MODIFIED GENERAL TRANSFER MATRIX METHOD FOR LINEAR ROTOR BEARING SYSTEMS , Dhenkanal, Orissa, India

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IV arn Autrala 9- July, 007 Abtract MODIFIED GENERAL TRANFER MATRIX METHOD FOR LINEAR ROTOR BEARING YTEM B B Maharath, R R Dah and T Rout Department of Mechancal Engneerng, IGIT, arang 769 6,

1 IV arn Autrala 9- July, 007 Abtract MODIFIED GENERAL TRANFER MATRIX METHOD FOR LINEAR ROTOR BEARING YTEM B B Maharath, R R Dah and T Rout Department of Mechancal Engneerng, IGIT, arang 769 6, Dhenkanal, Ora, Inda bbmaharath@yahoo.com A modfed general tranfer matrx method developed for the teady tate repone analy of lnear flexble rotor-bearng ytem n the frequency doman wth fxed matrx ze. In th paper, the modfcaton of the tranfer matrx method baed on Tmohenko Beam Theory are derved from the concept of contnuou ytem ntead of the conventonal lumped ytem concept and the paper tre to extend the tranfer matrx method to ft a ynchronou ellptcal orbt and a non-ynchronou mult-lobed whrlng orbt. To demontrate the applcablty of th method, a three-dk rotor-bearng ytem ued a a phycal model n the numercal analy.. INTRODUTION Dynamc charactertc of rotor-bearng ytem are obtaned by varou method uch a; tranfer matrx method (Lumped ytem and contnuou ytem), fnte element technque and dynamc tffne method conderng dfferent nfluencng parameter related to rotor, dk and bearng [-. In th work, an attempt ha been made to formulate the general tranfer matrx method baed on contnuou ytem model and upermpoed of vbraton of the haft n both the plan for dynamc repone and crtcal peed of rotor ytem.. TRANFER MATRIX OF HAFT The elatc relaton of the haft element baed on Tmohenko beam theory a hown n Fg. & are gven a follow : In the X-Z plane; X ρ ρ X ρ X + + Z G Z GE Tρ Y T Y + 0 G Z t Z ρa X ρω Y Y ρ + E Z t G t (a) P X Z

2 Fgure. A rotatng haft element. Fgure. Geometre of haft and dk unbalance. In the X-Y plane; Y ρ ρ Y ρ Y + + Z G ρa Y ρω X X ρ + + Z GE E Z t G t Tρ X X T 0 G Z Z + (b) nce the whrlng orbt ellptcal and ynchronou ( Ω w) a hown n Fg., the teady tate oluton can be wrtten a; X( z, t) X ( t) coωt + X ( t) nωt, Y( z, t) Y ( t) coωt + Y ( t) nωt () X, X, Y, Y are mode functon. Introducng general oluton can be wrtten a; z z X() z U e λ λ z λz + ju e and Y() z V e λ + jv e. X + P Y Z X jx and Y Y + jy, the U, U, V, V are real contant and λ the charactertc value wth repect to a pecfc natural mode. eparatng real and magnary part, t yeld a; ( λ + flλ + g)u + [ (hλ + k) + j(cλ + dλ) V 0 j( λ + flλ + g)u + [j(hλ + k) + (cλ + dλ) V 0 [( hλ + k) j(cλ + dλ) U + ( λ + flλ + g) V 0 j(hλ + k) (cλ + dλ) U + j( λ + f λ + g) V 0 () [ l nce U, U, V, V beng non-trval, the charactertc equaton can be obtaned by ettng the determnant of Eq. () to zero. It yeld a; {( λ + f λ + g) + [j(cλ + dλ) (hλ + k)}.{( λ + f λ + g) [j(cλ + dλ) (hλ + k)} 0 () l l P f ρω ρω +, E G ρω ρ ω h, k, E GE ρ ω ρaω g GE T Tρω c, d G By eparatng Eq. () nto two part a follow : {( + flλ + g) + [ j ( cλ + dλ) ( hλ + k) } 0 {( + flλ + g) [ j ( cλ + dλ) ( hλ + k) } 0 λ (a) λ (b)

3 The four complex root of Eq. (a) are λ a + jb for ~ and other four complex root of Eq. (b) are λ a + jb for ~. For λ a + jb, ~, U V, U -V For λ a + jb, ~, U -V, U V Thu, the four homogeneou oluton are a follow : X X Y Y a Z a Z ( Z) Ae cobz + Ae cobz a Z a Z B e nb Z + B e nb Z a Z a Z ( Z) Ae nbz + Ae nbz a Z a Z + B e cob Z B e cob Z az az ( Z) Ae n bz + Ae n bz az az B e cob Z B e cob Z a Z a Z ( Z) Ae cobz Ae cobz a Z a Z B e nb z B e nb Z (6) A and B are real contant ( ~ ). Dfferentatng Eq. (6) wth repect to z and ubttutng Z0, t become A [ W( Z 0) [ M 7 7 B 7 (7) W Z 0 X 0,X 0,Y 0,Y 0, [ ( ) [ ( ) ( ) ( ) ( ) t X, " " t [X,X,X,X X " " t [X,X,X,X Y, " " t [Y,Y,Y,Y Y " " t [Y,Y,Y,Y A [A, A, A, A, A, A 6, A 7, A t, B [B, B, B, B, B, B 6, B 7, B t t tranpoe of the array. A B [ M [ W( z 0) () 7 7 At Z L, A [ W( Z L) [ H 7 7 B (9) [W(ZL[X (L), X (L), Y (L), Y (L) t Now, [ W( Z L) [ H 7 7[ M 7 7[ W( Z 0) [ N 7 7 [ W( Z 0) (0)

4 and, The tate varable can be wrtten a; Q X X + jx, X X P M α, X" X ρω X GE GE G ρω ρω X" ρ ω α ρ ω β f I j f I ft f + QX MX G G A Y Y + jy, Y + (a) Qy P My ρω β, Y" Y GE GE G ρω β Y" fρiω jfρiω α G ( GA P) f GA Further, the tate vector can be wrtten a; X X + jx, Y Y + jy, α αc + jα, β βc + jβ M X MX + jmx, MY MY + jmy Q X MX + jqx, QY QY + jqy ρω ft + f + Q y + My (b) G A () ombnng Eq. () and () and eparatng real and magnary term, t yeld a; [ W ( Z L) 7 [ A 7 7[ l 7 () " " " " " " " " W Z L X,X,X,X,X,X,X,X Y,Y,Y,Y,Y,Y,Y,Y, } t [ ( ) {, [ { X,X,Y,Y, α, α, β, β, M At Z 0, [ W ( Z 0) [ A{ 0 } 7 X,MX,MY,MY,QX,QX,QY,QY, } t () ombnng Eq. () and (), t yeld { } 7 [A [N[A{ 0} 7 [T 7 7[0 () 7 T the tranfer matrx of the rotor egment (Z L). [ 7 7. TRANFER MATRIX OF THE DI The equlbrum condton, the relaton of the tate varable between the rght and left de of an unbalanced dk expreed a [-; R 7 L d [ T. TRANFER MATRIX OF THE BEARING From the force equlbrum, the relatonhp of the tate varable between the left and rght de can be wrtten a [-; R L [ Tb 7 7 (7) 7 7 (6)

5 . OVERALL TRANFER MATRIX The overall tranfer matrx of the rotor from one end to another can be wrtten a; { n} 7 [ U 7 7{ 0} 7 () U [T [T [T [T...[T [T [T [T { } (9) [ n bj n dj db b 0 6. OLUTION ANALYI nce the hear force and bendng moment are zero at both the free end of the haft, Eq. () become; [ [ [ n U U U 0 [ [ [ 0 U U U 0 t t {} {} (0) { X,X,Y,Y, α, α, β β } t, O { O,O,O,O,O,O,O,O } t, By deletng all element, whch are related to moment and hear force, t gve; { n } [U { 0} + [Ul (a) and [ U.{ 0} + [U {O} (b) ubcrpt O and n are labelled for tage. Eght multaneou equaton n Eq. (b) are olved to obtan the eght tate vector { 0 }. { } { } t 0 X,X,Y,Y, α, α, β, β O O O O O O O Then, the tate vector at any dered tage (p th tage) can be obtaned by ung Eq. () through matrx operaton for the determnaton of dynamc repone. { p } 7 [Up 7 7{ o} 7 () t t t } {{ } {O} { o 7 O } O 7. FINITE ELEMENT ANALYI A typcal flexble rotor-bearng ytem cont of a rotor compoed of dcrete dk and rotor egment, and dcrete elatc bearng a hown n Fg.. Each rotor element modelled a an eght degree of freedom element wth two rotaton and two tranlaton at each end n each plane. The co-ordnate ( q, l ~ ) e are the tme-dependent and pont dplacement of the fnte rotor element. t e e e e e e t q q,q,q,q,q,q,q, q { v,w, θ, φ,v,w, θ φ } () { } { } t e e 6 e 7, The Lagrangan equaton of moton for the fnte rotor element at the contant peed can be wrtten a; e e e e e e e & & () [M {q } ω[g {q } + [ {q } [F

6 The Lagrangan equaton of moton of the unbalanced rgd dk wth gyrocopc effect at the contant angular peed can be wrtten a; d d d d d d {[M + [M }{q & } ω[g {q& } {F } () d {F } t r d d u z ω u y ω d d d [F nωt + [Fc coωt u d ω y nωt + u ω coωt z d d u z m e nβ, u y m e coβ The governng equaton of the bearng can be wrtten a; b b b b b b [O{q & } + [ {q& } + [ {q } {F } b b xx xy v& xx xy v or + { F } b yx yy b w& yx yy b w (6) The aembled undamped ytem equaton of the form : & (7) [M {q } + { ω[g + [ }{q & } + [ {q } [F The teady tate oluton { q } { q } coωt + { q } nωt c () Dfferentatng any eparatng cone and ne term, t yeld ([ ω [M ) ω ([ [G ) {q c } {F c} ω ([ [G ) ([ ω [M ) {q } {F } (9) Then {q ω ω c } ([ [M ) ([ [G ) {F c} (0) {q } ω ([ [G ) ([ ω [M ) {F } The oluton of Eq.(0) provde { q } c and { } q and back ubttuton n Eq.() determne the unbalance repone of the rotor ytem at the requred poton.. NUMERIAL ANALYI In order to llutrate the accuracy of the theoretcal analy, a three dk rotor ytem mounted on flud-flm bearng condered a phycal model a hown n Fg. [dk ma (M d d ).7 kg, polar ma moment of nerta ( I p ).0 x 0 - kg m, tranvere ma d moment of nerta ( I T ). x 0 - kg m, drect tffne coeffcent ( xx yy ) x0 7 N/m, cro-coupled tffne coeffcent ( yx xy ) x 0 6 N/m, drect dampng and cro dampng coeffcent are xx yy x 0 N/m/ec, xy yx 0, repectvely. Two cae are condered n the numercal analy. ae- : Bearng wthout dampng ( xx, xy, yy, yx ) ae- : Bearng wth tffne and dampng ( xx, xy, yy, yx, xx, yy )

7 9. DIUION & ONLUION The ampltude of repone are calculated by the abolute value of the major ax of the ellptcal orbt. The calculated dynamc repone at dk- are plotted a old curve n Fg.6 & 7 for the frt natural mode regon baed on modfed tranfer matrx method. The dahed curve are calculated from the fnte element method by x element. It hown that two curve for both the cae are cloe to each other. The cro-tffne make the crtcal peed plt nto two value for t aymmetry, a evaluated n cae- at. Hz and. Hz. The crtcal peed can be located from the maxmum repone peak of the frequency repone curve. For aymmetrcal bearng due to cro-tffne and dampng, the whlng orbt become ellptcal rather than crcular. ynchronou repone due to unbalance ma and non-ynchronou repone due to the journal moton wth frequency beng three tme of the rotatng peed condered ( Ω ω) and the numercal reult are produced n Fg. & 9. The modfed general tranfer matrx method a veratle technque for the determnaton of dynamc charactertc of any rotor-bearng ytem conderng varou nfluencng parameter related to rotor, dk and bearng n the frequency doman. Fgure. Repone of a ynchronou whrlng orbt. Fgure. A fnte rotor element. Fgure. A three-dk rotor bearng ytem (h0.0 m, d 0.0 m). Fgure 6. Dynamc repone at dk- (cae-).

8 Fgure 7. Dynamc repone of dk-, cae-. Fgure. ynchronou whrlng orbt of dk-, cae-. Fgure 9. Non-ynchronou whrlng orbt of dk-, cae-. 0. REFERENE [ B. B. Maharath, Dynamc behavour analy of lnear rotor-bearng ytem ung the complex tranfer matrx technque, Int. Journal of Acoutc and Vbraton, 0, -0 (00). [ B. B. Maharath, et al, Dynamc behavour analy of a dual-rotor ytem ung the tranfer matrx method, Int. Journal of Acoutc and Vbraton, 0, - (00). [ B. B. Maharath, and A.. Behera, A tranfer matrx method for dynamc behavour of rotor-bearng ytem, Tran. of anadan ocety of Mechancal Engneerng, 99-ME-, June (00). [ B. B. Maharath, and A.. Behera, Tranfer matrx analy of hgh peed rotor by dtrbuted element, Tran. of Mechancal Engneerng, Autrala, Me-, -0 (00). [ T. omeya, Data book of Journal-Bearng, prnger-verlag, Tokyo, 9.

8 Waves in Uniform Magnetized Media

8 Waves in Uniform Magnetized Media 8 Wave n Unform Magnetzed Meda 81 Suceptblte The frt order current can be wrtten j = j = q d 3 p v f 1 ( r, p, t) = ɛ 0 χ E For Maxwellan dtrbuton Y n (λ) = f 0 (v, v ) = 1 πvth exp (v V ) v th 1 πv th