. Automatic balancers present a modular possibility to coun- teract variable rotor unbalances during operation. Two or more balancing masses, usually spheres, can orbit in a uid- lled annular cavity whose axis of symmetry coincides with the rotor axis. At supercritical speeds the masses { driven by the rotor de ection { tend towards stationary po- sitions inside the cavity opposing the primary rotor unbalance. Related to the phenomenon of rotating shafts being captured at resonances due to insucient drive power, automatic ball balancers inhibit operation speed bands with non-synchronous vibrations where the rotor surpassed the resonance but the balls continue to orbit with the eigenfrequency with respect to the inertial system. As a result, the balancing masses do not take stationary positions inside the cavity and the rotor is excited not only by the primary unbalance but also by the sub-synchronously orbiting balancing masses. The width of the operation speed band ex- hibiting non-synchronous behaviour depends on the balancing masses, the orbit radius, external damping of the rotor and viscous damping of the balls due to the uid inside the cavity. For a planar oscillator in isotropic supports with a balancer containing two balancing balls, an ex- plicit correlation between the stability border and the uid damping is presented. In order to parameterize the uid damping model, the drag on spheres in annular cavities is examined and a proposed relation based on the cavity geometry and the uid properties is presented. Keywords: automatic balancing, stability, sub-synchronous vibration 1 Introduction Rotating machinery can encounter variable mass distribution during operation, leading to unbalance excitation, which cannot be addressed by conventional bal- ancing at the beginning of the product life cycle. Examples for aforementioned systems are washing machines, centrifuges for medical research and optical disc drives with varying content as well as grinding machines whose discs are worn unevenly. To reduce variable unbalance excitations, automatic ball balancers (ABB) can be equipped to the rotor, counteracting the uneven mass distribution with freely 1 PreprintPreprint : This is a preprint of a contribution presented at the XLVIII International Summer School-Conference "Advanced Problems in Mechanics", June 21-26, 2020, St. Petersburg, Russia. The published version is available at doi:10.1007/978-3-030-92144-6 17. optical system disc loading trayABB unitbalancing balls Fig. 1. Partially disassembled DVD Writer with ABB unit (pried open) to counterbal- ance variable disc unbalances. movable masses (balls) inside an annular cavity whose symmetry axis coincides with the rotor axis, see Fig. 1 for an example application in an optical disc drive. Driven by inertial e ects resulting from the common de ection of the rotor and the ABB, the balancing balls move towards the orbital position with maximum distance to the axis of rotation. Resulting from this passive functional pricipal, ABBs only achieve the desired e ect of balancing at super-critical operation, where the rotor displacement and the unbalance excitation have undergone the typical phase shift when passing through resonance. In addition to the critical speed, which has to be surpassed for the intended operation of ABBs, a speed domain of unstable operation, which prevents the desired positioning of the balls just above the critical speed, has to be surpassed as well. The extent of this domain depends on the design parameters of the rotating system, in particular external damping and the selected uid1with which the cavity is lled and is environing the balancing balls. In the presented study, an explicit calculation of the unstable speed range for isotropically supported planar rotor systems with one ABB unit incorporating two balancing balls is derived. Section 2 describes the equations of motion of the planar system and Section 3 recapitulates the transformations used to gain a generalized eigenvalue problem whose frequency-dependent eigenvalues provide information about the stability of the operation, as presented in [1]. In Section 4 the Routh-Hurwitz criterion is applied to the eigenvalue problem and with the assumption of isotropic support of the rotor, an explicit relation between the viscous damping of the uid, the external damping, the eigenfrequency of the rotor system and the balancing ball mass and orbit radius is deducted. Finally, a conclusion and outlook is given in Section 5. 1Air is used in the most rudimentary design, which has low viscous damping. 2 PreprintPreprint : This is a preprint of a contribution presented at the XLVIII International Summer School-Conference "Advanced Problems in Mechanics", June 21-26, 2020, St. Petersburg, Russia. The published version is available at doi:10.1007/978-3-030-92144-6 17. OGOGOGOGOGOGOGOGOGOGOGOGOGOGOGOGOG EEEEEEEEEEEEEEEEE xy OROROROROROROROROROROROROROROROROROM m;Jmb;Jbm0m0m0m0m0m0m0m0m0m0m0m0m0m0m0m0m0  "0"0"0"0"0"0"0"0"0"0"0"0"0"0"0"0"0'i'i'i'i'i'i'i'i'i'i'i'i'i'i'i'i'iiiiiiiiiiiiiiiiii rrrrrrrrrrrrrrrrr """"""""""""""""" ky bykxbx Fig. 2. Translational oscillator with primary unbalance and one counterbalancing mass of a balancer with eccentricity. 2 Equations of Motion LetmandJbe the mass and mass moment of inertia of a planar rotor in thex{yplane, Fig. 2, excited by the primary unbalance mass m0located"0 away from the center of mass OMof the balanced rotor. The rotor is mounted orthotropically described by linear sti nesses kxandkyand viscous damping coecients bxandby, respectively. The rotation angle of the rotor is indicated by the position of the primary unbalance and describes the orientation of the rotor xed coordinate system {. Due to manufacturing and assembly tolerances, the raceway of the balancing balls may encounter an eccentric geometric center OG, described by angle and eccentricity E. The balancing balls of mass mband radiusrcan move inside the cavity on an orbit with radius ". The rotor- xed orbital position of ball iwith respect to the primary unbalance is described by 'i. Considering an ABB with two identical balls sharing the same orbit, the potential and kinetic energy U=1 2 kxx2+kyy2 (1) T=1 2 m _x2+ _y2 +J_2+m0 _x2 0+ _y2 0 +2X i=1mb _x2 i+ _y2 i +Jb_2 i 2(2) 3 PreprintPreprint : This is a preprint of a contribution presented at the XLVIII International Summer School-Conference "Advanced Problems in Mechanics", June 21-26, 2020, St. Petersburg, Russia. The published version is available at doi:10.1007/978-3-030-92144-6 17. can be formulated. Inserting the no-slip rolling condition, the coordinates of the centres of mass can be expressed in terms of x,y,and'i x0=x+"0cos(); xi=x+"cos(+'i) +Ecos(+ ); y0=y+"0sin(); yi=y+"sin(+'i) +Esin(+ ); i=" r'i: In accordance with [2,3,4] and others, the viscous e ect on the balls due to the uid are modelled in a linear manner by the torque MD= _'i: (3) In combination with the translatoric damping forces and an external damping torque Fbx=bx_x; Fby=by_y; M 0= 0_; evaluation of the Euler{Lagrange equations yields, with M=m+m0+P2 i=1mi, a system of ve di erential equations Mx+bx_x+kxx= m0"0sin () +2X i=1Embsin ( +) +mb"sin (+'i)!  + m0"0cos ()_+2X i=1Embcos ( +)_+mb" _+ _'i cos (+'i)! _ +2X i=1mb" _+ _'i cos (+'i) _'i+mb"sin (+'i) 'i (4) My+by_y+kyy=  m0"0cos () +2X i=1Embcos ( +) +mb"cos (+'i)!  + m0"0sin ()_+2X i=1Embsin ( +)_+mb" _+ _'i sin (+'i)! _ +2X i=1mb" _+ _'i sin (+'i) _'imb"cos (+'i) 'i (5) 4 PreprintPreprint : This is a preprint of a contribution presented at the XLVIII International Summer School-Conference "Advanced Problems in Mechanics", June 21-26, 2020, St. Petersburg, Russia. The published version is available at doi:10.1007/978-3-030-92144-6 17. J+m0"2 0+2X i=1Jb+mbE2+mb"2+ 2Emb"cos ( 'i)!  + 0+2X i=12Emb"sin ( 'i) _'i! _=M0 + m0"0sin () +2X i=1mbEsin ( +) +mb"sin (+'i)! x  m0"0cos () +2X i=1Embcos ( +) +mb"cos (+'i)! y 2X i=1 Emb"cos ( 'i)Jb" r+mb"2 'i+Emb"sin ( 'i) _'2 i(6) Jb"2 r2+mb"2 'i+ _'i=Jb" rEmb"cos ( 'i)mb"2  +mb" Esin ( 'i)_2+ sin (+'i) xcos (+'i) y 8i2f1;2g:(7) 2.1 Identi cation of the viscous drag coecient Assuming a quasi steady state with the uid ow velocity v owequal to the speed of the rotor and small orbit velocities _ 'iof the balls with respect to the rotor, a Stokes ow regime can be assumed, leading to a reciprocal relation between the Reynolds number Re =2v owr  uid2 _'i"r  uid; (8) with uid being the kinematic viscosity of the uid, and the resulting drag coecientcd. Therefore the product cdRe can be represented by a scalar, which depends on the geometry of the annular cavity and the relative ball size. Inserting the drag force FD=1 2 uidr2cdv2 owsign(v ow); (9) with uidbeing the uid density, into Eq. (3) leads to =cdRe 4 uid uidr"2: (10) An appropriate value for cdRe can be determined empirically, see [5,6], or by CFD analysis for the speci c geometry of the annular cavity and the balancing balls. One suitable estimation results from the widely studied drag forces on a sphere in contact with a at wall, e.g. cdRe = 322, which was identi ed empirically in [7]. 5 PreprintPreprint : This is a preprint of a contribution presented at the XLVIII International Summer School-Conference "Advanced Problems in Mechanics", June 21-26, 2020, St. Petersburg, Russia. The published version is available at doi:10.1007/978-3-030-92144-6 17. 3 Condition for stable balancing Following the procedure presented in [1], Equations (4), (5) and (7), evaluated fori=f1;2g, can be transformed utilizing the Krylov-Bogoliubov-Mitropolsky method [8]. The coordinates xandyare substituted by x=ex=Axcos()Bxsin(); y=ey=Aysin() +Bycos(); _ex=_Axsin()_Bxcos(); _ey=_Aycos()_Bysin();(11) ex=@_ex=@t; ey=@_ey=@t; yielding the additional conditions _Ax=_Bxsin(_t)=cos(_t); _By=_Aysin(_t)=cos(_t): (12) Considering the steady state, i.e. _=!,= 0, '1= 0 and '2= 0, a system of six ordinary di erential equations (ODEs) is formed in terms of Ax,Bx,Ay, By,'1and'2. Approximation of the ODEs by their integral mean over one revolution removes the trigonometric terms in !t. Following a linear perturbation analysis, i.e. substituting Ax=A0x+Ax; Bx=B0x+Bx; Ay=A0y+Ay; By=B0y+Bx; ' 1='01+'1; ' 2='02+'2; expanding the trigonometric terms in '1and'2and neglecting terms of higher order, the system of linear ODEs of rst order A_z+Bz= 0withz= [Ax;Bx;Ay;By;' 1;' 2]T, A=2 666666641 0 0 0mb"sin('01) Mmb"sin('02) M 01 0 0 mb"cos('01) Mmb"cos('02) M 0 0 1 0mb"sin('01) Mmb"sin('02) M 0 0 0 1mb"cos('01) Mmb"cos('02) M!cos('01) 2!sin('01) 2!cos('01) 2!sin('01) 2 mb"0 !cos('02) 2!sin('02) 2!cos('02) 2!sin('02) 20 mb"3 77777775; and B=2 666666664bx 2M! 2kx 2M!0 0mb!"cos('01) 2Mmb!"cos('02) 2M ! 2+kx 2M!bx 2M0 0mb!"sin('01) 2Mmb!"sin('02) 2M 0 0 by 2M! 2ky 2M!mb!"cos('01) 2Mmb!"cos('02) 2M 0 0 ! 2+ky 2M!by 2Mmb!"sin('01) 2Mmb!"sin('02) 2M !2sin('01) 2!2cos('01) 2!2sin('01) 2!2cos('01) 20 0 !2sin('02) 2!2cos('02) 2!2sin('02) 2!2cos('02) 20 03 777777775 is obtained. By solving the eigenvalue problem of the system  B+A v= 0 (13) 6 PreprintPreprint : This is a preprint of a contribution presented at the XLVIII International Summer School-Conference "Advanced Problems in Mechanics", June 21-26, 2020, St. Petersburg, Russia. The published version is available at doi:10.1007/978-3-030-92144-6 17. of order six and determining if all eigenvalues have negative real parts <()<08; (14) the stability of the system can be obtained for a given angular frequency !of the rotor. However, numerical methods show inaccuracies when '01='02==2 or'01='02=, i.e. when the ABB eccentricity is zero and the primary unbalance is very low or very high, respectively. This follows from the resulting poor condition of the matrices. Alternatively, the sixth order characteristic poly- nomial of Eq. (13) can be tested with the Routh{Hurwitz stability criterion [9]. With the variation of one design parameter of the ABB, a stability chart can be obtained, depicting the stable operating speeds, see Fig. 3. Since the choice of uid is barely subject to boundary conditions, the variation of the uid proper- ties is well suited to in uence the behaviour of the ABB unit. Hence, the stability maps are presented in the {!space. a) b) Fig. 3. Stability chart examples for variable viscous parameter for the case of (a) isotropic and (b) orthotropic ( kx<ky) supports. As can be seen from the graphs, stable balancing can only be achieved above the natural frequency if the viscous parameter of the liquid is suciently high. In the case of orthotropic supports, the stability area is divided in two regions, as only forward whirling motions of the rotor can be balanced [10]. 4 Explicit stability border for the isotropic eccentricity free case For a given rotor and ABB con guration with two balls, the stability chart can be readily obtained as presented above. Assuming isotropic supports, i.e. kx=ky=k,bx=by=band therefore k=M!2 0, and no eccentricity of the orbit raceway, the balancing balls will be positioned symmetrically in order to counterbalance the primary unbalance, resulting in '02='01; = 2'01: (15) 7 PreprintPreprint : This is a preprint of a contribution presented at the XLVIII International Summer School-Conference "Advanced Problems in Mechanics", June 21-26, 2020, St. Petersburg, Russia. The published version is available at doi:10.1007/978-3-030-92144-6 17. Simplifying the matrices in Eq. (13) under the assumptions stated above, the characteristic equation can be written in a factorised form 0 =16M4(mb")2!4h M2 !2!2 02+!2(2M+b)2i h 4(mb")4 2!6+ 22!2+!42 sin2('01) cos2('01) +  2(mb")2!4 M !2!2 0 + 3b +M2  !2!2 02 +4M2!2 (mb")2 2!2+!2 0 + (M+b) + !2b2i : (16) From the rst term in brackets follow the roots =b 2M+i!2 0!2 2!_=b 2M+i!2 0+!2 2!(17) with negative real parts. The second term in brackets is a fourth order polynomial a44+a33+a22+a1+a0with the coecients being rewritable as a4=4!2 4(mb")4!2sin2('01) cos2('01) +M2 2 ; a3=4M !2 2(mb")2!2+ (mb")2!2 0+ b ; a2=M2 2 !2!2 02+ !2 20(mb")4!4sin2('01) cos2('01) +6(mb")2 !2b+ 2b2 ; (18) a1=2(mb")2M !4(!!0) (!+!0); a0=4(mb")4!8sin2('01) cos2('01): Sincea4>0, it is necessary and sucient [9] that a0>0,a1>0,a2>0,a3>0 and 0<a1(a3a2a4a1)a0a2 3: (19) Froma1>0 follows the familiar condition ! > ! 0for automatic balancing at super critical speeds. Di erentiation of the right hand side of Eq. (19) with respect to'01yields d d'01a1(a3a2a4a1)a0a2 3=psin(4'01) (20) d2 d'2 01a1(a3a2a4a1)a0a2 3=4pcos(4'01) (21) with 0<p= 16(mb")4M2 2!12 b 2 b+ 3(mb")2!2 + 9(mb")2!2 0 b+ (mb")2(!2+!2 0) : The roots of Eq. (20) deliver possible extrema of the stability criterion at '01= j=4,j2f2;3;4g, depicted in Fig. 4. Evaluation of Eq. (21) shows that '01= 8 PreprintPreprint : This is a preprint of a contribution presented at the XLVIII International Summer School-Conference "Advanced Problems in Mechanics", June 21-26, 2020, St. Petersburg, Russia. The published version is available at doi:10.1007/978-3-030-92144-6 17. m03/4  3/4 m1 m2m0m1m2m0m1 m2 a) b) c) Fig. 4. Balancing ball positions resulting in extrema of the stability criterion. Primary unbalance magnitude: (a) Medium (b) low (c) high. 3=4 is a minimum of the criterion, therefore the stability of the system is guaranteed if Eq. (19) holds for '01= 3=4, which reads after factorization 0<8(mb")2M2 2!6 b(!2!2 0)!2 0(mb")2(!2+!2 0) h M2 2 !2!2 02+ 3(mb")2!3+ !b2i : Since the second braced term is always greater than zero, the criterion requires the rst braced term to be positive and thus >(mb")2!2 0 b!2+!2 0 !2!2 0: (22) From Eq. (22) the limit min= lim !!1 =(mb")2!2 0 b(23) can be derived, describing a minimum required value for the viscous drag param- eter , in order to be able to pass the unstable operation of the ABB, see Fig. 5. With Eq. (10) these requirements can be expressed as a minimum dynamic uid viscosity. The assumptions of absence of raceway eccentricity and speci c orbit positions of the counterbalancing masses are justi ed by the fact that these values are not subject to design, but are caused by manufacturing inaccuracies and the variable magnitude of the primary unbalance inherent to the application. Fur- thermore, sensitivity analyses of the stability considerations using Eq. (13) show only marginal in uences on the stability border. 5 Conclusion and outlook The problem of sub synchronous behaviour may be a minor issue in the design of ABB units. Since rotating machinery, which run above the rst critical frequency, are operated with sucient clearance to avoid substantial unbalance excitation. 9 PreprintPreprint : This is a preprint of a contribution presented at the XLVIII International Summer School-Conference "Advanced Problems in Mechanics", June 21-26, 2020, St. Petersburg, Russia. The published version is available at doi:10.1007/978-3-030-92144-6 17. 0.5 0.4 0.3 0.2 min Fig. 5. Stability chart for an ABB in isotropic supports showing the minimum viscous parameter value required to surpass unstable operation The frequency band of unstable behaviour, as shown in the stability maps, is often avoided due to already existing constraints concerning the operation speed. Nevertheless, the presented explicit determination of the stability border for ABB units in isotropic supports can simplify the estimation of stability problems signi cantly. An interesting result emerges from the minimum uid viscosity required to sur- pass the sub synchronous behaviour at all. This is most relevant, when air lled ABB units are utilized in order to avoid the design of sealing and lling pro- cedures. Due to the low viscosity, ABB designs with air lled cavities are most prone to unstable behaviour if external damping is insucient. However, the presented model neglects the dissipative rolling friction between the balls and the raceway. This damping behavior leads to stabilization and thus smaller areas of instability. In addition, no suitable factorization of the characteristic equation of the orthotropically supported ABB system was found, which would allow an analogous evaluation of the criterion to derive explicit representations for the stability limits.