Champneys, AR

Friswell, MI

Rodrigues, DJ

Wilson, RE

English

We present a nonlinear analysis of the dynamics of an automatic ball balancer (ABB) for rotors which are both eccentric and misaligned. The ABB consists of two or more ball bearings which are free to travel around a circular race at a fixed distance from the shaft. The balls, after a transient response, find a steady state which balances the rotor. Following the previous work of Green et al. at Bristol, we have included the effect of shaft misalignment which causes the rotor to precess. This can be countered by having two ABB races at different axial locations along the shaft. Mathematically, we use a Lagrangian approach to derive the equations of motion for the system. It is found that, contrary to the case of flexible rotors that are subject to eccentricity and shaft bending, there is no choice of co-ordinate system which leads to autonomous governing equations. Simulations are then computed which illustrate the role of the ball damping coefficient

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                          Rodrigues, D. J., Champneys, A. R., Friswell, M. I., & Wilson, R. E. (2006).Automatic balancing of a rigid rotor with misaligned shaft.Link to publication record in Explore Bristol ResearchPDF-documentUniversity of Bristol - Explore Bristol ResearchGeneral rightsThis document is made available in accordance with publisher policies. Please cite only the publishedversion using the reference above. Full terms of use are available:http://www.bristol.ac.uk/pure/about/ebr-terms.htmlTake down policyExplore Bristol Research is a digital archive and the intention is that deposited content should not beremoved. However, if you believe that this version of the work breaches copyright law please contactopen-access@bristol.ac.uk and include the following information in your message:• Your contact details• Bibliographic details for the item, including a URL• An outline of the nature of the complaintOn receipt of your message the Open Access Team will immediately investigate your claim, make aninitial judgement of the validity of the claim and, where appropriate, withdraw the item in questionfrom public view.Automatic balancing of a rigid rotor with misaligned shaftD.J. Rodrigues1, a, A.R. Champneys1, b, M.I. Friswell2, c, R.E. Wilson1, d1Department of Engineering Mathematics, University of Bristol, Queens Building,University Walk, Bristol BS8 1TR2Department of Aerospace Engineering, University of Bristol, Queens Building,University Walk, Bristol BS8 1TRaD.J.Rodrigues@bristol.ac.uk, bA.R.Champneys@bristol.ac.uk,cM.I.Friswell@bristol.ac.uk, dRe.Wilson@bristol.ac.uk.Keywords: Automatic balancing, Euler angles, Rigid rotor, Shaft misalignment.Abstract. We present a nonlinear analysis of the dynamics of an automatic ball balancer (ABB) for rotorswhich are both eccentric and misaligned. The ABB consists of two or more ball bearings which are freeto travel around a circular race at a fixed distance from the shaft. The balls, after a transient response,find a steady state which balances the rotor. Following the previous work of Green et al. at Bristol, wehave included the effect of shaft misalignment which causes the rotor to precess. This can be countered byhaving two ABB races at different axial locations along the shaft. Mathematically, we use a Lagrangianapproach to derive the equations of motion for the system. It is found that, contrary to the case of flexiblerotors that are subject to eccentricity and shaft bending, there is no choice of co-ordinate system whichleads to autonomous governing equations. Simulations are then computed which illustrate the role of theball damping coefficient.IntroductionOne of the primary causes of vibration in rotating machinery is mass imbalance, this occurs when theprincipal axis of the moment of inertia is not coincident with the axis of rotation. For a rigid rotor theimbalance is usually eliminated by adding (or subtracting) correction masses in two distinct planes insuch a way as to realign and recentre the principal axis. However when using this method, the rotor hasto be rebalanced every time its mass distribution changes. This limitation motivates the study of self-compensating balancing devices, in which masses automatically redistribute themselves so as to eliminateany imbalance.One such device is the automatic ball balancer (ABB), which consists of a series of balls that are freeto travel, around a race filled with viscous fluid. This idea was initially proposed in 1904 [1], and sincethen ABB’s have been incorporated into many products including optical disc drives, machine tools andwashing machines. The first study of an ABB was carried out by Thearle in 1932 [2], who demonstratedthe existence of a stable steady state at rotation speeds above the first critical frequency. More recently,Chung et al. [3] have studied the stability of automatic balancing for an eccentric flexible rotor, Chao et al.[4] have looked at the non-planar effects of an optical disk drive fitted with an ABB and Green et al. [5]have focused on using numerical continuation to map the bifurcations between different attracting statesin a Jefcott rotor fitted with an ABB.The rest of this paper is outlined as follows. In Section 2 we present a fully geometric model fortwo-plane automatic balancing of a rigid rotor [6]. Our techniques are fully nonlinear and do not rely onsmall angle or constant spin speed assumptions, and so may be applicable to many other fields, such ascelestial mechanics and the dynamics of spinning spacecraft [7]. In Section 3 we focus on the two-planeautobalancing of a rotor with shaft misalignment and simulate responses for different values of the balldamping parameter. Finally, in Section 4 we draw conclusions and discuss how we would make the systemamenable to bifurcation analysis.Two-plane automatic balancingReference frames and generalized co-ordinates. In describing the dynamics of a rigid body it is usefulto consider the relation between an xyz frame fixed to the body and an inertial XY Z space frame, seeFigure 1. Firstly a translation is needed to place the XY Z axes onto the X ′Y ′Z ′ axes which is parallel toXY Z but with the same origin as the xyz body axes. Then the orientation of the xyz body axes, relativeto the X ′Y ′Z ′ frame, can be specified through the use of Euler angles where we will use the conventionset out below.PSfrag replacementsXYZX′,Y ′Z′x′y′,z′x′′y′′z′′xyz,φφθθψψro=(X,Y,Z)Fig. 1: Definition of co-ordinate system.Here, a rotation by an angle φ about the X ′ axis results in the primed co-ordinate system x′y′z′. Thisintermediate sytem is then rotated by an angle θ about the y ′ axis resulting in the double primed systemx′′y′′z′′. Finally, a rotation ψ about the z ′′ axis gives the unprimed xyz co-ordinate system. The full rotationR can be concisely written in matrix form asR = RψRθRφ, (1)whereRφ =1 0 00 cosφ sinφ0 − sinφ cosφ , Rθ =cos θ 0 − sin θ0 1 0sin θ 0 cos θ , Rψ = cosψ sinψ 0− sinψ cosψ 00 0 1 . (2)The use of three separate axes eliminates any polar singularity, furthermore in the small angle limit, φ andθ reduce to projection angles, which are commonly seen in the rotordynamics literature [6]. Thus the fullco-ordinate transformation from xyz to XY Z is given byrXY Z = ro + RTrxyz, (3)where rXY Z and rxyz are the position vectors in the XY Z and xyz systems respectively, andro = [X,Y, Z] is the position vector of the xyz systems origin. Thus the generlized co-ordinatesq = [X,Y, Z, φ, θ, ψ]T can be used to describe the motion of any rigid body.Model. The model shown in Figure 2 consists of a balanced rotor of massM and moment of inertia I =diag[Ix, Iy, Iz] with respect to the xyz axes. The mass imbalance is introduced through two point massesm1, m2, fixed to the rotor in planes z1, z2, with eccentricities ε1, ε2, and angular positions α1, α2 so thatboth static and couple imbalance are present. This setup is dynamically equivalent to the most general rigidrotor, which has both an offset centre of mass and a misaligned principal axis. The autobalancing devicehas two races R1 and R2, each fitted with n balls, which are represented as points of mass mi, in plane zi,with eccentricity εi and angular position αi, where i = 3 . . . n + 2, and i = n + 3 . . . 2n + 2 correspondto the balls in R1 and R2 respectively. The supports S1 and S2, have rest positions [0, 0, l1] and [0, 0,−l2],with stiffness Kj = diag[kxj , kyj , kzj ], and damping Cj = diag[cxj , cyj , czj ], j = 1, 2 with respect to theXY Z axes. PSfrag replacementsMm1m2miε1ε2εiz1 z2zil1l2S1S2R1R2yxzψ˙γαiFig. 2: Two-plane autobalancing rotor model.Lagrangian formulation of the equations of motion. We begin by writing the position vectors of thecentre of mass (COM), imbalances and balancing balls in theXY Z co-ordinate system. Using (3) we haverM =XYZ , rmi =XYZ + RTεi cosαiεi sinαizi , i = 1 . . . 2n+ 2. (4)Without loss of generality we can take α1 = 0 and α2 = γ where γ is the constant phase between the twoimbalances.Next we write the angular velocity in the body axis, since it is in these axes in which the inertia matrixtakes a diagonal form. As ωφ is parallel to the space X axis, its components in the body axis are given byapplying the complete rotation R. Now ωθ lies along the intermediate y′ axis and so we apply only the lastrotation Rψ, finally ωψ lies along the body z axis and so needs no further work. Thus the components ofω in the body axis are given byω = Rφ˙00 + Rψ0θ˙0 +00ψ˙ ,= φ˙ cos θ cosψ + θ˙ sinψ−φ˙ cos θ sinψ + θ˙ cosψφ˙ sin θ + ψ˙ . (5)The kinetic energy T is now given by (see also the appendix)T =12ωTIω +12M r˙2M +122n+2∑i=1mir˙2mi. (6)The potential V comes from the elastic energy due to the deflection rS1 and rS2 of the supports S1 and S2thusV =122∑j=1rTSjKrSj , (7)whererSj =XYZ + RT00l˜j−00l˜j , l˜1 = l1, l˜2 = −l2. (8)Similarly Rayleigh’s dissapation function F can be written asF =122∑j=1r˙TSjCr˙Sj +122n+2∑i=3cbα˙i, (9)where the ball damping coefficient cb, arises from the drag on the balls as they pass through a viscous fluidin the race.The equations of motion can now be generated by applying Lagrange’s equationsddt(∂L∂q˙k)−∂L∂qk= Qk, (10)where L = T − V is the Lagrangian, and q = [X,Y, Z, φ, θ, ψ, αi]Ti = 3 . . . 2n + 2, are the generalisedco-ordinates. The generalised forces not arising from a potential Qk, are given by − ∂F∂q˙k except in theequation for ψ where we must also add the driving torque τ(ψ˙).Balancing of a misaligned rotorModel parameters. We have used the computer algebra system Maple to derive the equations of motion(omitted here) from (10). The equations are then transfered without resorting to linearization into Matlabwhere they are solved numerically. The model rotor used for simulation has parameters M = 1 kg,Ix = Iy = 1 kg m2, Iz = 0.5 kg m2, l1 = l2 = 0.5 m. The COM is confined to the XY plane, andthe spin speed ψ˙ = ω, is assumed constant with ω = 40 rad s−1. Both supports are identical, with stiffnessKi = diag[k, k, k], k = 50 N m−1 and damping Ci = diag[c, c, c], c = 5 N s m−1.The imbalance masses have values m1 = m2 = 0.04 kg, e1 = e2 = 0.1 m, z1 = −z2 = 0.1 m andα2 = γ = pi, which corresponds to a rotor whose imbalance occurs solely through misalignment. TheABB has parameters mi = 0.01 kg, ei = 0.1 m, i = 3 . . . 6, z3 = z4 = 0.25 m, z5 = z6 = −0.25 m,corresponding to two races with two balls each. Finally the damping of the balls in the race cb, is theparameter which is to be varied.Simulations. Numerical simulations which illustrate the influence of the ball damping coefficient cb,are shown in Figure 3. The top graphs show the vibration amplitude of the first support A1 = |rS1|, andthe ball angles are shown underneath.In the underdamped regime shown on the left, the balls initially oscillate about the unstable balancedpositions, before destabilising completely at around t = 50 s. The resulting vibrations are an order ofmagnitude worse than the motion without the ABB. For high damping on the right, these oscillations areprevented, but the balls take a long time to reach the balanced state. Ideally we desire critical dampingwhere the balls reach the equilibrium position in the minimum time without oscillation, as illustrated inthe middle graphs.00.511.522.53x 10−3A1, m0 20 40 60 80−6−4−20246time, secsangle, rad0 20 40 60 80time, secs0 20 40 60 80time, secsPSfrag replacementscb = 1× 10−6Nms cb = 1× 10−4Nms cb = 0.1Nmsα3α3α4α4α5α5α6α6Fig. 3: Influence of ball damping cb on the support vibrations and ball positionsConclusionWe have presented a fully nonlinear geometric model for the two-plane balancing of a rigid rotor. Inprevious studies [3,5] of flexible and planar rotors, autonomous equations are derived by formulating theLagrangian in the rotating frame. However, in the present case of a non-planar rigid rotor, the spin angleψ = ωt, is taken about a body z axis which is not parallel to the space Z axis. As a consequence, it is notpossible to use the rotating frame to forge equations, which are both autonomous, and satisfy the conditionthat the rotor moves in the horizontal Z = 0 plane.We have provided simulations for the two-plane automatic balancing of a misaligned rotor. It is shownthat the two-race mechanism can eliminate not only vibrations due to radial motion, but also those due toangular displacement. However automatic balancing can only be achieved in regions of parameter spacewhere the balanced state is stable, in particular an appropriate choice of the ball damping coefficient cb, isessential for a successful working device.It is known [6] that spin speed ω, moment of inertia I, and support damping C, all influence the stabilityof balanced operation. This dependence will be investigated in future studies, where we invisage usingnumerical continuation, to compute bifurcation diagrams showing the boundaries of stability in variousparameter planes [5].References[1] Lee J. and Van Moorhem W. K., Analytical and experimental analysis of a self compensating dynamic balancerin a rotating mechanism, ASME Journal of Dynamic Systems, Measurement and Control 118 (1996) 468-475.[2] Thearle E., A new type of dynamic-balancing machine, Trans. ASME 54(APM-54-12) (1932) 131-141.[3] Chung J. Jang I., Dynamic response and stability analysis of an automatic ball balancer for a flexible rotor.Journal of Sound and Vibration 259(1) (2003) 31-43.[4] Chao C.-P. Huang Y.-D. Sung C.-K., Non-planar dynamic modeling for the optical disk drive spindles equippedwith an automatic balancer. Mechanism and Machine Theory 38 (2003) 1289-1305.[5] Green K. Champneys A. R. and Lieven N. J., Bifurcation analysis of an automatic dynamic balancer for eccent-ric rotors. Journal of Sound and Vibration 291 (2006) 861-881.[6] Sperling L. Ryzhik B. Linz Ch. and Duckstein H., Simulation of two-plane automatic balancing of a rigid rotor,Mathematics and computers in simulation 58 (2002) 351-365.[7] Genta G. Delprete C. and Busa E., Some considerations on the basic assumptions in rotordynamics, Journal ofSound and Vibration 227(3) (1999) 611-645.APPENDIXThe explicit forms for the kinetic energy T , potential V and Rayleigh’s dissapation function F which arederived in Section 2 are given hereT =12Ix(φ˙ cos θ cosψ + θ˙ sinψ)2+12Iy(−φ˙ cos θ sinψ + θ˙ cosψ)2+12Iz(φ˙ sin θ + ψ˙)2+12M(X˙2 + Y˙ 2 + Z˙2)122n+2∑i=1mi{ [X˙ + εi(− sin θθ˙ cos (ψ + αi)− cos θ sin (ψ + αi)(ψ˙ + α˙i))+ zi cos θθ˙]2+[Y˙ + εi(cos θθ˙ sinφ cos (ψ + αi) + sin θ cosφφ˙ cos (ψ + αi)− sin θ sinφ sin (ψ + αi)(ψ˙ + α˙i)− sinφφ˙ sin (ψ + αi)+ cosφ cos (ψ + αi)(ψ˙ + α˙i))+ zi(sin θθ˙ sinφ− cos θ cosφφ˙) ]2+[Z˙ + εi(− cos θθ˙ cosφ cos (ψ + αi) + sin θ sinφφ˙ cos (ψ + αi)+ sin θ cosφ sin (ψ + αi)(ψ˙ + α˙i)+ cosφφ˙ sin (ψ + αi)+ sinφ cos (ψ + αi)(ψ˙ + α˙i))+ zi(− sin θθ˙ cosφ− cos θ sinφφ˙) ]2}, (A-1)V =122∑j=1{kxj(X + sin θ l˜j)2+ kyj(Y − cos θ sinφ l˜j)2+ kzj(Z − (1− cos θ cosφ) l˜j)2 }, (A-2)F =122∑j=1{cxj(X˙ + cos θθ˙ l˜j)2+ cyj(Y˙ +(sin θθ˙ sinφ− cos θ cosφφ˙)l˜j)2czj(Z˙ −(sin θθ˙ cosφ+ cos θ sinφφ˙)l˜j)2 }+122n+2∑i=3cbα˙i. (A-3)

31-43.[4] Chao C.-P. Huang Y.-D. Sung C.-K., Non-planar dynamic modeling for the optical disk drive spindles equipped with an automatic balancer.

A new type of dynamic-balancing machine,

Analytical and experimental analysis of a self compensating dynamic balancer in a rotating mechanism,

Bifurcation analysis of an automatic dynamic balancer for eccentric rotors.

Simulation of two-plane automatic balancing of a rigid rotor, Mathematics and computers in simulation 58

Some considerations on the basic assumptions in rotordynamics,

Automatic balancing of a rigid rotor with misaligned shaft

We present a nonlinear analysis of the dynamics of an automatic ball balancer (ABB) for rotors which are both eccentric and misaligned. The ABB consists of two or more ball bearings which are free to travel around a circular race at a fixed distance from the shaft. The balls, after a transient response, find a steady state which balances the rotor. Following the previous work of Green et al. at Bristol, we have included the effect of shaft misalignment which causes the rotor to precess. This can be countered by having two ABB races at different axial locations along the shaft. Mathematically, we use a Lagrangian approach to derive the equations of motion for the system. It is found that, contrary to the case of flexible rotors that are subject to eccentricity and shaft bending, there is no choice of co-ordinate system which leads to autonomous governing equations. Simulations are then computed which illustrate the role of the ball damping coefficient.We present a nonlinear analysis of the dynamics of an automatic ball balancer (ABB) for rotors which are both eccentric and misaligned. The ABB consists of two or more ball bearings which are free to travel around a circular race at a fixed distance from the shaft. The balls, after a transient response, find a steady state which balances the rotor. Following the previous work of Green et al. at Bristol, we have included the effect of shaft misalignment which causes the rotor to precess. This can be countered by having two ABB races at different axial locations along the shaft. Mathematically, we use a Lagrangian approach to derive the equations of motion for the system. It is found that, contrary to the case of flexible rotors that are subject to eccentricity and shaft bending, there is no choice of co-ordinate system which leads to autonomous governing equations. Simulations are then computed which illustrate the role of the ball damping coefficient

Rodrigues, DJ.

https://research-information.bris.ac.uk/files/3007743/rodriguesbathsv.pdf

Automatic balancing of a rigid rotor with misaligned shaft

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