad orietatio. Istability boudaries are obtaied aalytically as closed-form expressios usig a first order perturbatio method, ad these aalytical results compare well with umerical results. Although there is essetially oe idepedet excitatio frequecy (rig-set rotatio eed v ), it is coupled to the odal diameters m, by proectios of the rig force oto the vibratio modes i Galerki discretizatio. As a result, the modal coordiate equatios have the parametric excitatio frequecies ( m + ) v ad ( m ( ) v. Summatio combiatio istabilities of two kids occur correodig to two differet values of v : oe at lower frequecy ( p + p ) /( m + ) ad aother at higher frequecy ( pm + p ) /( m ( ). Differece type istabilities do ot exist for this problem. The stiffess, orietatio, ad relative acig betwee rig-sets gover the occurrece ad width of the istability regios. Equally aced, idetical rig-sets ad diametrically opposed, idetical rig-sets are show to suppress several of the istabilities. Simple rules relatig the odal diameters of the suppressed istabilities ad the umber of rig-sets are obtaied ad demostrate the advatages symmetry ca play i physical systems. The effect of fixed rig-sets together with movig rig-sets, ad the effects of a rotatig rig ad time-varyig stiffess rig-sets have also bee ivestigated i related studies [, ]. REFERENCES. Seidel, B.S. ad E.A. Erdelyi, O the vibratio of a thick rig i its ow plae. Joural of Egieerig for idustry, 96. 6: p. -.. Evese, D.A., Noliear flexural vibratios of thi circular rigs. Joural of Applied echaics, 966. : p Rao, S.S. ad V. Sudararaa, I-plae flexural vibratios of circular rigs. Joural of Applied echaics, (): p Kirkhope, J., Simple frequecy expressio for the i-plae vibratio of thick circular rigs. Joural of the Acoustical Society of America, (): p Kirkhope, J., I-plae vibratio of a thick circular rig. Joural of Soud ad Vibratio, 977. (): p Kuhl, W., easuremets to the theories of resoat vibratios of circular rigs of arbitrary wall thickess. Akustiche Zeitschrift, 9. 7: p Licol, J.W. ad E. Volterra, Experimetal ad theoretical determiatio of frequecies of elastic toroids. Egieerig echaics, 967. : p Rao, S.S., Three-dimesioal vibratios of a rig o a elastic foudatio. The Aeroautucal Joural of the Royal Aeroautical Society, 97. 7: p Wu, X. ad R.G. Parker, Vibratios of rigs o multiple discrete rig supports. (i preparatio).. Huag, S.C. ad W. Soedel, Effect of coriolis acceleratio o the free ad forced i-plae vibratios of rotatig rigs o elastic foudatio. Joural of Soud ad Vibratio, 97. (): p Huag, S.C. ad W. Soedel, Reose of rotatig rigs to harmoic ad periodic loadig ad compariso with the iverted problem. Joural of Soud ad Vibratio, 97. (): p etriki, A.V. ad.v. Tochili, Steady-state vibratios of a elastic rig uder movig load. Joural of Soud ad Vibratio,. (): p. -.. Vagipuram Cachi, S. ad R.G. Parker, Parametric istability of a circular rig subected to movig rigs. Joural of Soud ad Vibratio, 6. 9: p Vagipuram Cachi, S. ad R.G. Parker, Parametric istability of a rotatig rig with movig, time-varyig rigs. Joural of Vibratio ad Acoustics, 6. : p. -. m