Study Of The Nonlinear Periodic Motions And Analytical Bifurcation Characteristics In Rotor-bearing-seal Systems

Rotating disks,shafts and bearings are important machine components widely used in mechanical,textile,transportation,aerospace,chemical and electrical engineering.Rotors are key components in rotating machines and their safety,stability and reliability in rotor systems have significant influence on the safe operation of the entire mechanical systems.As China's economics develop so fast,industry has put new requirements to the research of rotor dynamics,especially high speed nonlinear rotor systems.As survey proofs,nonlinear restoring forces and nonlinear sealing forces of rotor system have huge impact on the dynamic characteristics of rotor-bearing-seal systems.Currently,we don't have high accurate analytical method for periodic motions in nonlinear rotor system.Numerical methods have difficulties for obtaining accurate and continuous nonlinear bifurcation trees.Research for unstable periodic motions of rotor systems is not adequate.This paper focuses on building accurate theoretical models,predicting the periodic orbits,determining the analytical bifurcation points,solving the independent periodic solutions,performing numerical simulations and experimental verification of rotor-bearing-seal systems.This research will be mainly on the nonlinear periodic motions and bifurcation characteristics of the rotor-bearing-seal systems.The contents are summarized as:Firstly,aiming at the odd order subharmonic and superharmonic resonant phenomenon which traditional methods have difficulties to explain,discrete mapping method is implemented to a nonlinear elastic supported rotor system.This discrete mapping method discretizes the continuous nonlinear rotor system in small time intervals and forms a mapping chain of rotor sub motions.Together with periodicity conditions,the close loop mapping structure is constructed for periodic motions of nonlinear elastic supported rotor systems.Computer programs will be coded for the computation of the periodic solutions and vibration characteristics of the rotor system.Based on the chosen parameters,the periodic solutions can convergent to high accuracy.Displacement and velocity are studied with the variation of rotating speed.The elastic support rotor system has a sequential periodic motion from independent period-1motion to chaos which is demonstrated by the series of independent period-1 motion to independent period-3 motion to independent period-5 motion.Stability and bifurcation conditions are determined for the sequent nonlinear periodic motions.Results indicate the periodic orbits are complex in such an elastic supported rotor system.High order steady state periodic solutions convergent slowly.There are many Saddle Node bifurcations.All the odd number periodic motions consist of stable and unstable solutions and the stable and unstable solutions are connected,and they form close loops in certain rotating speed.The odd order subharmonic and superharmonic resonant phenomenon exists in such a sequence of independent periodic motions.Based on the general Reynolds equation,the hydrodynamics of oil film in journal bearings are investigated.The influence of nonlinear journal displacement and velocity on the increment of oil film forces is studied through database analysis,and the nonlinear model of oil film forces with thirteen variables is obtained.The linear oil film forces,database oil film model and thirteen-variable oil film forces are discussed.The effects of journal displacement and velocity on the oil film force increments and residuals are obtained.Based on such nonlinear oil film force model,a rotor-journal bearing system is built.The periodic motions,eigenvalue dynamics and period-doubling characteristics of such a rotor-journal bearing system are analyzed.The results from forward frequency sweep are different from the ones from backward frequency sweep.The bifurcation points will not change by sweeping frequency forward.But the period-doubling bifurcation will turn to saddle node bifurcation when sweeping frequency backward.Additionally,the analytical bifurcation trees of period-1 to period-4 motion are predicted in the rotor-journal bearing system.Thus,from period-doubling bifurcation,the period-1 motion is generated to period-2 motion.And from period-doubling bifurcation,the period-2 motion is further generated to period-4motion.But period-4 motion is turned to period-2 motion by saddle node bifurcation and period-2 motion is also turned to period-1 motion.As for the coupling effects of brush and rotor,contact mechanical model of bristle and disk is achieved based on elasticity.The expression of the brush restoring forces is obtained through the interpolation of the cycled brush forces.The air flow sealing forces are neglected since it is much smaller than the restoring forces of the bristle.The rotor-seal system model is built based upon the such analysis.The nonlinear vibration of such a rotor-seal system is studied with different rotating speed and eccentricities.Nonlinear displacement and velocity varying with rotating speed are obtained with different eccentricities.The influence of period-doubling bifurcation and Saddle node bifurcation on the nonlinear periodic motions of rotor-seal system is discussed.The saddle node bifurcation parameter map and period-doubling bifurcation parameter map are also presented.Results indicate the nonlinear resonant peaks arise in the corresponding linear resonance speed range.When eccentricity increases,the number of bifurcation points increases and speed ranges of unstable periodic motions becomes wide.The period-doubling bifurcation parameter map can be used for designing rotor system with smaller vibration and saddle node bifurcation map can be used to avoid nonlinear jumping phenomenon.A rotor-journal bearing-seal system is constructed based on the thirteen-variable nonlinear oil film forces and nonlinear seal forces.Aiming at the complicated analytical solutions,finite Fourier series with coefficients slowing varying with time is applied to simulate the nonlinear orbits of such a rotor-bearing-seal system.The original rotor-bearing-seal system is transformed to a new dynamic system of coefficients of Fourier series.The periodic solutions of original nonlinear system are turned to solve the equilibrium points of the dynamic system of coefficients.High precision and convergent program is designed for the complex vibration characteristics.The constant and harmonic terms of the periodic motions are obtained.Analytical solutions of periodic motions are achieved through general harmonic balance method.One route from period-1 motion to chaos is presented and the other route from period-3 motion to chaos is also investigated too.Based on generalized harmonic balance method,the route from period-1 motion to chaos is demonstrated by the bifurcation tree of period-1motion to period-2m motion and the route from period-3 motion to chaos is demonstrated by the bifurcation tree of period-3 to period-3×2m motion.For period-3and period-6 motion,the stable and unstable bifurcation branches connected with each other and they form close loop in limited speed range.Finally,mid-point scheme is studied and the corresponding equations and accuracy are obtained.Mid-point integration method is applied to verify the obtained results.From the elastic support characteristics,a testing rig with elastic rubber ring is designed for the elastic supported rotor system.For sliding bearing-rotor system,the testing rig with oil film bases and oil film shafts is constructed.The theoretical results are verified by the experimental results.Chaotic phenomenon of rotor system is discussed and the routes toward from periodic motions to chaos are released.An analytical method for bifurcation and chaos of such rotor-bearing-seal system is presented in the paper.There are three kinds of routes into chaos in such rotor-bearing-seal system.One starts from Saddle node bifurcation.The route evolves from period-m motions to period-(2m+1)till chaos.Second route starts from period-doubling bifurcation or Hopf bifurcation which generates continuous bifurcation trees from period-1 motions to period-2m motion till chaos.Last one starts from Saddle node bifurcation but the route is produced by period-doubling bifurcation or Hopf bifurcation.This route is also continuous and with the bifurcation tree of period-n motions to period-n×2m motion till chaos.This is the first time to discover such routes toward chaos in nonlinear rotor-bearing-seal system with high accuracy in China.