Rev. Fac. Ing. - Self Unv. balancng Tarapacá, system vol. for Nº rotatng, 005, mechansms pp. 59-64 SEF BAANCING SSTEM FOR ROTATING MECHANISMS Marco Antono Meraz Andrés ánez Carlos Jménez Raúl Pchardo Recbdo el de juno de 004, aceptado el 0 de abrl de 005 ABSTRACT A self balancng system analyss s presented whch utlzes freely movng balancng bodes (balls) rotatng n unson wth a rotor to be balanced. Usng agrange s Equaton, we derve the non-lnear equatons of moton for an autonomous system wth respect to the polar coordnate system. From the equatons of moton for the autonomous system, the equlbrum postons and the lnear varatonal equatons are obtaned by the perturbaton method. Because of resstance to moton, eccentrcty of race over whch the balancng bodes are movng and the nfluence of external vbratons, t s mpossble to attan a complete balance. Based on the varatonal equatons, the dynamc stablty of the system n the neghborhood of the equlbrum postons s nvestgated. The results of the stablty analyss provde the desgn requrements for the self balancng system. Keywords: Self balancng system, varatonal, Raylegh dspaton functon. RESUMEN Se presenta el análss de un sstema de autobalance el cual utlza bolas lbres de movmento rotando con el rotor que será balanceado. Se usa la ecuacón de agrange para dervar un sstema de ecuacones no lneales para un sstema autónomo con respecto a un sstema de coordenadas polares. De las ecuacones de movmento, se obtenen ecuacones lnealzadas varaconalmente y poscones de equlbro por el método de perturbacón. A causa de la resstenca al movmento, la excentrcdad y el movmento de los cuerpos lbres que son provocados por la nfluenca de vbracones externas, hace mposble obtener un balanceo completo. Basado en el método varaconal, se nvestga el comportamento dnámco del sstema en la frontera de la poscón de equlbro. os resultados del análss de establdad proveen los requermentos de dseño para el sstema de autobalance. Palabras clave: Sstema de autobalance, método varaconal, funcón de dspacón de Raylegh. INTRODUCTION The rotaton of unbalanced rotor produces vbraton and ntroduces addtonal dynamc loads. Partcular angular speeds encountered n presently bult modern rotatng machnery, mpose rgorous requrements concernng the unbalance of rotatng mechansms. In the system, however, where the dstrbuton of masses around the geometrc axs of rotaton vares durng the operaton of a machne or each tme the machne s beng restarted, the conventonal balancng method becomes mpractcable. Therefore, self balancng methods are practced n such systems where the role of fxed balancng bodes s performed, ether by a body of lqud or by a specal arrangement of movable balancng bodes (balls or rollers) whch are sutably guded for free movement n predetermned drectons. In the case when a body of lqud s self balancng the attanable degree of balance does not exceed 50% of ntal unbalance of rotatng parts []. In fact, however, there are a lot of reasons renderng the attanment of such a hgh degree of balance practcally mpossble. Self balancng systems are used to reduce the mbalance n washng machnes, machnng tools and optcal dsk drves such as CD-ROM and DVD drves. In self balancng systems, the basc research was ntated by Thearle [], [], Alexander [4] and Cade [5]. Analyss for varous self balancng systems can be encountered n references [6-9]. Equatons obtaned are for nonautonomous systems, these equatons have lmtatons on complete stablty analyss. Chung and Ro [9] studed the stablty and dynamc behavor of an ABB for the Jeffcott rotor. They derved the equatons of moton for an autonomous system by usng the polar coordnates nstead of the rectangular Department of Mechancal Engneerng, Insttuto Tecnológco de Puebla, Puebla, Méxco, ameraz69@aol.com Rev. Fac. Ing. - Unv. Tarapacá, vol. Nº, 005 59 06-Meraz 59 05/0/005, :49
Antono Meraz, Andrés ánez, Carlos Jménez, Raúl Pchardo coordnates. Hwang and Chung [0] appled ths approach to the analyss of an ABB wth double races. Chung and Hang [] studed ABB for a rotor wth a flexble shaft. In that case they adopted Stodola-Green rotor nstead of the Jeffcott model. In ths study, authors got a smlar analyss for a flexble shaft and two self balancng systems on the ends. Descrbng the rotor centre wth polar coordnates, the non-lnear equatons of moton for an autonomous system are derved from agrange s equaton. After a balanced equlbrum poston and lnearzed equatons n the neghborhood of the equlbrum poston are obtaned by the perturbaton method and theoretcally t shows that after crtcal speed rotor can be balanced. The system has a small lubrcaton on ts balls and they are collocated themselves by nertal moton upper frst natural frequency. a) b) EQUATIONS OF MOTION Fg. Schematc representaton of self balancng system, a) Front vew, b) Euler angles. Fg. Self balancng system on the ends of rotor. The rotor wth double self balancng system s shown n Fg., where the shaft s supportng two self balancng systems on the ends. It s assumed that the shaft mass s neglgble compared to the rotor mass. The Z coordnate system s a space-fxed nerta reference frame end the ponts C and G of both rotors are centrod and mass centre respectvely. Descrbng the rgd body rotatons of the rotor wth respect to the and -axs, Euler angles are used, whch gve the orentaton to the rotor-fxed xyz-coordnate system relatve to the space-fxed Z-coordnate system. In ths case, the Euler angles of ωt, α and β are used as shown n Fg.. A rotaton through an angle ωt about the Z-axs results n the prmed system. Smlarly rotaton α about x -axs and a rotaton β about y -axs results double prmed and xyz-coordnate systems respectvely. In matrx form: and rotaton matrces: x x x x x () β α ω Pont O may be regarded as projecton of the centrod C onto the axs O Z. The ball balancer conssts of a crcular rotor wth a groove contanng balls and a dampng flud. The balls move freely n the groove and the rotor spns wth angular velocty ω. It s assumed that deflecton of the shaft s small so may be assumed that he center C moves n the -plane. As shown n Fg., the centrod C may be defned by the polar coordnates r and θ. The mass centre can be defned by eccentrcty ε and angle ωt, for the gven poston of the centrod and the angular poston of the ball B s gven by the ptch radus R and the angle φ. β α ω cos β 0 sn β 0 0 sn β 0 cos β 0 0 0 cosα snα 0 snα cosα cosωt snωt 0 snωt cosωt 0 0 0 () 60 Rev. Fac. Ing. - Unv. Tarapacá, vol. Nº, 005 06-Meraz 60 05/0/005, :49
Self balancng system for rotatng mechansms Iˆ + Jˆ + ZKˆ x x + y j + z k x x + y j + z k x x + yj + zk () We are supposng that two balls at begnnng of ths study, the knetc energy T s gven by: T M dr dt G m dr B drb + + ω T Jω (0) dt dt n whch all components are unt vectors along assocated drectons respectvely. Frst step s consderng the knetc energy of the rotor wth the self balancng system. The poston vector of the mass centre G can be expressed usng the rotaton matrces: where β α ω cos β 0 sn β 0 0 sn β 0 cos β 0 0 0 cosα snα 0 snα cosα cosωt snωt 0 snωt cosωt 0 0 0 (4) rg βαω roc / Z + rcg (5) roc / Z r(cosθ Iˆ + snθ J ˆ) rcg ε (6) Usng a common generalzed coordnate ψ defned by ψ ωt θ (7) After matrx product the poston vector of the mass centre, r G : rg [ r cos β cosψ r snα sn β sn ψ ] r cosα snψ j + [ r sn β cosψ + r snα cos β sn ψ ] k And the poston vector of the Ball: r [ r cos β cosψ r snα sn β snψ + R cos φ ] B [ r cosα snψ + R sn φ ] j + [ r sn β cosψ + r snα cos β sn ψ ] k (8) (9) where J s the nerta Matrx and ω s the angular velocty vector of the rotor; m s the mass of ball and M s the mass of rotor: Jx 0 0 J 0 J y 0 0 0 Jz ω ( ω cosα sn β + α cos β) + ( ω sn α + β) j + ( ω cosα cosβ + α sn β)k () () n whch J s the mass moment of nerta about x,y,zaxs. Neglectng gravty and the torsonal and longtudnal deflectons of the shaft, the potental energy, or the stran energy, results form the bendng deflectons of the shaft. As shown n Fg., the shaft can be regarded as a beam wth loads on ends, whch s fxed at Z/4 from ends. The shaft deflectons n the and drectons: D D r cosθ r snθ () For the gven rotaton angles α and β, the rotaton angles about the - and -axs: Φ Φ α cosωt β cosα snωt α snωt + β cosα cosωt (4) Snce the deflecton and slope at Z/4, n the Z-plane are D and Φ whle those n the Z-plane are D and Φ, the deflecton curves of the shaft n the Z- and Z- planes: D Φ D Φ δ Z Z D + Φ D Φ δ Z Z (5) Rev. Fac. Ing. - Unv. Tarapacá, vol. Nº, 005 6 06-Meraz 6 05/0/005, :50
Antono Meraz, Andrés ánez, Carlos Jménez, Raúl Pchardo The stran energy V due to the shaft bendng: V EI 0 δ Z + δ Z dz (6) where E s oung s modulus and I s the area moment of nerta of the shaft cross-secton. By the way, Raylegh s dsspaton functon F for two dscs can be represented by: ( ) + ( + ) + t r F c r + r θ c α β D φ (7) where c t and c r s the equvalent dampng coeffcent for translaton and rotaton respectvely and D s the vscous drag coeffcent of the balls n the dampng flud. The equatons of moton are derved from agrange s equaton: d dt T T q q V + q k k k k F + 0 (8) q In ths formulaton q k are the generalzed coordnates. For the gven system, the generalzed coordnates are r, ψ, α,β and φ, ; therefore, the dynamc behavor of the self balancng system s governed by +4 ndependent equatons of moton. Under the assumpton that r, ψ, α,β are small and ts products too, the equatons of moton are smplfed and lnearzed n the neghborhood by perturbaton method : r r + r ψ ψ + ψ α α + α β β + β φ φ + φ (9) In ths case each above equaton has two components; the coordnates for equlbrum postons and ther small perturbatons. It s consdered r 0 n equlbrum poston. And the lnearzed equatons of moton: EI ( M + m) r + ctr + ( M + m) ω r ( ) α snψ β cosψ mr φ sn φ + ψ + ωφ cos φ + ψ ω φ sn φ ψ 0 ( ) ( + ) (0) ( ) + M + m ωr ctω r α cosψ β snψ mr φ cos φ + ψ ωφ sn φ + ψ ω φ cos φ + ψ ( ) ( ) ( ) 0 () J + mr sn φ α mr β cos φ sn φ + crα + Jz J ωβ r ψ mr ( ) sn ω β cos φ φ sn + 4 EI + ( JZ J ) ω + mr ω sn φ α 0 mr α cos φ sn φ + J + mr cos φ β ( JZ J ) ωα + crβ ψ r cos mr ω α J cos φ sn φ + 4EI + ( J ω ω cos Z J ) + mr φ β 0 ( ) ( + ) () () mr φ + Dφ mr r sn φ + ψ ω r cos φ ψ + mr r ( + ) ω sn φ ψ 0,, It s assumed n the above 4 equatons: r α β 0 Mε cos φ, snφ 0 mr SIMUATION (4) (5) The mass moments of nerta, J J and J Z are gven by: J J MR JZ MR 4 (6) 6 Rev. Fac. Ing. - Unv. Tarapacá, vol. Nº, 005 06-Meraz 6 05/0/005, :5
Self balancng system for rotatng mechansms The balanced equlbrum poston can be represented: r α β 0 φ φ tan mr Mε (7) Small perturbatons of the generalzed coordnates from the balanced poston can be wrtten as: t r t α r e α e t β β e t t φ φ φ e φ e (8) Fg. Possble equlbrum poston for varatons of rotatng speed. and λ s an egenvalue. Substtutng equatons (7) and (8) nto equatons (0)-() and usng the Ptagoras dentty equaton, the condton that equatons (8) have non-trval solutons can be expressed as the characterstc equaton gven as k c k λ 0 k 0 (9) where the coeffcents c k (k0,,.) are functons of ω, M, m, R,, ε, E, I, D, c t and c r. The explct expressons of c k are omtted of ths paper. The Routh-Hurwtz crtera provde a suffcent condton for the real parts of all roots to be negatve. The followng geometry parameters are consdered: ω ct ζt 4 MEI ζ r o cr 4 48EI M JEI (0) And ω o s the reference frequency; ζ t and ζ r are dmensonless dampng factors for translaton and rotaton. In ths paper the stablty of the balancer are studed for the varatons of the non-dmensonal system parameters such as ω/ω o versus ε/r. There are some parameters to be consdered: /R, and m/m ε/r D/mR ω o ζ t ζ r 0.0. Next fgure shows two dfferent areas of equlbrum and non-equlbrum balls poston takng ε/r and ω/ω o as reference. ou can see that equlbrum area s very lttle. Fg. 4 Schematc representaton of two postons of rotor. a) Equlbrum poston, b) Nonequlbrum poston. CONCUSIONS The balancng bodes of self balancers do not assume postons whch ensure complete balancng a rotor. Effectve postons of a balancng body dffer by φ ι from equlbrum poston. Other reasons may also appear such as the rubbng of balancng bodes aganst the sdes of drums wthn they are dsposed, rregulartes of shape or axally asymmetrcal weght dstrbuton of rollng balancng bodes. The postons Rev. Fac. Ing. - Unv. Tarapacá, vol. Nº, 005 6 06-Meraz 6 05/0/005, :5
Antono Meraz, Andrés ánez, Carlos Jménez, Raúl Pchardo errors are relatve large ones and the larger they are the hgher s the coeffcent of resstance to rollng moton and the hgher s the rato ω/ω o (when s greater than ). In order to reduce these errors t would be necessary to change the method of gudng the balancng bodes, for example ar cushon, bodes suspended by magnetc or electrostatc forces. To obtan the balancng, ω greater than the frst natural frequency. The flud dampng D and the dsspaton for translaton c t are essental to obtan balancng, but dsspaton for rotaton c r s not. The stablty of the system have been analyzed wth the lnear varatonal equatons and the Routh-Hurwtz crtera. REFERENCES [] J.N. MacDuff and J.R. Currer. Vbraton Control, McGraw-Hll, New ork, 958. [] E.. Thearle, Automatc dynamc balancers (Part. leblanc), Machne Desgn : 9-4, 950. [] E.. Thearle, Automatc dynamc balancers (Part. rng, pendulum, ball balancers), Machne Desgn : 0-06, 950. [4] J.D. Alexander. An automatc dynamc balancer. Proceedngs of nd Southeastern Conference vol. : 45-46, 964. [5] J.W. Cade. Self-compensatng balancng n rotatng mechansms, Desgn News: 4-9, 965. [6] Majewsk Tadeusz. Synchronous elmnaton of Vbraton n the Plane, Part : Method Effcency and ts stablty. Journal of Sound and Vbraton (000), (): pp 57-584. [7] Majewsk Tadeusz. Poston Error Occurrence n Self Balancers Used n Rgd Rotors of Rotatng Machnery, Mechansm and Machne Theory, Vol., No. : pp 7-78, 988. [8] C. Rajalngham and S. Rakheja. Whrl suppresson n hand-held power tool rotors usng guded rollng balancers, Journal of Sound and Vbraton 7: 45-466, 998. [9] J. Chung and D.S. Ro. Dynamc analyss of an automatc dynamc balancer for rotatng mechansms, Journal of Sound and Vbraton 8: 05-056, 999. [0] C.H. Hwang and J. Chung. Dynamc analyss of an automatc ball balancer wth double races, JSME Internatonal Journal 4: 65-7, 999. [] J. Chung and I. Jang. Dynamc response and stablty analyss of an automatc ball balancer for a flexble rotor, Journal of Sound and Vbraton 59: -4, 00. 64 Rev. Fac. Ing. - Unv. Tarapacá, vol. Nº, 005 06-Meraz 64 05/0/005, :5