The Stability of Automatic Ball Balancers (Paper ... - Michael I Friswell

7th IFToMM-Conference on Rotor Dynamics, Vienna, Austria, 25-28 September 2006The Stability of Automatic Ball BalancersKirk Green Michael I. Friswell Alan R. Champneys Nicholas A.J. LievenDept. of TheoreticalPhysics, VrijeUniversiteit, 1081 HVBristol Laboratory for Advanced Dynamics Engineering (BLADE)University of Bristol, Queen’s Building, University Walk, Bristol BS8 1TR, UKAmsterdam, TheNetherlandsk.green@few.vu.nl m.i.friswell@bris.ac.uk a.r.champneys@bris.ac.uk nick.lieven@bris.ac.ukABSTRACTOne realisation of an automatic balancer uses two or more balls that are free to travel in a race, filled with aviscous fluid, at a fixed distance from the shaft centre. The objective is for the balls to position themselves sothat they counteract any residual unbalance. The fact that no external force is required to achieve balance, that isthe system is passively controlled, means that the balancer is potentially able to cope with a time-varyingunbalance. Typical applications include optical disc drives and machine tools. However, the usefulness of thisdevice depends on the balanced steady state solution being achievable and stable. This paper describes adynamic model of a Jeffcott rotor with an automatic balancer and provides a non-linear analysis of its dynamicsto determine steady states and their bifurcations as parameters are varied. The pseudospectra of the linearizationof the system about a balanced steady state solution are computed. This approach allows the eigenvalues that aremost sensitive to perturbation to be quantified. Furthermore, how the sensitivity of the eigenvalues influences thetransient response may be determined. These tools will help to design reliable and robust automatic balancers.KEY WORDSball balancer, bifurcation analysis, pseudospectra, Jeffcott rotor1 INTRODUCTIONUnbalance is a major cause of vibration in rotating machinery. In most cases standard methods of balancingare able to reduce the vibration to satisfactory levels. Sometimes, however, the unbalance varies with time, forexample due to the fouling of blades. Here some sort of active or semi-active system of balancing is required.This motivates the use of self-compensating, automatic dynamic balancers (ADB), that require no external forcesto achieve balance. ADBs have been applied to optical disc drives to obtain higher operating speeds without anyreduction in the tracking performance [2, 9, 10], and to balance machine tools, such as, lathes, angle grinders andcutting tools [12]. Lee and Van Moorhem [11] reviewed the early history of ADBs. The ADB of interest hereconsists of two or more masses that are free to travel around a race, filled with a viscous fluid, at a fixed distancefrom the centre of rotation of the rotor. Chung and Ro [3] and Adolfsson [1] derived the equations of motion andperformed a linear stability analysis using perturbation methods to identify stability regions of the full nonlinearproblem. However, due to the highly nonlinear nature of the problem, the ADB mechanism can make thevibration response much worse for some parameter values. This has limited the successful commercialapplication of the devices [13]. A linear stability analysis, used alone, is an inadequate tool for understandingsuch nonlinear behavior.In this paper, we provide a detailed nonlinear bifurcation analysis of the dynamics of the ADB by usingnumerical continuation tools [4]. The equations of motion are solved in the rotating frame, since the problem isthen a time-independent (autonomous) system that is very amenable to a full nonlinear bifurcation analysis. Theapproach finds, and then follows, steady states and periodic solutions, irrespective of their stability, and is able todetect their local bifurcations. These bifurcations mark the stability boundaries of the steady state equilibrium in1 Paper-ID 74

the full nonlinear system [5]. We also consider the transient response of a machine fitted with an ADBmechanism. The transient behavior in linear dynamical systems can be very different to that predicted by a linearstability analysis, where the eigenvalues determine the asymptotic behavior. Often, in stable non-normalsystems, transients may exhibit an undesired dynamic response, such as, a large transient growth (or apparentinstability) preceding a long, highly oscillatory, exponential decay or, alternatively, a rapid collapse beforeeventual asymptotic decay to the stable equilibrium [6]. This behavior is of particular concern when consideringa linearization about an equilibrium of a system with strong nonlinearities. In this case, a large transient growthmay result in a trajectory of the full nonlinear system coming close to regions of state space where thelinearization breaks down. Thus the eigenvalues of the linearization do not always correctly predict the behaviorof the full nonlinear system, particularly in a finite time interval [8]. Hence an analysis and understanding of thesensitivity to perturbation and the ensuing transient response is necessary for the robust design of a practicaldevice.2 EQUATIONS OF MOTIONFigure 1 shows a schematic of an ADB with two balancing masses. Movement is confined to the planardirections X and Y. The equations of motion describing this system are derived using Lagrange’s equation [5].The equations describing the displacement in the X and Y directions are given by2MX + c X + k X −M εψ cos ψ−MεψsinψX Xn∑ i i i i ii=12{ ( ) ( ) ( ) ( )}+ m X −R ψ + φ sin ψ+ φ − R ψ + φ cos ψ+ φ = 0n2MY + cYY + kYY −M εψ sin ψ+ M εψcosψ+ Mg + ∑ migi=1n(2)2+ ∑ mi{ Y + R( ψ+φ i) cos( ψ+φi) −R( ψ+φ i) sin ( ψ+φ i)} = 0i=1where ψ is the angular position of the rotor. Finally, the equation of motion of the ith ball is given by2−mR i ⎡X sin ( ψ+φi) −Y ⎣cos ( ψ+φ i) ⎤⎦+ mR i ( ψ+φ i) + Diφ i + mgR i cos( ψ+φ i)= 0 . (3)These equations can be simplified by assuming equal masses m of the balancing balls, an equal viscous dragforce D acting on each ball, a constant angular velocity of the rotor ω ( ψ =ω t ) , and isotropic suspension of therotor c (damping constant) and k (spring constant). Finally, we neglect the effect of gravity on the motion of theballs, that is we set g=0.Furthermore, we can write the equations in dimensionless form by introducing the following nondimensionalstatesX YX = , Y , t nt,R= R=ω (4)and the dimensionless parametersω m ε c DΩ= , µ= , δ= , ς= , β= .ω22n M R kM mR ωnwhere ω n = k/M is the natural frequency of the rotor.Finally, we obtain a time independent set of equations by transforming to a rotating frame of reference, using( ) ( ) ( ) ( )X = xcos Ωt − ysin Ω t , Y = xsin Ω t + ycosΩ t . (6)This results in the following second-order dimensionless system describing the ADB in the rotating frame:(1)(5)2 Paper-ID 74

( n )⎛1+ nµ 0 ⎞⎛x⎞ ⎛ 2ς −2Ω 1+ µ ⎞⎛x⎞ ⎛ K −2Ως⎞⎛x⎞⎜ ⎟⎜ ⎟+ ⎜⎟⎜ ⎟+⎜ ⎟⎜ ⎟⎝ 0 1+ nµ ⎠⎝y⎠ ⎝2Ω ( 1+ nµ ) 2ς⎠⎝y⎠ ⎝2ΩςK ⎠⎝y⎠( Ω+φ)22 n ⎛⎞⎛ ⎞δΩ ⎜ i φi ⎟⎛cosφi⎞= +µ,⎜ 0 ⎟ ∑2 ⎟⎜ ⎟⎝ ⎠sini=1⎜ φ−φ ii ( Ω + φi)⎟⎝ ⎠⎝⎠(7)2 2( x x y) ( y y x) φ i +βφ i = −Ω −2Ω sinφi − −Ω + 2Ωcos φi.In this paper, we consider two balancing masses, that is, n = 2 . Therefore, our system describing the ADB,when written as a system of first-order equations, is eight-dimensional. This system has three steady statesolutions. Namely, a balanced steady state where the radial vibration is reduced to zero; a ‘coincident’ steadystate where the balls occupy the same position in the race, resulting in a non-zero radial vibration; and an ‘inline’steady state where the two balls lie on opposite sides of the race in a line with the centre of mass of therotor [5]. Here, we concentrate on the balanced steady state solution given by,x = y = 0 ,−1⎛−δ⎞φ 1 = cos ⎜ ⎟⎝2µ⎠ , φ 2 =−φ 1. (8)A conclusion we can immediately make for the balanced steady state to exist is that, 2µ>δ. In words, thebalancing balls must have a large enough mass (with respect to the mass of the rotor) in order to overcome themass eccentricity (or equivalently the residual unbalance).Yφ 1mφ 2k XDRC RC M00 1100 11M00 11ɛψc Xk Yc Y0Figure 1: Automatic dynamic balancer.X3 BIFURCATION ANALYSISBifurcation analysis is used to ‘map’ out how the stability of a system depends on changes in its parameters.We use the numerical continuation package AUTO [4] to investigate the stability of the steady states detailedabove as the dimensionless parameters for the mass µ, eccentricity δ, external damping ζ , and internal dampingβ are varied. Figure 2 shows the stability regions in ( Ω , µ ) -space (a), ( Ω,δ)-space (b), ( Ως , )-space (c), and( Ωβ , )-space (d). In each case, when fixed, the other parameters are µ = 0.05, β = δ = ζ = 0.01. Dark shadedregions in Fig. 2 correspond to parameter values where the balanced state is stable. Lighter regions correspond toparameter values where the coincident equilibrium is stable. No shading corresponds to regions of instability.Dashed curves indicate lines of Hopf bifurcations, marked by a H. In other words, crossing one of these curves,due to the variation in the parameters, will result in an oscillatory instability. In particular, the Hopf curvesbordering the regions with dark shading result in an oscillatory instability of the stable balanced state and a3 Paper-ID 74

subsequent stable periodic solution. The solid curves indicated in Fig. 2, labeled SN, indicate lines of saddlenodebifurcations. As a line of saddle-node bifurcations is crossed a coincident equilibrium is born (whencrossing into the light shaded region) or destroyed (when crossing into a region of instability). Finally, the curvein Fig. 2(a) marked PF indicates a line of pitch-fork bifurcations in which an additional steady state is born ordestroyed. For example, as µ is increased from zero, the balanced state is born at the curve PF. This balancedstate, born at PF, is unstable for Ω< 1.3 but stable otherwise, that is, crossing from the light shaded region tothe dark shaded region for large values of Ω.0.08µ0.06H 1(a)H 2 + H 1PF 1,2 +0.1δ(b)H 1H 10.04SN 2 ±0.050.02SN 2 ±SN 2 ± SN 2 ±PF 1,2 −H 100.5 1 1.5 2 2.5Ω00 1 2 3Ω0.5(c)0.5(d)ζβH 1H 10 1 2 3H 10.250.25SN 2 ± SN 2 ±SN 2 ±SN 2 ±H 1000 1 2 3ΩΩFigure 2: Two-parameter bifurcation diagrams showing regions of stability in parameter space of the balancedsteady state (dark shading), and the coincident steady state (light shading); see text for full details.A main conclusion to be drawn from Fig. 2 is that for rotation speeds of approximately Ω> 2.5 the balancedstate is always achievable. Furthermore, Fig. 2(a) shows a ‘wedge’ of stability of the balanced state for Ω = 1,the rotation speed corresponding to the first resonant frequency ω n . This is the rotation speed at which oneobserves greatest radial vibration in a standard eccentric (Jeffcott) rotor. The fact that balance may be achievablefor rotation speeds at and close to the resonant frequency ω n is of great physical interest.However, we must note that the ADB is highly nonlinear. This has the consequence that while a balancedsteady state may be achievable, it may not be the unique solution. As an example, Fig. 3 shows three coexistingsolutions for Ω= 4 , µ = 0.05 , δ = 0.01 , ζ = 0.01 and β = 0.01. Figure 3(a) shows the dynamics of the rotorwith the ADB, as a black trajectory, for initial conditions x( 0) = x( 0) = y( 0) = y ( 0)= 0, φ 1( 0) = φ 2( 0)= 0and φ ( 0) = − φ ( 0)= π 2. The ADB is shown to balance the eccentric rotor, that is, the vibration1 22 2r= x + y is seen to go to zero. For comparison, the grey trajectory shows the dynamics of the eccentricrotor without the ADB. However, two more dynamic responses can also be observed for the same parametervalues. These are shown in Figs. 3(b) and (c). In both cases, the attracting dynamics is a periodic oscillation ofthe rotor with the ADB. This results in a greater vibration than the rotor without the ADB. To obtain Figs. 3(b)4 Paper-ID 74

ε( A) { z ( A E): E }Λ = ∈Λ + ≤εwhere Λ is the spectrum (set of eigenvalues) of A+E.Figure 4 shows the ε-pseudospectra of the balanced steady state computed using the Matlab package EIG-TOOL [16]. Parameters were fixed at µ = 0.05, δ=ς = β = 0.01 , and from Fig. 4(a) to (d) the rotation speedΩ= 2.5, 3.0, 3.5, 4.0 , respectively. From outermost to innermost, the contours shown in each panel of Fig. 40 1correspond to ε-pseudospectra for ε= 10 , 10 − 2, 10 − 3, and 10 − . These contours are centered around theeigenvalues, indicated by a black dot. Note that the linearized system is eight-dimensional, the additional foureigenvalues and corresponding pseudospectra are symmetric counterparts of those shown for I ( z) > 0. Thesmallest value of ε for which a contour touches the imaginary axis is also known as the stability radius of thelinearized system. From Fig. 4(a) to (d) the stability radii are given by ε= 1.32075× 10 −3 , ε= 1.64948× 10 −3 ,ε= 1.40770× 10 −3 , and ε= 1.076729× 10 −3 . The first three of these correspond to pseudospectra contourssurrounding the eigenvalue closest to the real axis. However, in Fig. 4(d) the stability radius is given by thepseudospectra contour surrounding the third eigenvalue from the real axis. In other words, as the rotation speedΩ is increased, the eigenvalue most sensitive to perturbation changes from being the one with least imaginarypart to being one with a large imaginary part. Furthermore, the order of the sensitivity of the other eigenvaluesalso changes [6]. This change in sensitivity affects the transient dynamics of trajectories of the system as thesteady state is approached.I(z)0(a)I(z)0(b)−2−2−4−4−6−6−1 0 R(z) 1−1 0 R(z) 1I(z)0−2(c)0(d)I(z)−2−4−4−6−6−1 0 R(z) 1−1 0 R(z) 1Figure 4: Pseudospectra of the balanced steady state of the linearized system describing the ADB; see text forfull details.6 Paper-ID 74

Figure 5 shows transient dynamics of the norm of the matrix exponentialAte of the linearized system A(around the balanced steady state solution) for parameter values corresponding to those used in Fig. 4. In eachpanel a large transient growth is observed which decays to the stable, balanced steady state solution. Thepseudospectra of Fig. 4 give us a lower bound on the size of this transient growth. Specifically, if an ε-pseudospectra contour stretches a distance η into the right-half complex plane, a lower bound on the size of thetransient growth is given by η/ε . In other words, as the system becomes more non-normal, one observes alarger transient growth. This is evident in Fig. 5 where an increase in rotation speed Ω (which increases the nonnormalityof the system) is shown to lead to an increase in the initial transient growth. Furthermore, fromFig. 5(a) to (d) the transient is seen to undergo oscillatory decay at differing frequencies. The frequency of thisdecay is directly related to the sensitivity of the eigenvalues, identified above. Specifically, as the rotation speed20‖e At ‖(a)1000 250 500 750 t 100020‖e At ‖(b)1000 250 500 750 t 100020‖e At ‖(c)1000 250 500 750 t 100020‖e At ‖(d)1000 250 500 750 t 1000Figure 5: Transient behaviour of the norm of the matrix exponential of the linearized system describing theADB.7 Paper-ID 74

Ω is increased the eigenvalues most sensitive to perturbation change from those with small imaginary part tothose with large imaginary part. Moreover, this leads to a change from transient dynamics which exhibit a slowfrequency response to transient dynamics which exhibit rapid, high frequency oscillations. Note that the mostsensitive eigenvalue need not be the rightmost, which for our system is the eigenvalue with least imaginary part.Hence, unlike for normal systems, the transient dynamics do not decay at a rate defined by the real part of therightmost eigenvalue but, if it is more sensitive, may decay at a rate defined by the leftmost eigenvalue; seeFig. 5(d). In [6] it is shown that increasing the rotation speed Ω further results in larger transient growths andhigher frequency oscillations to the balanced state. Furthermore, in [6] we show how this transient growth anddecay is directly related to the dynamics of the full nonlinear system. In particular, we show how increasedinitial transient growth may cause the trajectory to escape the basin of attraction of the steady state, resulting inpossible attraction to more complex, periodic dynamics.5 CONCLUSIONSThis paper has demonstrated the nonlinear bifurcation analysis of an automatic dynamic balancingmechanism for a simple Jeffcott rotor model. Bifurcation diagrams show that dynamic balance can be achievedprovided the mass of the balancing balls is sufficiently high. However it is also clear that any automaticbalancing system must be carefully designed, since there are large regions of parameter space where there are nostable solutions. Of particular interest is the region where the shaft speed is close to a critical speed, since theresponse to any residual unbalance will be high at these speeds. In this case obtaining a stable solution requires avery careful choice of system parameters, and work is continuing to increase the size of these stable regions.Transient dynamics also play a crucial role in a complete understanding of this highly nonlinear mechanism. Inmany applications it is important to know which frequencies may be excited by a perturbation. In particular, thedegree and nature of the transient response to perturbation can be highly unpredictable. The transient responseand sensitivity to perturbation of a rotor fitted with an ADB has been investigated by computing thepseudospectra of the first-order linearization. The results have shown that the system exhibits a large transientgrowth phenomena and that this may be predicted using the pseudospectrum. Much more work is required,utilizing the methods presented in this paper, in order to optimize the design for a particular physical application.Experimental work is also in progress to validate the nonlinear models of the rotor and ADB.ACKNLOWLEDGEMENTThe authors acknowledge the support of the Engineering and Physical Sciences Research Council throughgrant GR/S35684/01, ‘Pseudospectra, Uncertainty, Vulnerability and Bifurcation in Structural Mechanics’.Friswell acknowledges the support of a Royal Society-Wolfson Research Merit Award.REFERENCES[1] Adolfsson, J. (2001): Passive Control of Mechanical Systems: Bipedal Walking and Autobalancing.PhD Thesis, Royal Institute of Technology, Stockholm.[2] Chao, P.C.P., Huang, Y.-D. and Sung, C.-K. (2003): Non-planar dynamic modeling for the optical diskdrive spindles equipped with an automatic balancer. Mechanism and Machine Theory, 38, pp. 1289-1305.[3] Chung, J. and Ro, D.S. (1999): Dynamic analysis of an automatic dynamic balancer for rotatingmechanisms. Journal of Sound and Vibration, 228, pp. 1035-1056.[4] Doedel, E., Champneys, A., Fairgrieve, T., Kuznetsov, Yu., Sandstede, B. and Wang, X. (1997): AUTO97: Continuation and bifurcation software for ordinary differential equations.http://indy.cs.concordia.ca/auto/.[5] Green, K., Champneys, A.R. and Lieven, N.A.J. (2006): Bifurcation analysis of an automatic dynamicbalancing mechanism for eccentric rotors. Journal of Sound and Vibration, 291, pp. 861-881.[6] Green, K., Champneys, A.R. and Friswell, M.I. (2006): Analysis of the transient response of anautomatic dynamic balancer for eccentric rotors. International Journal of Mechanical Sciences, 48(3),pp. 274-293.[7] Green, K., Champneys, A.R., Friswell, M.I. and Muñoz, A.M. (2006): Investigation of a multi-ballautomatic dynamic balancing mechanism for eccentric rotors. Philosophical Transactions Royal SocietyA, submitted.[8] Higham, D.J. and Trefethen, L.N. (1992): Stiffness of ODEs. BIT, 33, pp. 285-303.[9] Huang, W.-Y., Chao, C.-P., Kang, J.-R. and Sung, C.-K. (2002): The application of ball- type balancersfor radial vibration reduction of high-speed optic drives. Journal of Sound and Vibration, 250, pp. 415-430.8 Paper-ID 74

[10] Kim, W. and Chung, J. (2002): Performance of automatic ball balancers on optical disc drives. IMechEJournal of Mechanical Engineering Science, 216, pp. 1071-1080.[11] Lee, J. and Van Moorhem, W.K. (1996): Analytical and experimental analysis of a self-compensatingdynamic balancer in a rotating mechanism. Transactions of ASME, 118, pp. 468-475.[12] Rajalingham, R. and Rakheja, S. (1998): Whirl suppression in hand-held power tool rotors using guidedrolling balancers. Journal of Sound and Vibration, 217, pp. 453-466.[13] SKF Autobalance Systems. Dynaspin. http://dynaspin.skf.com[14] Trefethen, L.N. (1999): Computation of pseudospectra. Acta Numerica, 8, pp. 247-295.[15] van de Wouw, N., van den Heuvel, M.N., van Rooij, J.A. and Nijmeijer, H. (2005): Performance of anautomatic ball balancer with dry friction. International Journal of Bifurcation and Chaos, 15, pp. 65-82.[16] Wright, T.G. (2002): EIG-TOOL: a graphical tool for nonsymmetric eigenproblems. Oxford University.http://www.comlab.ox.ac.uk/pseudospectra/eigtool.9 Paper-ID 74