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An example of the chaotic picture, in theform of orbit and Poincare's map, is shown in Figure 4, where u/u1"1)77. The Poincare'smap was obtained by recording data from period 2001 to period 10,000 and the maximumLyapunov exponent was computed to be 3)08. Figure 5 is the orbit and Poincare's map foru/u1"3)54. The attractor is a clear indication of chaotic motion in which stretching andfolding can be clearly seen and the information dimension was calculated to be 1)21. Inorder to see if there is any rub happening between the rotor and the left bearing, the orbit atthe left-bearing position is also shown in Figure 5. The chaotic motion remains untilu/u1"6)06209 and at u/u1"6)06210 the motion suddenly becomes periodic. Figure 6shows two orbits at the disk position and the left-bearing position for u/u1"5)66. Figure 7is the Poincares map and the orbit at the disk position and the orbit at the left-bearingposition for u/u1"6)06. An interesting phenomenon that can be seen is that the disk is speed and the motion settles into periodic "nally. This is di!erent from the route ofintermittence found in the rub-impact system[10]. In conclusion, when the rotating speed isused as the control parameter the motion enters into the chaotic region in thequasi-periodic route and leaves in a route of intermittence.We now take imbalance as the control parameter to see various forms of vibration andthe route to or out of chaos. In this case, we "x u/u1"3)54. Figure 9 is the Poincare'smapat the disk position, and orbits at both the disk position and the left-bearing position foru"0)0220947498313 mm and the motion can be found periodic. Figure 10 is foru"0)0220947498314 mm and the motion becomes chaotic. The maximum Lyapunovexponent was calculated as 0)3302 in this case. It can be seen that the route to chaos isa kind of period-to-chaos, or crisis, a route mentioned recently in several publications [10,14]. If we further increase the imbalance, various forms of chaotic vibrations can be found.Figure 11 is the Poincare's map and the orbit at the disk position whereu"0)1699999999 mm. The attractor is very loose and the information dimension wascalculated to be 1)38. Figure 12 is for u"0)17 mm where the motion has become The fourth order Runge}Kuttamethod is used to integrate the governing equations. Veryrich forms of periodic, quasi-periodic and chaotic vibrations can be observed. Three types ofroute to or out of chaos, that is, period-to-chaos, quasi-periodic route and intermittence, arefound. The analytical results are very important for diagnosing the pedestal looseness inrotating machinery.ACKNOWLEDGMENTSThis research is supported "nancially by a project from the Ministry of Science andTechnology in China (Grant No. PD9521908Z2), National Natural Science Foundation ofChina (Grant No. 19990510) and Tsinghua University Basic Research Foundation (GrantNo. Jc1999045).REFERENCES1. P. GOLDMAN and A. MUSZYNSKA 1991 Rotating Machinery and <ehicle Dynamics, AmericanSociety of Mechanical Engineers DE-<ol. 35,11}17. Analytical and experimental simulation ofloose pedestal dynamic e!ects on a rotating machine vibrational response.2. A. MUSZYNSKA and P. GOLDMAN 1995 Chaos Solitons & Fractals 5, 1683}1704. Chaoticresponses of unbalanced rotor-bearing stator systems with looseness or rubs.3. E. J. GUNTER,L.E.BARRETT and P. E. ALLAIRE 1977 Journal of ¸ubrication ¹echnology,¹ransactions of the American Society of Mechanical Engineers 99,57}64. Design of nonlinearsqueeze-"lm dampers for aircraft engines.4. F. CHU and Z. ZHANG 1998 Journal of Sound and <ibration 210,1}18. Bifurcation and chaos ina rub-impact Je!cott rotor system5. Y. B. KIM and S. T. NOAH 1991 Journal of Applied Mechanics, ¹ransactions of the AmericanSociety ofMechanical Engineers 58, 545}553. Stability and bifurcation analysis of oscillators withpiecewise-linear characteristics: a general approach.6. F. CHU and R. HOLMES 1998 Computer Methods in Applied Mechanics and Engineering 164,363}373. E$cient computation on nonlinear responses of a rotating assembly incorporating thesqueeze-"lm damper.7. C. KAAS-PETERSEN 1985 Journal of Computational Physics 58, 395}408. Computation ofquasi-periodic solutions of forced dissipative systems.