Nonlinear Dynamics of Flexible Rotors Supported on Journal Bearings—Part II: Numerical Bearing Model

2017 ◽  
Vol 140 (2) ◽  
Author(s):  
Mohammad Miraskari ◽  
Farzad Hemmati ◽  
Mohamed S. Gadala
Keyword(s):  
Nonlinear Stability ◽  
Nonlinear Dynamic ◽  
Shooting Method ◽  
Monodromy Matrix ◽  
Nonlinear Coefficient ◽  
Journal Bearings ◽  
Flexible Rotor ◽  
Dynamic Coefficients ◽  
Rotor Bearing ◽  
Bearing System

The nonlinear stability of a flexible rotor-bearing system supported on finite length journal bearings is addressed. A perturbation method of the Reynolds lubrication equation is presented to calculate the bearing nonlinear dynamic coefficients, a treatment that is suitable to any bearing geometry. A mathematical model, nonlinear coefficient-based model, is proposed for the flexible rotor-bearing system for which the journal forces are represented through linear and nonlinear dynamic coefficients. The proposed model is then used for nonlinear stability analysis in the system. A shooting method is implemented to find the periodic solutions due to Hopf bifurcations. Monodromy matrix associated to the periodic solution is found at any operating point as a by-product of the shooting method. The eigenvalue analysis of the Monodromy matrix is then carried out to assess the bifurcation types and directions due to Hopf bifurcation in the system for speeds beyond the threshold speed of instability. Results show that models with finite coefficients have remarkably better agreement with experiments in identifying the boundary between bifurcation regions. Unbalance trajectories of the nonlinear system are shown to be capable of capturing sub- and super-harmonics which are absent in the linear model trajectories.

Nonlinear Dynamics of Flexible Rotors Supported on Journal Bearings—Part I: Analytical Bearing Model

2017 ◽  
Vol 140 (2) ◽  
Author(s):  
Mohammad Miraskari ◽  
Farzad Hemmati ◽  
Mohamed S. Gadala
Keyword(s):  
Journal Bearing ◽  
Nonlinear Coefficient ◽  
Journal Bearings ◽  
Flexible Rotor ◽  
Dynamic Coefficients ◽  
Flexible Rotors ◽  
Rotor Bearing ◽  
Bearing Force ◽  
Derivatives Of ◽  
Bearing System

To determine the bifurcation types in a rotor-bearing system, it is required to find higher order derivatives of the bearing forces with respect to journal velocity and position. As closed-form expressions for journal bearing force are not generally available, Hopf bifurcation studies of rotor-bearing systems have been limited to simple geometries and cavitation models. To solve this problem, an alternative nonlinear coefficient-based method for representing the bearing force is presented in this study. A flexible rotor-bearing system is presented for which bearing force is modeled with linear and nonlinear dynamic coefficients. The proposed nonlinear coefficient-based model was found to be successful in predicting the bifurcation types of the system as well as predicting the system dynamics and trajectories at spin speeds below and above the threshold speed of instability.

Correlation of Parameters to Instability and Chaos of a Horizontal Flexible Rotor Ball Bearing System

2013 ◽  
Author(s):  
T. C. Gupta ◽  
K. Gupta
Keyword(s):  
Fixed Point ◽  
Shooting Method ◽  
Ball Bearing ◽  
Monodromy Matrix ◽  
Flexible Rotor ◽  
Energy Functions ◽  
Fixed Point Algorithm ◽  
Time Period ◽  
Varying Compliance ◽  
Bearing System

The higher order effects from ball bearing nonlinearities cause complex vibration characteristics in rotor ball bearing systems. The sources of nonlinearities are internal radial clearance, Hertzian contact forces between balls and races and varying compliance effect. The same authors in their earlier work have identified the sets of parameters corresponding to instability and chaos for a horizontal flexible rotor supported on deep groove ball bearing. To the best of author’s knowledge, there is not much work reported in the literature on the dynamic analysis for instability and chaos, which is based on energy functions and bearing loads. Extending the preceding research work in the present paper by using a typical set of parameters and specifications of rotor ball bearing system, a correlation of parameters to instability and chaos is attempted using different energy functions associated with the dynamical system. A generalized Timoshenko beam finite element formulation is used to model the flexible rotor shaft. To achieve the convergence of solution with smaller number of elements, shape functions are derived from the exact solutions of governing differential equations of Timoshenko beam element. The sources of excitation are rotating unbalance and parametric excitation due to varying compliance of ball bearing during motion. For the bearing used in the present paper, the ratio of these excitation frequencies comes out to be an irrational number. Therefore, the dynamic response would be quasi-periodic with time period equal to infinity. To extend the use of non-autonomous shooting method to derive quasi-periodic solution, the fixed point algorithm (FPA) proposed in the literature is used to deduce the time period for non-autonomous shooting algorithm. The shooting method otherwise is used only to derive periodic solutions. Thus the non-autonomous shooting method coupled with fixed point algorithm (FPA) is used to compute the quasi-periodic solution, which also gives the monodromy matrix. The eigenvalues of the monodromy matrix, called Floqoet multipliers, give information about instability. The chaotic nature of the dynamic response is established by the maximum value of Lyapunov exponent. Once the instability and chaos is confirmed based on computed values of Floquet multipliers and Lyapunov exponents, the nature of the work done (positive or negative) by different conservative and non-conservative forces and moments during motion are analyzed and the fundamental causes, which make the system response unstable and / or chaotic, are established.

Numerical Analysis of Nonlinear Behaviors of a Flexible Rotor Dynamic System with Turbulent Journal Bearings Support

2011 ◽  
Vol 142 ◽  
pp. 7-11
Author(s):  
Yan Jun Lu ◽  
Yong Fang Zhang ◽  
Xiao Yong Ma ◽  
Xu Liu
Keyword(s):  
Period Doubling ◽  
Journal Bearings ◽  
Flexible Rotor ◽  
Chaotic Motions ◽  
System A ◽  
Runge Kutta Method ◽  
Lubricant Flow ◽  
Rotor Bearing ◽  
The Rich ◽  
Bearing System

For description of rotor-bearing system, a symmetrical flexible rotor supported by two turbulent journal bearings is modeled. The analysis of the rotor-bearing system is implemented under the assumptions of turbulent lubricant flow and a long bearing approximation. The bifurcation and chaos behaviors of the system are investigated for various rotational speeds. The motion equations are solved by the self-adaptive Runge-Kutta method. The numerical results show that the bifurcation of nonlinear responses of the system varies with the rotational speed of the rotor. It is found that the rich and complex dynamic behaviors of the system include period-1, period-doubling, quasi-periodic and chaotic motions etc.

Dynamic Characteristics of Rotor-Bearing System with a Labyrinth Seal

2017 ◽  
Vol 739 ◽  
pp. 169-181
Author(s):  
Shiuh Hwa Shyu ◽  
Yu Wei Chen
Keyword(s):  
Dynamic Characteristics ◽  
Labyrinth Seal ◽  
Perturbation Approach ◽  
Flexible Rotor ◽  
The Finite Element Method ◽  
Governing Equations ◽  
Dynamic Coefficients ◽  
Rigid Disks ◽  
Rotor Bearing ◽  
Bearing System

In this paper, the dynamic characteristics of rotor-bearing system with a labyrinth seal are investigated. The dynamic coefficients of labyrinth seal and the whirl speeds of rotor-bearing system are analyzed. The simplified rotor-bearing model consists of a rigid disk and a flexible shaft. The rotor-bearing system supported by two ball bearings, which are modeled as the spring-damper systems. A labyrinth seal supports one of the rigid disks. The perturbation approach has been applied to the governing equations of the bulk-flow model to calculate the dynamic coefficients of labyrinth seal. The Finite Element Method (FEM) is utilized to model the flexible-rotor system. The numerical results show the labyrinth seal effect on the dynamic characteristics of rotor-bearing system, which depend on a great variety of parameters such as geometry of the labyrinth seal.

Nonlinear Dynamic Behavior of a Rotor Bearing System With Nonlinear Viscous Damping Suspension

2017 ◽  
Author(s):  
Shuai Yan ◽  
Bin Lin ◽  
Jixiong Fei ◽  
Pengfei Liu
Keyword(s):  
Dynamic Behavior ◽  
Nonlinear Dynamic ◽  
Vibration Isolation ◽  
Lubricating Oil ◽  
Viscous Damping ◽  
Oil Viscosity ◽  
Speed Range ◽  
Rotor Bearing ◽  
Nonlinear Dynamic Behavior ◽  
Bearing System

Nonlinear damping suspension has gained attention owing to its excellent vibration isolation performance. In this paper, a cubic nonlinear viscous damping suspension was introduced to a rotor bearing system for vibration isolation between the bearing and environment. The nonlinear dynamic response of the rotor bearing system was investigated thoroughly. First, the nonlinear oil film force was solved based short bearing approximation and half Sommerfeld boundary condition. Then the motion equations of the system was built considering the cubic nonlinear viscous damping. A computational method was used to solve the equations of motion, and the bifurcation diagrams were used to display the motions. The influences of rotor-bearing system parameters were discussed from the results of numerical calculation, including the eccentricity, mass, stiffness, damping and lubricating oil viscosity. The results showed that: (1) medium eccentricity shows a wider stable speed range; (2) rotor damping has little effect to the stability of the system; (3) lower mass ratio produces a stable response; (4) medium suspension/journal stiffness ratio contributes to a wider stable speed range; (5) a higher viscosity shows a wider stable speed range than lower viscosity. From the above results, the rotor bearing system shows complex nonlinear dynamic behavior with nonlinear viscous damping. These results will be helpful to carrying out the optimal design of the rotor bearing system.

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