Bifurcation Theory Applied to Oil Whirl in Plain Cylindrical Journal Bearings

1984 ◽  
Vol 51 (2) ◽  
pp. 244-250 ◽  
Author(s):  
C. J. Myers
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Equilibrium Position ◽  
Small Amplitude ◽  
Bifurcation Theory ◽  
Equations Of Motion ◽  
Journal Bearings ◽  
The Self ◽  
Oil Whirl ◽  
Nonlinear Equations Of Motion

An analysis of the self-excited oscillations of a rotor supported in fluid film journal bearings is presented. It is shown that Hopf bifurcation theory may be used to investigate small-amplitude periodic solutions of the nonlinear equations of motion for rotor speeds close to the speed at which the steady-state equilibrium position becomes unstable. A numerical investigation supports the findings of the analytic work.

Hopf Bifurcation in Gas Journal Bearings

1993 ◽  
Vol 46 (7) ◽  
pp. 392-398 ◽  
Author(s):  
K. Czołczyn´ski
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Bifurcation Theory ◽  
Journal Bearing ◽  
Equations Of Motion ◽  
Journal Bearings ◽  
Fredholm Alternative ◽  
Stiffness Coefficients ◽  
Nonlinear Equations Of Motion ◽  
Gas Journal Bearing

This paper reviews a numerical investigation of the problem of small self-excited vibrations in gas journal bearings. The method of analysis is based on the Hopf bifurcation theory, in which the approximate periodic solutions of nonlinear equations of motion are computed using the Fredholm alternative. This theory enables us to construct the bifurcating periodic solutions and to determine their stability. The equations of motion of the investigated gas journal bearing have been formulated after estimating the damping and stiffness coefficients of a gas film. For this purpose, a new method of identification has been proposed.

On The Steady State and Dynamic Performance Characteristics of Floating Ring Bearings

1981 ◽  
Vol 103 (3) ◽  
pp. 389-397 ◽  
Author(s):  
Chin-Hsiu Li ◽  
S. M. Rohde
Keyword(s):  
Steady State ◽  
Linear Theory ◽  
High Speed ◽  
Dynamic Performance ◽  
Equations Of Motion ◽  
Nonlinear Behavior ◽  
Journal Bearings ◽  
Transient Problem ◽  
The Stability ◽  
Bearing System

An analysis of the steady state and dynamic characteristics of floating ring journal bearings has been performed. The stability characteristics of the bearing, based on linear theory, are given. The transient problem, in which the equations of motion for the bearing system are integrated in real time was studied. The effect of using finite bearing theory rather than the short bearing assumption was examined. Among the significant findings of this study is the existence of limit cycles in the regions of instability predicted by linear theory. Such results explain the superior stability characteristics of the floating ring bearing in high speed applications. An understanding of this nonlinear behavior, serves as the basis for new and rational criteria for the design of floating ring bearings.

scholarly journals Effects of Stator Stiffness, Gap Size, Unbalance, and Shaft’s Asymmetry on the Steady-State Response and Stability Range of an Asymmetric Rotor with Rub-Impact

2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Zhaoli Zheng ◽  
Yonghui Xie ◽  
Di Zhang ◽  
Xiaolong Ye
Keyword(s):  
Steady State ◽  
Nonlinear Equations ◽  
Equations Of Motion ◽  
Impact Behavior ◽  
Gap Size ◽  
Steady State Response ◽  
Arc Length ◽  
State Response ◽  
Asymmetric Rotor ◽  
Nonlinear Equations Of Motion

The asymmetric rotor and the rub-impact behavior are important sources of instability and may cause severe vibrations. However, the dynamics of the rotor-bearing system simultaneously considering the two factors has not gained sufficient attention in available investigations. In this paper, the steady-state response and stability of an asymmetric rotor with rub-impact were evaluated. The asymmetric rotor was modeled by beam elements with asymmetric cross section, and the nonlinear equations of motion were established in the rotating frame. The multiharmonic balance (MHB) method was employed to obtain the linearized form of the nonlinear equations of motion. Either the asymmetry of rotor or rub-impact can result in instability and make the problem difficult to solve. Thus, the arc-length method was utilized to trace the branch of the solutions. In order to improve the calculation speed and accurately predict the solution, the alternating frequency/time domain (AFT) was adopted to calculate the iteration of the arc-length method. Based on the proposed method, the effects of stator stiffness, gap size, unbalance, and asymmetric in shaft on the steady-state response and stability were obtained.

Modal Analysis of Ballooning Strings With Small Sag

1999 ◽  
Author(s):  
Rongjun Fan ◽  
Sushil K. Singh ◽  
Christopher D. Rahn
Keyword(s):  
Steady State ◽  
High Speed ◽  
Natural Frequencies ◽  
Rotation Speed ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Theoretical Predictions ◽  
Nonlinear Equations Of Motion

Abstract During the manufacture and transport of textile products, yarns are rotated at high speed and form balloons. The dynamic response of the balloon to varying rotation speed, boundary excitation, and disturbance forces governs the quality of the associated process. Resonance, in particular, can cause large tension variations that reduce product quality and may cause yarn breakage. In this paper, the natural frequencies and mode shapes of a single loop balloon are calculated to predict resonance. The three dimensional nonlinear equations of motion are simplified via small steady state displacement (sag) and vibration assumptions. Axial vibration is assumed to propagate instantaneously or in a quasistatic manner. Galerkin’s method is used to calculate the mode shapes and natural frequencies of the linearized equations. Experimental measurements of the steady state balloon shape and the first two natural frequencies and mode shapes are compared with theoretical predictions.

scholarly journals Bifurcations and Dynamics of the Rb-E2F Pathway Involving miR449

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-20
Author(s):  
Lingling Li ◽  
Jianwei Shen
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Bifurcation Theory ◽  
Critical Value ◽  
Asymptotically Stable ◽  
Delay Model ◽  
Dynamical Behaviors ◽  
Theoretical Results

We focused on the gene regulative network involving Rb-E2F pathway and microRNAs (miR449) and studied the influence of time delay on the dynamical behaviors of Rb-E2F pathway by using Hopf bifurcation theory. It is shown that under certain assumptions the steady state of the delay model is asymptotically stable for all delay values; there is a critical value under another set of conditions; the steady state is stable when the time delay is less than the critical value, while the steady state is changed to be unstable when the time delay is greater than the critical value. Thus, Hopf bifurcation appears at the steady state when the delay passes through the critical value. Numerical simulations were presented to illustrate the theoretical results.

Dynamic Behavior of Unbalanced Rigid Shaft Supported on Turbulent Journal Bearings—Theory and Experiment

1990 ◽  
Vol 112 (2) ◽  
pp. 404-408 ◽  
Author(s):  
H. Hashimoto ◽  
S. Wada
Keyword(s):  
Dynamic Behavior ◽  
Equations Of Motion ◽  
Journal Bearings ◽  
Operating Conditions ◽  
Experimental Results ◽  
Fluid Film ◽  
Force Components ◽  
The Stability ◽  
Nonlinear Equations Of Motion ◽  
Analytical Expressions

In this paper, the combined effects of turbulence and fluid film inertia on the dynamic behavior of an unbalanced rigid shaft supported horizontally on two identical aligned short journal bearings are investigated theoretically and experimentally. Utilizing analytical expressions for the dynamic fluid film force components considering the effects of turbulence and fluid film inertia, the nonlinear equations of motion for the rotor-bearing systems are solved by the improved Euler’s forward integration method. The journal center trajectories with unbalance eccentricity ratio of εμ = 0, 0.1 and 0.2 are examined theoretically for Reynolds number of Re = 2750, 4580, and 5500, and the theoretical results are compared with experimental results. From the theoretical and experimental results, it was found that the fluid film inertia improves the stability of unbalanced rigid shaft under certain operating conditions.

The Effects of Fluid Inertia Forces on the Dynamic Behavior of Short Journal Bearings in Superlaminar Flow Regime

1988 ◽  
Vol 110 (3) ◽  
pp. 539-545 ◽  
Author(s):  
H. Hashimoto ◽  
S. Wada ◽  
M. Sumitomo
Keyword(s):  
Dynamic Behavior ◽  
Equations Of Motion ◽  
Reynolds Numbers ◽  
Journal Bearings ◽  
Operating Conditions ◽  
Fluid Inertia ◽  
Oil Film ◽  
Euler’S Method ◽  
Onset Velocity ◽  
Nonlinear Equations Of Motion

The effects of fluid inertia on the dynamic behavior of oil film journal bearings are theoretically investigated. The dynamic oil film forces considering the combined effects of turbulence and fluid inertia are analytically obtained under the short bearing assumption. Based on the linearized analysis, the whirl onset velocity for a balanced rigid rotor supported horizontally in the oil film journal bearings are determined initially in the case of the length-to-diameter ratio of λ = 0.5 for Reynolds numbers of Re = 2750, 4580, and 5500. Moreover, the nonlinear equations of motion for the rotor are solved by the improved Euler’s method, and the relations between the transient journal motion and the pressure distribution corresponding to the above Reynolds numbers are examined. It is found that the fluid inertia significantly affects the dynamic behavior of turbulent journal bearings under certain operating conditions.

An Application of Short Bearing Theory to Dynamic Characteristic Problems of Turbulent Journal Bearings

1987 ◽  
Vol 109 (2) ◽  
pp. 307-314 ◽  
Author(s):  
Hiromu Hashimoto ◽  
Sanae Wada ◽  
Jin-ichi Ito
Keyword(s):  
Dynamic Characteristic ◽  
Equations Of Motion ◽  
Computation Time ◽  
Journal Bearings ◽  
Oil Film ◽  
Euler’S Method ◽  
Linearized Stability ◽  
Rotor Bearing ◽  
Time Required ◽  
Nonlinear Equations Of Motion

The dynamic characteristic problems of turbulent journal bearings are theoretically analyzed. The oil film forces considering the turbulent effects are analytically obtained under the short bearing assumption. The linearized stability analysis is developed for a perfectly rigid rotor supported horizontally in the two identical aligned oil film journal bearings and the whirl onset velocities for rotor-bearing systems are determined for various Reynolds numbers. In addition, the nonlinear equations of motion for rotor-bearing systems are solved by the improved Euler’s method and the journal center trajectories are examined for both cases with and without out-of-balance forces. The results obtained from the short bearing theory are compared with those from the finite bearing theory for the length-to-diameter ratio λ = 0.25 and 0.5. It is found that the turbulence has significant effects on the dynamic behavior of rotor-bearing systems and the short bearing theory has an advantage of reducing a computation time required for the calculation of journal center trajectories with an acceptable accuracy.

Ship Maneuverability Analysis Using Steady-State Techniques

1991 ◽  
Vol 28 (03) ◽  
pp. 163-180
Author(s):  
Volf Asinovsky ◽  
Kiang-Ning Huang ◽  
Mark C. Oakes
Keyword(s):  
Steady State ◽  
Equations Of Motion ◽  
Second Step ◽  
Hydrodynamic Characteristics ◽  
Kinematic Parameters ◽  
Step Method ◽  
Analytical Functions ◽  
Ship Maneuverability ◽  
Nonlinear Equations Of Motion ◽  
The Relationship

This paper discusses the application of steady-state techniques to maneuverability, position-keeping and track-keeping analyses in the ship design process. A two-step method of calculating the diagram of steering is described. The first step gives an approximate solution based on a preliminary analysis of the relationship between the drift angle and the angular velocity of ship in the steady turn. The second step solves the nonlinear equations of motion using an iterative process. In the second step of the solution, the hull's hydrodynamic characteristics are introduced into the equations of motion directly in numerical form without a preliminary approximation by analytical functions. Assumptions about the kinematic parameters of motion are not used. This results in increased accuracy of the calculations. The hydrodynamic characteristics of the hull, rudder and appendages, and their interaction are considered separately. Position and track-keeping analysis using an approach similar to that used for the calculation of the diagram of steering is discussed.

Steady-State Responses in Systems of Nearly-Identical Torsional Vibration Absorbers

2003 ◽  
Vol 125 (1) ◽  
pp. 80-87 ◽  
Author(s):  
A. S. Alsuwaiyan ◽  
Steven W. Shaw
Keyword(s):  
Steady State ◽  
Torsional Vibration ◽  
Equations Of Motion ◽  
Vibration Absorbers ◽  
Fully Nonlinear Equations ◽  
State Response ◽  
Steady State Responses ◽  
Nonlinear Equations Of Motion ◽  
The Mathematical Model ◽  
State Responses

In this paper we consider the steady-state response of a rotor fitted with a system of nearly identical torsional vibration absorbers. The absorbers are of the centrifugal pendulum type, which provide an effective mean of attenuating torsional vibrations of the rotor at a given order. The model considered employs absorbers that are tuned close to the order of the excitation, with an intentional mistuning that is selected by design, and imperfections among the absorbers which arise from manufacturing, wear, and other effects. It is shown that these systems can experience localized responses in which the response amplitude of one or more absorbers can become relatively large as compared to the response of the corresponding system with identical absorbers. The results are based on an exact steady-state analysis of the mathematical model, and they show that the strength of the localization depends on the average level of absorber mistuning (a design parameter), the magnitude of the relative imperfections among the absorbers, and the absorber damping. It is found that the most desirable situation is one in which the relative imperfections are kept as small as possible, and that this becomes more crucial when the levels of mistuning and damping are very small. The results of the analysis are confirmed by simulations of the fully nonlinear equations of motion of the rotor/absorber system. It is concluded that the presence of localization should be accounted for in absorber designs, since its presence makes the absorbers less effective in terms of vibration reduction and, perhaps more significantly, it can drastically reduce their operating range, since such absorbers typically have limited rattle space.

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