Characteristics of the Nonlinear Hysteresis Loop for Rotor-Bearing Instability

Volume 3B: 15th Biennial Conference on Mechanical Vibration and Noise — Acoustics, Vibrations, and Rotating Machines ◽

10.1115/detc1995-0503 ◽

1995 ◽

Author(s):

Jenq-Shan Guo ◽

M. L. Adams

Keyword(s):

Hopf Bifurcation ◽

Hysteresis Loop ◽

Natural Frequency ◽

Journal Bearing ◽

Integration Scheme ◽

Lubrication Equation ◽

Nonlinear Hysteresis ◽

Saddle Node ◽

Excited Vibration ◽

Bearing Loads

Abstract The nonlinear hysteresis loop of a journal-bearing supported rotordynamic system is characterized by the Hopf-bifurcation and Saddle-Node speeds. The nonlinearity of this system occurs in the journal-bearing fluid-film forces, and requires the imbedding of a solution for the Reynolds lubrication equation within a numerical integration scheme of the coupled motion equations in order to perform proper simulations. These show that under light bearing static load, the Saddle-Node and Hopf bifurcation coalesce to a single speed at essentially two times the self-excited vibration frequency (i.e., the lowest natural frequency). At higher bearing loads, the classical instability threshold speed (i.e., Hopf bifurcation) occurs at progressively higher rotor speeds. However, the disappearance speed (Saddle Node of the periodic orbit) of the nonlinear limit cycle occurs at progressively lower rotor speeds, asymptotically approaching approximately 1.725 times the lowest natural frequency. By adding a rotor unbalance force to the model, exploratory simulations have been made to determine the extent to which chaos signal processing in the normal speed-up or coast-down vibration of actual machines could be used to locate this lower speed bound of the instability hysteresis loop.

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Improvement of Tilting-Pad Journal Bearing Operating Characteristics by Application of Eddy Grooves

Lubricants ◽

10.3390/lubricants9020018 ◽

2021 ◽

Vol 9 (2) ◽

pp. 18

Author(s):

Eckhard Schüler ◽

Olaf Berner

Keyword(s):

High Speed ◽

Journal Bearing ◽

Load Capacity ◽

Fluid Film ◽

Operating Characteristics ◽

Fluid Film Bearings ◽

Tilting Pad ◽

Bearing Load ◽

Bearing Loads ◽

Tilting Pad Journal Bearing

In high speed, high load fluid-film bearings, the laminar-turbulent flow transition can lead to a considerable reduction of the maximum bearing temperatures, due to a homogenization of the fluid-film temperature in radial direction. Since this phenomenon only occurs significantly in large bearings or at very high sliding speeds, means to achieve the effect at lower speeds have been investigated in the past. This paper shows an experimental investigation of this effect and how it can be used for smaller bearings by optimized eddy grooves, machined into the bearing surface. The investigations were carried out on a Miba journal bearing test rig with Ø120 mm shaft diameter at speeds between 50 m/s–110 m/s and at specific bearing loads up to 4.0 MPa. To investigate the potential of this technology, additional temperature probes were installed at the crucial position directly in the sliding surface of an up-to-date tilting pad journal bearing. The results show that the achieved surface temperature reduction with the optimized eddy grooves is significant and represents a considerable enhancement of bearing load capacity. This increase in performance opens new options for the design of bearings and related turbomachinery applications.

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Experimental and Theoretical Rotordynamic Characteristics of a Hybrid Journal Bearing

Journal of Tribology ◽

10.1115/1.2832446 ◽

1997 ◽

Vol 119 (1) ◽

pp. 132-141 ◽

Cited By ~ 17

Author(s):

J. T. Sawicki ◽

R. J. Capaldi ◽

M. L. Adams

Keyword(s):

Journal Bearing ◽

Operating Conditions ◽

Dynamic Force ◽

Test Results ◽

Force Measurements ◽

Lubrication Equation ◽

Load Cells ◽

Theoretical Predictions ◽

Stiffness And Damping

This paper describes an experimental and theoretical investigation of a four-pocket, oil-fed, orifice-compensated hydrostatic bearing including the hybrid effects of journal rotation. The test apparatus incorporates a double-spool-shaft spindle which permits independent control over the journal spin speed and the frequency of an adjustable-magnitude circular orbit, for both forward and backward whirling. This configuration yields data that enables determination of the full linear anisotropic rotordynamic model. The dynamic force measurements were made simultaneously with two independent systems, one with piezoelectric load cells and the other with strain gage load cells. Theoretical predictions are made for the same configuration and operating conditions as the test matrix using a finite-difference solver of Reynolds lubrication equation. The computational results agree well with test results, theoretical predictions of stiffness and damping coefficients are typically within thirty percent of the experimental results.

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Nonlinear Vibration Analysis of a Flexible Rotor Supported by a Flexure Pivot Tilting Pad Journal Bearing in Comparison to a Fixed Profile Journal Bearing with Experimental Verification

Journal of Engineering for Gas Turbines and Power ◽

10.1115/1.4050367 ◽

2021 ◽

Author(s):

Nuntaphong Koondilogpiboon ◽

Tsuyoshi Inoue

Keyword(s):

Hopf Bifurcation ◽

Journal Bearing ◽

Flexible Rotor ◽

Maximum Test ◽

Operating Region ◽

Stable Operating ◽

Subcritical Hopf Bifurcation ◽

Tilting Pad ◽

Calculation Results ◽

Tilting Pad Journal Bearing

Abstract In this paper, an efficient numerical method consisting of the real mode component mode synthesis (CMS) model reduction, shooting method with parallel computing, and Floquet analysis was developed for nonlinear rotordynamics analysis of a flexible rotor supported by a 4-lobe flexure pivot tilting pad journal bearing (FPTPJB) in load-on-pad (LOP) and load-between-pad (LBP) orientations in comparison to a fixed profile journal bearing (JB) of the same pad geometry. The method used the rotor's finite elements and bearing forces obtained from directly solving the Reynolds equation to determine the limit cycles and Hopf bifurcation types. For the investigated rotor and bearing parameters, the numerical results indicated that the onset speed of instability (OSI) of FPTPJB is considerably higher than that of JB of the same orientation. Also, FPTPJB in LOP orientation yielded higher OSI than the LBP one, whereas the OSI of JB in LOP orientation was substantially higher than the LBP counterpart. Nonlinear calculation results indicated that all bearing types and orientations gave subcritical Hopf bifurcation. The FPTPJB in LOP orientation produced the largest stable operating region, whereas the JB in LBP configuration yield the smallest one. The experiment showed subcritical Hopf bifurcation occurred at speed close to the calculated OSI in all cases except FPTPJB in LOP orientation that the OSI is higher than the maximum test rig speed. The whirling orbit had the same frequency as the first critical speed and precessed in the direction of shaft rotation.

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Numerical Investigation for Characteristics and Oil-Air Distributions of a Tilting-Pad Journal Bearing Under Different Loads

Volume 10A: Structures and Dynamics ◽

10.1115/gt2020-15151 ◽

2020 ◽

Author(s):

Aoshuang Ding ◽

Xuesong Li

Keyword(s):

Film Thickness ◽

Journal Bearing ◽

Viscosity Ratio ◽

Cavitation Model ◽

Tilting Pad ◽

Sst Model ◽

Bearing Load ◽

Simulation Results ◽

Bearing Loads ◽

Tilting Pad Journal Bearing

Abstract This paper analyses the flow characteristics and oil-air distributions of oil flows in a tilting-pad journal bearing under different bearing loads. This titling-pad journal bearing is working at 3000 rpm rotation speed and its minimum film thicknesses have been measured under different loads from 180 kN to 299 kN. Based on the previous researches of this bearing under 180 kN, the gaseous cavitation and low-turbulence flow exists in this bearing flow. A suitable gaseous cavitation model and the SST model with low-Re correction are used in the film flow simulations. With the rotor and pads assumed to be rigid, the dynamic mesh and motion equations are applied to simulate the motions of the rotor and the rotations of the pads. Based on the simulation results under different bearing loads, the simulated minimum film thicknesses agrees well with the measured data. It indicates that the simulation results can catch the film geometries and flows correctly. With the load increasing, the rotor moves closer to the loaded pads and the minimum film thickness decreases. Taking the effect of boundary layers into consideration, the turbulence has a negative relationship with the film thickness and decreases in the loaded area under higher bearing load. It can be verified by the simulated lower turbulent viscosity ratio distributions in the loaded pads. In the unloaded area, both the film thickness and turbulence viscosity ratio are positively related to the bearing loads. Thus, the higher bearing load may lead the flow to be more different in the loaded and unloaded area, and the turbulence in the loaded pads may transfer to laminar in the end. As for the oil-air distributions, in the unloaded pads, with the bearing load increasing, the simulated air volume fraction increases in the unloaded pads with lower pressure. It should be caused by the higher film thickness of the unloaded pads under higher loads. In sum, the flow turbulence and cavitation process changes with the bearing load. With a higher load, the cavitation becomes more in the unloaded pads and the flow changes sharper from the high-turbulence unloaded area to the low-turbulence loaded area. As the simulation results is in good accordance with the experimental data, the SST model with low-Re correction and the gaseous cavitation model are verified to be suitable for bearing film simulations under different loads.

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Complex Behaviors of Epidemic Model with Nonlinear Rewiring Rate

Complexity ◽

10.1155/2020/7310347 ◽

2020 ◽

Vol 2020 ◽

pp. 1-16

Author(s):

Ding Fang ◽

Yongxin Zhang ◽

Wendi Wang

Keyword(s):

Hopf Bifurcation ◽

Numerical Simulations ◽

Bifurcation Analysis ◽

Epidemic Model ◽

Multiple Equilibria ◽

Homoclinic Bifurcation ◽

Propagation Model ◽

Bifurcation Curve ◽

Saddle Node Bifurcation ◽

Saddle Node

An SIS propagation model with the nonlinear rewiring rate on an adaptive network is considered. It is found by bifurcation analysis that the model has the complex behaviors which include the transcritical bifurcation, saddle-node bifurcation, Hopf bifurcation, and Bogdanov–Takens bifurcation. Especially, a bifurcation curve with “S” shape emerges due to the nonlinear rewiring rate, which leads to multiple equilibria and twice saddle-node bifurcations. Numerical simulations show that the model admits a homoclinic bifurcation and a saddle-node bifurcation of the limit cycle.

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Adaptive Control of Electromagnetic Suspension System by HOPF Bifurcation

Mathematical Problems in Engineering ◽

10.1155/2013/928719 ◽

2013 ◽

Vol 2013 ◽

pp. 1-5

Author(s):

Aming Hao ◽

Xiaolong Li ◽

Longhua She

Keyword(s):

Adaptive Control ◽

Hopf Bifurcation ◽

Large Scale ◽

Stability Range ◽

Hopf Bifurcation Point ◽

Maglev System ◽

Large Scale Disturbance ◽

Excited Vibration ◽

Parameter Variance ◽

The Stability

EMS-type maglev system is essentially nonlinear and unstable. It is complicated to design a stable controller for maglev system which is under large-scale disturbance and parameter variance. Theory analysis expresses that this phenomenon corresponds to a HOPF bifurcation in mathematical model. An adaptive control law which adjusts the PID control parameters is given in this paper according to HOPF bifurcation theory. Through identification of the levitated mass, the controller adjusts the feedback coefficient to make the system far from the HOPF bifurcation point and maintain the stability of the maglev system. Simulation result indicates that adjusting proportion gain parameter using this method can extend the state stability range of maglev system and avoid the self-excited vibration efficiently.

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Bifurcation analysis method of nonlinear traffic phenomena

International Journal of Modern Physics C ◽

10.1142/s0129183115501119 ◽

2015 ◽

Vol 26 (10) ◽

pp. 1550111 ◽

Cited By ~ 3

Author(s):

Wenhuan Ai ◽

Zhongke Shi ◽

Dawei Liu

Keyword(s):

Hopf Bifurcation ◽

Traffic Flow ◽

Bifurcation Analysis ◽

Phase Plane ◽

Flow Model ◽

Analysis Method ◽

Equilibrium Solutions ◽

Saddle Node Bifurcation ◽

Traffic Flow Model ◽

Saddle Node

A new bifurcation analysis method for analyzing and predicting the complex nonlinear traffic phenomena based on the macroscopic traffic flow model is presented in this paper. This method makes use of variable substitution to transform a traditional traffic flow model into a new model which is suitable for the stability analysis. Although the substitution seems to be simple, it can extend the range of the variable to infinity and build a relationship between the traffic congestion and the unstable system in the phase plane. So the problem of traffic flow could be converted into that of system stability. The analysis identifies the types and stabilities of the equilibrium solutions of the new model and gives the overall distribution structure of the nearby equilibrium solutions in the phase plane. Then we deduce the existence conditions of the models Hopf bifurcation and saddle-node bifurcation and find some bifurcations such as Hopf bifurcation, saddle-node bifurcation, Limit Point bifurcation of cycles and Bogdanov–Takens bifurcation. Furthermore, the Hopf bifurcation and saddle-node bifurcation are selected as the starting point of density temporal evolution and it will be helpful for improving our understanding of stop-and-go wave and local cluster effects observed in the free-way traffic.

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COHERENCE RESONANCE–INDUCED STOCHASTIC NEURAL FIRING AT A SADDLE-NODE BIFURCATION

International Journal of Modern Physics B ◽

10.1142/s0217979211101673 ◽

2011 ◽

Vol 25 (29) ◽

pp. 3977-3986 ◽

Cited By ~ 25

Author(s):

HUAGUANG GU ◽

HUIMIN ZHANG ◽

CHUNLING WEI ◽

MINGHAO YANG ◽

ZHIQIANG LIU ◽

...

Keyword(s):

Hopf Bifurcation ◽

Bifurcation Point ◽

Firing Pattern ◽

Neuronal System ◽

Neural Firing ◽

Coherence Resonance ◽

Saddle Node Bifurcation ◽

Hopf Bifurcation Point ◽

Firing Patterns ◽

Saddle Node

Coherence resonance at a saddle-node bifurcation point and the corresponding stochastic firing patterns are simulated in a theoretical neuronal model. The characteristics of noise-induced neural firing pattern, such as exponential decay in histogram of interspike interval (ISI) series, independence and stochasticity within ISI series are identified. Firing pattern similar to the simulated results was discovered in biological experiment on a neural pacemaker. The difference between this firing and integer multiple firing generated at a Hopf bifurcation point is also given. The results not only revealed the stochastic dynamics near a saddle-node bifurcation, but also gave practical approaches to identify the saddle-node bifurcation and to distinguish it from the Hopf bifurcation in neuronal system. In addition, many previously observed firing patterns can be attribute to stochastic firing pattern near such a saddle-node bifurcation.

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