Mode-Locking , Quasi-Period and Chaos of Rotors Mounted on Nonlinear Bearings

This study is to investigate the dynamics of a rotor, with a single degree of freedom (SDOF), mounted on nonlinear bearings. This system has piecewise-linear stiffness and is subjected to a forcing excitation due to residual mass imbalance as well as a parametric one due to an axial periodic thrust. The frequencies for each individual parametric and forcing excitations are not equivalent, neither do they have a ratio of two simple integers. By using the fourthorder Runge–Kutta method a J-integral model, this strongly nonlinear system can be estimated for various parameters. The J-integral bifurcation can be analyzed by using the Poincare maps, the frequency spectra, the response waveforms, and the Lyapunov exponents in order to illustrate the jump phenomenon, the frequency-locking, and the routes to chaos. Furthermore, the intra-systematic relationship can be determined by the frequencies of spontaneous sidebanding clusters.


INTRODUCTION
The nonlinear dynamics of the rotor-bearing sys- tems is an important and practical problem because those systems frequently exist in many fields.The piecewise-linear systems having components with clearance have been introduced by the following ways.Firstly, the clearance factor is necessary at the stages of designing or assembling.Secondly, it can be caused by wear since the system has experienced intermittent motion for contacting and separating from other components.Last, the system might be connected through a backlash.These systems usually exhibit phenomena such as multiple solu- tion regimes, superharmonic, subharmonic, quasi- periodic, and chaotic solutions.
The existance of nonlinear oscillations in the field of high-pressure and high-speed turbomachinary, motivates a lot of research studies.Experimental works have been conducted.For examples, obser- vations including the subcritical superharmonic response, the supercritical subharmonic response, the spontaneous sidebanding, and the chaotic phenomenon have been shown in waveforms, fre- quency spectra, phase plane diagrams, and water- fall charts by Ehrich (1988, 1991, 1992a,b, 1995).
The conventional methods for analyzing piece- wise-linear systems which have been subjected to the harmonic force can be analyzed as follows: (a) Padmanabhan and Singh (1995) have adapted an analytical method by using the parametric con- tinuation scheme based on the shooting method.Although the occurrence of a period-doubling bifurcation which can be determined by the eigen- values of the period-two has been resolved; how- ever, the periodic solutions are hardly obtained.(b) Choi and Noah (1988, 1992), Kim and Noah (1991), Lau and Zhang (1992), and Narayanan and Sekar (1995) have used the harmonic balance method to determine the coefficients ofsolutions either directly or iteratively; however, these coefficients corre- sponding to the synchronous or multiple forcing frequency do not seem to converge easily to finite terms.Furthermore, the influence of initialcondi tions on steady-state solutions has been no longer unambiguous.Neither have these solutions been identical to the ones for the original equations of motion.Consequently the solutions were approx- imate ones only.(c) Mahfouz and Badrakhan  (1990a,b) have used the numerical integration method to solve two nonlinear systems which have merely the forcing excitations with harmonic fre- quencies.And they can determine the harmonic, the subharmonic, and the chaotic motions for various parameters.
This study is to investigate the dynamics of a SDOF rotor having piecewise-linear stiffness.Also, the system is subjected to a parametric excitation and a forcing excitation with a nonmultiple fre- quency ratio.This system is described by a second- order differential equation and is solved by the fourth-order Runge-Kutta method in order to determine the stationary J-integral solutions for various parameters.The J-integral bifurcation can be analyzed by using the Poincar6 maps, the fre- quency spectra, the response waveforms, the phase portraits, and the Lyapunov exponents in order to illustrate the jump phenomenon, the frequencylocking, and the routes to chaos.

EQUATIONS OF MOTION
In a SDOF rotor as shown in Fig. 1, a typical one-mode oscillation is considered, and it can be governed by the following equation d2y dy m-7 + c-7 + k, (1 -/*2 sin (co2 t) )y + F(y) p sin(col t), in which piecewise-linear restoring force F(y) is generated by a clearance c, and is given by 0 for y _< e, (2) F(y)-k2(y-sgny) for Yl >, where m is the lump mass of a rotor, c is the linear damping coefficient, kl and k2 are the shaft stiffness and the restoring stiffness respectively, co and co2 are the forcing and the parametric frequencies, res- pectively, and p and k/*2 are amplitudes of the forcing and the parametric excitations respectively.The parametric excitation is caused by an axial

MODE-LOCKING
193 thrust, and the forcing excitation is caused by an unbalance force of the rotor.
(1) and (2) gives 2 + 2 + (1 #: sin(u2r))x +f(x) #1 sin (4) where The initial conditions play an important role of determining the small orbit and large orbit; more- over, the multi-solutions for this kind system may coexist.In real systems, the actual disturbances cannot always be expressed in terms of the initial conditions.By using computer analysis with various initial conditions, the steady-state solutions of this system can be obtained and classified by J- integral.In general, the calculation of the J-integral for various initial conditions gives the number of values corresponding to the subharmonic number about the steady-state response.The large ampli- tudes should be distinguished from those small amplitudes which can be used to determine the occurrences of large orbits or small ones.
(5) and the superscript dot denotes the differentiation with respect to r.
Since the above equation has a strong nonlinear restoring force and the forcing frequency has a nonmultiple value via the parametric frequency, there has been no analytical method of solution.
Thus, the fourth-order Runge-Kutta method is utilized.In the process of numerical integration, the value of N is chosen from 200 to 600 for the time interval Ar 27r/(uN).The first forcing periods, which have long terms, were not recorded in order to avoid transient solutions.At the same time, a frequency spectrum is determined for the 400 periods state-steady solutions; also, the Poincar6 map is obtained by sampling the stationary solu- tions over 2000 points (x, 2) for one forcing period (T= 27r/ul).
An effective integral of the response waveforms has been verified as a method for determining whether or not a transient solution can be lead to one of the steady solutions (Kang et al., 1998).This integral value monitored on the computer has been utilized in this study is 2r/ul J-"]t.\"/-..l 2 dr.J0 (6) 3. MODE-LOCKED MOTION AND QUASI-PERIODIC MOTION The rotor system concerned in this study is char- acterized by two frequencies belonging to forcing and parametric excitations.The pattern of points on a Poincar6 map show the dependence on the numerical relationship between the two frequencies.
If the ratio of the two frequencies can be expressed as the ratio of two integers (that is, as a "rational fraction,"), then the Poincar6 section will consist of a finite number of points.This type of motion is often called frequency-locking, mode-locked, or phase-locking motion because one of the frequencies is locked over a finite control parameter range, and is a multiple integer of the other.
If the frequency ratio is irrational, the points on a Poincar6 map will fill a continuous curve in the Poincar6 plane.The motion can be described as quasi-periodic because the motion never repeats itself.However, the motion is not chaotic; rather, it is consisted of two or more periodic components whose presence can be determined by measuring the frequency spectrum of motion.In order to detect the difference between quasi-periodic and chaotic motions, the Lyapunov exponents are calculated by using the algorithm devised by Wolf et al.   (1985).When the system has chaotic behavior, the calculated value of the most positive Lyapunov exponent is plotted as a function of the total average time which can be measured in units of the forcing period.

NUMERICAL EXAMPLES
For a rotor system of parameters at /)2 3, k--7, #1 =0.25, #2--0.17,and ,-0.07;J-integral versus rotating speed u is shown in Fig. 2. Small orbits occur in a range of 1.122 < ul < 1.6, large orbits occur in a range of 0.8 < ul < 1.5, and the coexisting motion occurs in a range of 1.1 < Ul < 1.5.For the small orbit, one J-integral value indicates P1 motion.For the large orbit, a single J-integral value appeared in a range of 0.8 < ul < 1.16 indicates P1 motion; two J-integral values appeared in a range of 1.16 < ul < 1.2 indicate P2 motion; four J-integral values appeared in a range 1.2 < ul < 1.208 indicate P4 motion; and a range of speed over 1.21 appeared many J-integral values indicates chaotic motion.When the behavior is chaotic, the J-integral values seem to be smeared over the complete range of observed values.In particular, irregular J-integrals are scattered in the range of 1.1 < u < 1.122 which then indicate tangle orbit.
Since the frequency ratio is irrational, the Poincar6 map shows an almost-closed curve for a periodic response.Figure 4(a)-(p) show the frequency spectra and the Poincar6 maps for various rotating speeds in Fig. 2. The Poincar6 maps of Fig. 4(a) and (b) are shown as closed curves corresponding to quasi-periodic motion, and eleven points corre- spond to period one motion for u2/ul value being 3// and 30/11, respectively.Figure 4(c), (d), (f), (h) show P1, P2, P4, and P16 motions corre- sponding to values of 1, 2, 4, 16 for the J-integral respectively.Figure 4(i) and (j) show both small P1 motion and chaotic motions.In the rotating range with chaotic behavior, P2 and P4 motions are shown in Fig. 4(k) and (1).The tangle P3 orbit and large P1 orbit are coexisting at ul 1.101 which can be shown in Fig. 4(m) and (n). Figure 4(o) and (p) show P3 motions on tangle orbit range.
The frequency spectra in Fig. 4 show that all peak frequencies can be described by (7) where An= lug-2 ul/n, with n integers, and r/l, n2=0, 1,2,... Thus, these motions are combinations of the nlth superharmonic components of ue, the (nz/n)th subharmonic components of uz 2u as well as the harmonic components of ul.One may denote these motions by period n or Pn because of the corre- sponding number of the peak frequencies in each cluster.
The Feigenbaum universality constant has been derived for one-dimensional maps; however, many high-dimensional systems exhibit perioddoubling bifurcation which leads to chaos and can be observed of having a value of (4.66920161...).

CONCLUSION
Oscillations of a strongly nonlinear rotor system which is subjected to both a forcing excitation due to an imbalance as well as a parametric excitation due to all axial periodic thrust has been investi- gated.The J-integral has been utilized to construct the bifurcations; thus, the small orbits and the large ones can be distinguished.Moreover, the jump phe- nomena, the subharmonics of various orders, the frequency-locking, the period n motion, the perioddoubling cascade which leads to chaos can be distinguished.A:vn [u2-2u[/n (dimensionless)   clearance (m) Economic and environmental factors are creating ever greater pressures for the efficient generation, transmission and use of energy.Materials developments are crucial to progress in all these areas: to innovation in design; to extending lifetime and maintenance intervals; and to successful operation in more demanding environments.Drawing together the broad community with interests in these areas, Energy Materials addresses materials needs in future energy generation, transmission, utilisation, conservation and storage.The journal covers thermal generation and gas turbines; renewable power (wind, wave, tidal, hydro, solar and geothermal); fuel cells (low and high temperature); materials issues relevant to biomass and biotechnology; nuclear power generation (fission and fusion); hydrogen generation and storage in the context of the 'hydrogen economy'; and the transmission and storage of the energy produced.As well as publishing high-quality peer-reviewed research, Energy Materials promotes discussion of issues common to all sectors, through commissioned reviews and commentaries.The journal includes coverage of energy economics and policy, and broader social issues, since the political and legislative context influence research and investment decisions.

F
FIGUREPhysical model of an SDOF rotor.

S
SU UB BS SC CR RI IP PT TI IO ON N I IN NF FO OR RM MA AT TI IO ON N Volume 1 (2006), 4 issues per year Print ISSN: 1748-9237 Online ISSN: 1748-9245 Individual rate: £76.00/US$141.00Institutional rate: £235.00/US$435.00Online-only institutional rate: £199.00/US$367.00For special IOM 3 member rates please email s su ub bs sc cr ri ip pt ti io on ns s@ @m ma an ne ey y. .cco o. .uuk k E ED DI IT TO OR RS S D Dr r F Fu uj ji io o A Ab be e NIMS, Japan D Dr r J Jo oh hn n H Ha al ld d, IPL-MPT, Technical University of Denmark, Denmark D Dr r R R V Vi is sw wa an na at th ha an n, EPRI, USA F Fo or r f fu ur rt th he er r i in nf fo or rm ma at ti io on n p pl le ea as se e c co on nt ta ac ct t: : Maney Publishing UK Tel: +44 (0)113 249 7481 Fax: +44 (0)113 248 6983 Email: subscriptions@maney.co.uk or Maney Publishing North America Tel (toll free): 866 297 5154 Fax: 617 354 6875 Email: maney@maneyusa.comFor further information or to subscribe online please visit w ww ww w. .mma an ne ey y. .cco o. .uuk k C CA AL LL L F FO OR R P PA AP PE ER RS S Contributions to the journal should be submitted online at http://ema.edmgr.comTo view the Notes for Contributors please visit: www.maney.co.uk/journals/notes/emaUpon publication in 2006, this journal will be available via the Ingenta Connect journals service.To view free sample content online visit: w ww ww w. .i in ng ge en nt ta ac co on nn ne ec ct t. .cco om m/ /c co on nt te en nt t/ /m ma an ne ey y E EN NE ER RG GY Y M MA AT TE ER RI IA AL LS S Materials Science & Engineering for Energy Systems