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 <strong>of</strong> 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 <strong>of</strong> the full nonlinear system. In particular, we show how increasedinitial transient growth may cause the trajectory to escape the basin <strong>of</strong> attraction <strong>of</strong> the steady state, resulting inpossible attraction to more complex, periodic dynamics.5 CONCLUSIONSThis paper has demonstrated the nonlinear bifurcation analysis <strong>of</strong> an automatic dynamic balancingmechanism for a simple Jeffcott rotor model. Bifurcation diagrams show that dynamic balance can be achievedprovided the mass <strong>of</strong> the balancing balls is sufficiently high. However it is also clear that any automaticbalancing system must be carefully designed, since there are large regions <strong>of</strong> 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 <strong>of</strong> system parameters, and work is continuing to increase the size <strong>of</strong> these stable regions.Transient dynamics also play a crucial role in a complete understanding <strong>of</strong> this highly nonlinear mechanism. Inmany applications it is important to know which frequencies may be excited by a perturbation. In particular, thedegree and nature <strong>of</strong> the transient response to perturbation can be highly unpredictable. <strong>The</strong> transient responseand sensitivity to perturbation <strong>of</strong> a rotor fitted with an ADB has been investigated by computing thepseudospectra <strong>of</strong> the first-order linearization. <strong>The</strong> 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 <strong>of</strong> the rotor and ADB.ACKNLOWLEDGEMENT<strong>The</strong> authors acknowledge the support <strong>of</strong> the Engineering and Physical Sciences Research Council throughgrant GR/S35684/01, ‘Pseudospectra, Uncertainty, Vulnerability and Bifurcation in Structural Mechanics’.<strong>Friswell</strong> acknowledges the support <strong>of</strong> a Royal Society-Wolfson Research Merit Award.REFERENCES[1] Adolfsson, J. (2001): Passive Control <strong>of</strong> Mechanical Systems: Bipedal Walking and Autobalancing.PhD <strong>The</strong>sis, Royal Institute <strong>of</strong> 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 <strong>The</strong>ory, 38, pp. 1289-1305.[3] Chung, J. and Ro, D.S. (1999): Dynamic analysis <strong>of</strong> an automatic dynamic balancer for rotatingmechanisms. Journal <strong>of</strong> 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 s<strong>of</strong>tware for ordinary differential equations.http://indy.cs.concordia.ca/auto/.[5] Green, K., Champneys, A.R. and Lieven, N.A.J. (2006): Bifurcation analysis <strong>of</strong> an automatic dynamicbalancing mechanism for eccentric rotors. Journal <strong>of</strong> Sound and Vibration, 291, pp. 861-881.[6] Green, K., Champneys, A.R. and <strong>Friswell</strong>, M.I. (2006): Analysis <strong>of</strong> the transient response <strong>of</strong> anautomatic dynamic balancer for eccentric rotors. International Journal <strong>of</strong> Mechanical Sciences, 48(3),pp. 274-293.[7] Green, K., Champneys, A.R., <strong>Friswell</strong>, M.I. and Muñoz, A.M. (2006): Investigation <strong>of</strong> a multi-ballautomatic dynamic balancing mechanism for eccentric rotors. Philosophical Transactions Royal SocietyA, submitted.[8] Higham, D.J. and Trefethen, L.N. (1992): Stiffness <strong>of</strong> ODEs. BIT, 33, pp. 285-303.[9] Huang, W.-Y., Chao, C.-P., Kang, J.-R. and Sung, C.-K. (2002): <strong>The</strong> application <strong>of</strong> ball- type balancersfor radial vibration reduction <strong>of</strong> high-speed optic drives. Journal <strong>of</strong> Sound and Vibration, 250, pp. 415-430.8 <strong>Paper</strong>-ID 74
[10] Kim, W. and Chung, J. (2002): Performance <strong>of</strong> automatic ball balancers on optical disc drives. IMechEJournal <strong>of</strong> Mechanical Engineering Science, 216, pp. 1071-1080.[11] Lee, J. and Van Moorhem, W.K. (1996): Analytical and experimental analysis <strong>of</strong> a self-compensatingdynamic balancer in a rotating mechanism. Transactions <strong>of</strong> ASME, 118, pp. 468-475.[12] Rajalingham, R. and Rakheja, S. (1998): Whirl suppression in hand-held power tool rotors using guidedrolling balancers. Journal <strong>of</strong> Sound and Vibration, 217, pp. 453-466.[13] SKF Autobalance Systems. Dynaspin. http://dynaspin.skf.com[14] Trefethen, L.N. (1999): Computation <strong>of</strong> 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 <strong>of</strong> anautomatic ball balancer with dry friction. International Journal <strong>of</strong> 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 <strong>Paper</strong>-ID 74
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