Nonlinear Dynamics Behaviors of a Rotor Roller Bearing System with Radial Clearances and Waviness Considered

A rotor system supported by roller beatings displays very complicated nonlinear behaviors due to nonlinear Hertzian contact forces, radial clearances and bearing waviness. This paper presents nonlinear bearing forces of a roller bearing under four-dimensional loads and establishes 4-DOF dynamics equations of a rotor roller bearing system. The methods of Newmark-β and of Newton-Laphson are used to solve the nonlinear equations. The dynamics behaviors of a rigid rotor system are studied through the bifurcation, the Poincar è maps, the spectrum diagrams and the axis orbit of responses of the system. The results show that the system is liable to undergo instability caused by the quasi-periodic bifurcation, the periodic-doubling bifurcation and chaos routes as the rotational speed increases. Clearances, outer race waviness, inner race waviness, roller waviness, damping, radial forces and unbalanced forces-all these bring a significant influence to bear on the system stability. As the clearance increases, the dynamics behaviors become complicated with the number and the scale of instable regions becoming larger. The vibration frequencies induced by the roller bearing waviness and the orders of the waviness might cause severe vibrations. The system is able to eliminate non-periodic vibration by reasonable choice and optimization of the parameters.

COMPENSATION CONTROL OF REAL-TIME UNBALANCE FORCE FOR ACTIVE MAGNETIC BEARING SYSTEM

Aiming at the stability and others properties of active magnetic bearing （AMB） system influenced by the periodic unbalance stimulation synchronous with rotor rotational speed, a new real-time adaptive feed-forward unbalance force compensation scheme is proposed based on variable step-size least mean square（LMS） algorithm as the feed-forward compensation controller. The controller can provide some suitable sinusoidal signals to com- pensate the feedback unbalance response signals synchronous with the rotary frequency, then reduce the fluctua- tion of the control currents and weaken the active control of AMB system. The variable step-size proportional to the rotational frequency is deduced by analyzing the principle of normal LMS algorithm and its deficiency in the application of real-time filtering of AMB system. Experimental results show that the new method can implement real-time unbalance force compensation in a wide frequency band, reduce the effect of unbalance stimulant force on the housing of AMB system, and provide convenience to improve rotational speed.

COUPLED DYNAMIC CHARACTERISTICS AND CONTROL OF ROTOR-CONICAL MAGNETIC BEARING SYSTEM

The coupled dynamic characteristics of the conical electromagnetic bearing are presented and their definitions are given. On the basis of the analyses of the characteristics, the dynamic model of five degrees of freedom (five-DOF) rotor-conical electromagnetic bearing system is made, and the influence of the coupled characteristics on the system optimal controller is analyzed.

A new nonlinear force model to replace the Hertzian contact model in a rigid-rotor ball bearing system

A new nonlinear force model based on experimental data is proposed to replace the classical Hertzian contact model to solve the fractional index nonlinearity in a ball bearing system. Firstly, the radial force and the radial deformation are measured by statics experiments, and the data are fitted respectively by using the Hertzian contact model and the cubic polynomial model. Then~ the two models are compared with the approximation formula appearing in Aeroengine Design Manual. In consequence, the two models are equivalent in an allowable deformation range. After that, the relationship of contact force and contact deformation for single rolling element between the races is cal- culated based on statics equilibrium to obtain the two kinds of nonlinear dynamic models in a rigid-rotor ball bearing system. Finally~ the displacement response and frequency spectrum for the two system models are compared quantitatively at different rotational speeds, and then the structures of frequency-amplitude curves over a wide speed range are compared qualitatively under different levels of radial clearance, amplitude of excitation, and mass of supporting rotor. The results demonstrate that the cubic polynomial model can take place of the Hertzian contact model in a range of deformation.

Seismic response of selective pallet racks isolated with friction pendulum bearing system

The concept of base isolation for storage racks is still a developing subject and has not been addressed in the recently updated seismic codes for storage racks.The friction pendulum bearing system(FPS)generates a natural period independent of the structure's seismic mass.This property makes FPS an ideal choice for isolating rack structures since merchandise may be placed on the racks in several possible arrangements.The results of a comprehensive parametric study aimed at evaluating the seismic performance of a four-shelf,two-bay selective pallet rack isolated with FPS is presented in this paper.The influence of radius of curvature of the FPS,seismic mass,and mass irregularity on the rack's seismic response is examined.The effect of variation of the coefficient of friction due to axial loading is studied by choosing friction values from two distinct friction characterization studies.The coefficient of friction calculated from these studies shows mild and significant variations,respectively,for the whole range of static axial loads expected on the isolator,and from different pallet mass arrangements considered.The optimum radius of curvature and the appropriate range of friction coefficient,for the defined range of static axial loads,to attain a desirable seismic performance are determined.

Bearing loads of dual rotor-bearing system under stationary and transient conditions

Aimed at aeroengine vibration failure, bearing loads of dual rotor-bearing system caused by unbalance are calculated under stationary and transient conditions. The three-dimensional （3-D） finite element method （FEM） model of dual rotor-bearing system was established. Applying the ro- tor dynamics function of Ansys 12.0, bearing loads were calculated under various unbalance force in stationary condition, and the transient vibration characteristics and the effect of acceleration on bearing loads were discussed. On the basis of simulation results, the influence disciplines of unbal- ance on bearing loads and theoretic reference for reducing bearing loads during start-up were ob- tained.

Instability analysis of unsymmetrical rotor-bearing systems

This paper presents the instability analysis of unsymmetrical rotor bearing system in accordance with the Campbell diagrams of the system, and concludes that the unstable regions stay in high speed regions with the increase of supporting stiffness and they decrease or disappear with the decrease of the inequality diametrical moments of the inertia and stiffness of shaft.

Seismic performance of cable-sliding friction bearing system for isolated bridges

During past strong earthquakes, highway bridges have sustained severe damage or even collapse due to excessive displacements and/or very large lateral forces. For commonly used isolation bearings with a pure friction sliding surface, seismic forces may be reduced but displacements are often unconstrained. In this paper, an alternative seismic bearing system, called the cable-sliding friction bearing system, is developed by integrating seismic isolation devices with displacement restrainers consisting of cables attached to the upper and lower plates of the bearing. Restoring forces are provided to limit the displacements of the sliding component. Design parameters including the length and stiffness of the cables, friction coefficient, strength of the shear bolt in a fixed-type bearing, and movements under earthquake excitations are discussed. Laboratory testing of a prototype bearing subjected to vertical loads and quasi-static cyclic lateral loads, and corresponding numerical finite element simulation analysis, were carried out. It is shown that the numerical simulation shows good agreement with the experimental force-displacement hysteretic response, indicating the viability of the new bearing system. In addition, practical application of this bearing system to a multi-span bridge in China and its design advantages are discussed.

RESEARCH ON CONTROL OF FLYWHEEL SUSPENDED BY ACTIVE MAGNETIC BEARING SYSTEM WITH SIGNIFICANT GYROSCOPIC EFFECTS

Traditional PID controllers are no longer suitable formagnetic-bearing-supported high-speed flywheels with significant gyroscopic effects. Becausegyroscopic effects greatly influence the stability of the flywheel rotor, especially at highrotational speeds. Velocity cross feedback and displacement cross feedback are used to overcomeharmful effects of nutation and precession modes, and stabilize the rotor at high rotational speeds.Theoretical analysis is given to show their effects. A control platform based on RTLinut and a PCis built to control the active magnetic bearing (AMB) system, and relevant results are reported.Using velocity cross feedback and displacement cross feedback in a closed loop control system, theflywheel successfully runs at over 20000 r/min.

MATLAB WEB SERVER AND ITS APPLICATION IN REMOTE COLLABORATIVE DESIGN OF MAGNETIC BEARING SYSTEMS

Acclimatizing itself to the development of network,Math Works Inc constructed a MATLAB Web Server environment by dint of which one can browse the calculation and plots of MATLAB through Internet directly.The installation and use of the environment is introduced.A code established on the platform of MATLAB,which deals with the modal analysis of magnetic bearing system(MBS) supporting rotors of five degrees of freedom and considering the coupling of thrust bearing with radical bearings is modified to work in the environment.The purpose is to realize a remote call of the code by users through Internet for the performance analysis of the system.Such an application is very important to the concurrent design of MBS and for the utilization of distributive knowledge acquisition resources in collaborative design.The work on modification and realization is described and the results are discussed.

ANALYSIS ON MOTION STABILITY OF A HIGH-SPEED ROTOR-BEARING SYSTEM

A non-linear dynamic model of one type of high-speed rotor system with gassupporting system is set up. The laws between the capacity force and the parameters of bearing, thestatic equilibrium position and rotating speed are studied on the basis of above model. Then, thefailure rotating, speed is given in the working state, and the relation between the minimum failurerotating speed and clearance of bearing is also studied. At last, the stability and failurecondition are discussed in different working conditions.

Research on a Rigid Rotor-Sliding Bearing System with a Squeeze Film Damper

In this paper, a dynamic model on a rigid rotor sliding bearing system with a SFD is established. The stability and bifurcation behaviors of the system are studied. On the basis of the differential equations of fluid momentum and mass continuity, the distribution pressure function is derived by taking oil film inertia force into consideration. Damping force, clearance excitation force, interference force of different frequencies and static load are also considered in the model. Finally, the governing equations of the stability and bifurcation behaviors of the system are solved by Floquet theory. Simulation of dynamic model shows that the rigid rotor sliding bearing system can maintain stability and exhibit a Hopf bifurcation phenomenon in a certain range.

Synchronous Vibration Suppression of Magnetic Bearing Systems without Angular Sensors

Active Magnetic Bearing(AMB)levitates rotor by magnetic force without friction,and it can provide active control force to suppress vibration while rotating.Most of vibration suppressing methods need angular speed sensors to obtain rotating speed,but in many occasions,angular speed sensor is difficult to install or is difficult to guarantee reliability.This paper proposed a vibration suppressing strategy without angular speed sensor based on generalized integrator and frequency locked loop(GI-FLL)and phase shift generalized integrator(PSGI).GI-FLL and high-pass filter estimate frequency from control current,PSGI is applied to generate compensating signal.Firstly,model of AMB system expressed by transfer function is established and effect of centrifugal force is analyzed.Then,principle and process of vibration suppressing strategy is introduced.Influence of parameters are analyzed by root locus and bode diagram.Simulation results display the process of frequency estimation and performance of displacement.Experiments are carried on a test rig,results of simulations and experiments demonstrate the effectiveness of proposed vibration suppressing strategy.

Effects of Dimensional Tolerances on the Friction Power Loss of Hydrodynamic Journal Bearing System

According to the dimensional tolerances on hydrodynamic journal bearing system, a nonlinear oil film force model was established,and the Reynolds' equation was solved by adopting finite difference method. In order to fulfill different dimensional tolerances in the system,adopting 2kfactor design and using the eccentricity ratio corresponding to the stability critical curve,the effects of the friction power loss brought by the dimensional tolerances of the dynamic viscosity,bearing width,bearing diameter and journal diameter were analyzed. The effect on dynamic characteristics of the hydrodynamic journal bearing system was quantitatively analyzed,and the nonlinear dynamic analysis, modeling and calculation methods were studied while considering the manufacturing tolerances. The results show that in contrast to the impacts of the tolerances in journal diameter,dynamic viscosity and bearing width,the bearing diameter tolerance would lead to the rise in the power loss, and the dimensional tolerances have different degrees of impacts on the journal bearing system. The friction power loss decreased as the eccentricity ratio increased, and when the eccentricity ratio was 0. 695 the power loss came to the minimum.The investigation would find the best solution and reduce energy consumption,then control varieties of nonlinear dynamical behavior effectively,and provide a theoretical basis for hydrodynamic journal bearing system in parameter design.

Effectof Film Force Nonlinearity on Unbalance Responseof a Jeffcot RotorJournal Bearing System

The effect of oil film force nonlinearity on unbalance response of Jeffcot rotor elliptical bearing system is studied. Linear analysis is done by linearizing the bearing oil film dynamic forces and expressing them with stiffness and damping constants. Nonlinear dynamic simulation is done by taking the nonlinearity of bearing oil film dynamic force into full consideration, with the latter being expressed by a special database which is generated beforehand, and based on non stationary Reynolds equation and Reynolds boundary condition of film rupture. The linear and nonlinear unbalance responses of a Jeffcot rotor supported on a pair of elliptical bearings is studied. The resulting dynamic characteristics and rotor and journal whirling orbits are compared. While good consistancy between nonlinear and linear predictions is found under very small unbalance, significant or even drastic differences are found whenever the unbalance is not small, indicating the necessity of including the nonlinearity of oil film dynamic forces, especially when response to large unbalance is to be predicted.

GA-BASED PID NEURAL NETWORK CONTROL FOR MAGNETIC BEARING SYSTEMS

In order to overcome the system non-linearity and uncertainty inherent in magnetic bearing systems, a GA（genetic algnrithm）-based PID neural network controller is designed and trained tO emulate the operation of a complete system （magnetic bearing, controller, and power amplifiers）. The feasibility of using a neural network to control nonlinear magnetic bearing systems with unknown dynamics is demonstrated. The key concept of the control scheme is to use GA to evaluate the candidate solutions （chromosomes）, increase the generalization ability of PID neural network and avoid suffering from the local minima problem in network learning due to the use of gradient descent learning method. The simulation results show that the proposed architecture provides well robust performance and better reinforcement learning capability in controlling magnetic bearing systems.

The Evolution of Reliability and Efficiency of Aerospace Bearing Systems

The worldwide air traffic underwent a rapid development in recent decades.?Between the early 70s and the late 90s of the last century civil air traffic?doubled every 15 years. The civil aviation market will continue to grow with 4% - 5%?each year within the next 20 years. This enormous growth represents major?challenges for airframers, engine makers, suppliers, airlines, air traffic management?and ground infrastructure. In addition, the public debate on the worldwide?civil air traffic is dominated by environmental and climate issues, even though only 2% of the man-made carbon dioxide (CO2) emissions are due to air transportation. Therefore the aerospace industry will have to focus on a low-emission and quite air traffic, and on the conservation of natural resources and our environment. The end-use consumer and environmental policy requirements?for aircrafts of the next generation translate into components with improved efficiency and reliability. Rolling bearings are one of these components which?significantly determine the reliability and mechanical efficiency of aerospace applications such as aircraft and rotorcraft engines and transmission systems.?They have to withstand very demanding operating conditions. Especially main shaft?bearings in modern aircraft engines experience high rotational speeds andtemperatures. Furthermore aerospace bearings have to meet the highest reliability?standards and require low-weight design solutions. These operating conditions?and requirements present a continuous challenge for improvements in all?fields of bearing technology. This article presents solutions in aspects of materials, design, analysis, and surface technologies in order to meet the environmental, reliability, and economical requirements of advanced aerospace bearing systems. State of the art bearing analysis and advanced bearing design solutions?contributing to lower friction power losses and increased systems efficiency?are discussed. Weight, functional, and maintenance benefits are presented with the example of highly integrated aircraft engine main shaft bearings. It is also shown that the progress in bearing materials and surface technology development is the basis for weight and friction energy reduction in aerospace?bearing systems.

Optimization of a Flexible Rotor-Sliding Bearing System With a Squeeze Film Damper

A dynamic model of a flexible rotor-sliding bearing system ( FRSBS ) with asqueeze film damper ( SFD) is established. Considered in the model are oil film inertia force,damping farce, clearance excitation force, interference force of different frequencies and staticload, as opposed to previous research. On the basis of this model, the optimal design of the systemis deeply studied. Simulation shows that the system optimization design can effectively improve thesystem stability.

INFLUENCE OF UNBALANCE ON LOW-FREQUENCY VIBRATION OF ROTOR-BEARING SYSTEM

The influence of unbalance on low-frequency vibration exists whenthere are nonlinear factors in oil film force. Based on the Muszpeka oil film mod-el , a theoretical proof is presented. Some new results are obtained based on the im-proved simulation meth

Bifurcation and chaos in nonlinear rotor-bearing system

Bifurcation and chaos in rigid Jefccott rotor bearing system are studied, by following the multi variable Floquet theory. By calculating the largest Lyapunov exponent, the chaotic motion and ″periodic window″ phenomena are found for a certain bifurcation parameter. The results show that the motion of the rotor system features a complicated nonlinear dynamics phenomena, such as period doubling bifurcation, saddle node bifurcation, secondary Hopf bifurcation and chaotic motion.