Investigation of Turbulence Effects on the Nonlinear Vibration of a Rigid Rotor Supported by Finite Length 2-Lobe and Circular Bearings

2019 ◽  
Vol 14 (12) ◽  
Author(s):  
Nuntaphong Koondilogpiboon ◽  
Tsuyoshi Inoue
Keyword(s):  
Nonlinear Vibration ◽  
Limit Cycles ◽  
Reynolds Equation ◽  
Equations Of Motion ◽  
Rigid Rotor ◽  
Reynolds Equations ◽  
Multiplier Analysis ◽  
Turbulence Effects ◽  
The Stability ◽  
Nonlinear Equations Of Motion

Abstract This study investigated the effect of turbulence on the nonlinear vibration of a symmetrical rigid rotor supported by two identical journal bearings. The bearings consisted of various length to diameter (L/D) ratio circular and 2-lobe bearings with differing pad preloads. Two turbulent (Ng–Pan–Elrod and Constantinescu model) and one laminar Reynolds equations were selected for comparison, and they were solved using a finite difference method to obtain nonlinear bearing forces. The nonlinear equations of motion for the rotor-bearing system were solved using a shooting method and arclength continuation to obtain limit cycles for each bearing configuration. Floquet multiplier analysis was then utilized to identify the stability of the obtained limit cycles. For the cases of the circular and 2-lobe bearing without pad preload, the turbulent Reynolds equations yielded a lower onset speed of instability and L/D ratio at which the bifurcation type changed from supercritical to subcritical than the laminar Reynolds equation. However, at higher pad preloads (preloads of 0.25 or 0.5), the turbulence effects increased the onset speed of instability, especially for L/D ratios > 0.7, and only supercritical bifurcation was observed. For all bearing configurations, the ratio of the limit cycle whirl frequency to shaft rotational speed for both turbulence bearing models was higher than that of the laminar bearing model, and the Ng–Pan–Elrod turbulence model always generated lower onset speed of instability than the Constantinescu model.

Nonlinear Analysis of Rigid Rotor System Supported by 2 Lobes Journal Bearing in Comparison With Circular Journal Bearing

2018 ◽  
Author(s):  
Nuntaphong Koondilogpiboon ◽  
Tsuyoshi Inoue
Keyword(s):  
Limit Cycles ◽  
Shooting Method ◽  
Reynolds Equation ◽  
Journal Bearing ◽  
Equation Of Motion ◽  
Rigid Rotor ◽  
Floquet Multiplier ◽  
Multiplier Analysis ◽  
Speed Range ◽  
Bearing Force

The effect of 2 lobes journal bearing parameters such as L/D ratio and pad preload on the bifurcation of the rigid rotor is investigated in comparison with the circular bearing. Nonlinear bearing force in the equation of motion is obtained by solving Reynolds equation using the finite difference method. Shooting method and Floquet multiplier analysis are employed to obtain limit cycles and their stability. The results show that, for some bearing parameters, multiple limit cycles coexist at a specific shaft rotational speed range. Comparing with the circular bearing of same L/D ratio, the 2 lobes bearing without pad preload decreases the onset speed of instability and also decreases speed range from the onset speed of instability (Hopf) point to the limit point of the bifurcation (saddle-node) in the subcritical bifurcation case. Increasing the pad preload only increases the onset speed of instability significantly in the small L/D ratio case. For both circular and 2 lobes bearing, increasing the L/D ratio decreases the onset speed of instability and tends to change the type of the bifurcation from supercritical to subcritical.

Nonlinear Analysis of a Rigid Rotor on Magnetic Bearings

1995 ◽  
Author(s):  
C. Nataraj
Keyword(s):  
Nonlinear Analysis ◽  
Equations Of Motion ◽  
Forced Response ◽  
Magnetic Bearings ◽  
Rigid Rotor ◽  
Stability Criteria ◽  
Safe Operation ◽  
Qualitative And Quantitative ◽  
Quantitative Results ◽  
Nonlinear Equations Of Motion

A simple model of a rigid rotor supported on magnetic bearings is considered. A proportional control architecture is assumed, the nonlinear equations of motion are derived and some essential nondimensional parameters are identified. The free and forced response of the system is analyzed using techniques of nonlinear analysis. Both qualitative and quantitative results are obtained and stability criteria are derived for safe operation of the system.

Analysis of the Nonlinear Dynamics of a Railway Vehicle Wheelset

1973 ◽  
Vol 95 (1) ◽  
pp. 28-35 ◽  
Author(s):  
E. Harry Law ◽  
R. S. Brand
Keyword(s):  
Lateral Displacement ◽  
Equations Of Motion ◽  
Vertical Displacement ◽  
Linear Analysis ◽  
Design Parameters ◽  
Railway Vehicle ◽  
Nonlinear Variation ◽  
Constraint Equations ◽  
The Stability ◽  
Nonlinear Equations Of Motion

The nonlinear equations of motion for a railway vehicle wheelset having curved wheel profiles and wheel-flange/rail contact are presented. The dependence of axle roll and vertical displacement on lateral displacement and yaw is formulated by two holonomic constraint equations. The method of Krylov-Bogoliubov is used to derive expressions for the amplitudes of stationary oscillations. A perturbation analysis is then used to derive conditions for the stability characteristics of the stationary oscillations. The expressions for the amplitude and the stability conditions are shown to have a simple geometrical interpretation which facilitates the evaluation of the effects of design parameters on the motion. It is shown that flange clearance and the nonlinear variation of axle roll with lateral displacement significantly influence the motion of the wheelset. Stationary oscillations may occur at forward speeds both below and above the critical speed at which a linear analysis predicts the onset of instability.

Non-Linear Transient Stability Analysis of a Rigid Rotor Supported on Journal Bearings With Rectangular Dimples

2015 ◽  
Author(s):  
Ram Turaga
Keyword(s):  
Stability Analysis ◽  
Surface Texture ◽  
Transient Stability ◽  
Equations Of Motion ◽  
Journal Bearings ◽  
Pressure Curve ◽  
Rigid Rotor ◽  
Non Linear ◽  
The Stability ◽  
Transient Stability Analysis

The influence of deterministic surface texture on the sub-synchronous whirl stability of a rigid rotor has been studied. Non-linear transient stability analysis has been performed to study the stability of a rigid rotor supported on two symmetric journal bearings with a rectangular dimple of large aspect ratio. The surface texture parameters considered are dimple depth to minimum film thickness ratio and the location of the dimple on the bearing surface. Journal bearings of different Length to diameter ratios have been studied. The governing Reynolds equation for finite journal bearings with incompressible fluid has been solved using the Finite Element Method under isothermal conditions. The trajectories of the journal center have been obtained by solving the equations of motion of the journal center by the fourth-order Runge-Kutta method. When the dimple is located in the raising part of the pressure curve the positive rectangular dimple is seen to decrease the stability whereas the negative rectangular dimple is seen to improve the stability of the rigid rotor.

The Influence of Orifice Restriction on the Stability of Rigid Rotor-Aerostatic Bearing System

2009 ◽  
Author(s):  
Cheng-Hsien Chen ◽  
Ding-Wen Yang ◽  
Yuan Kang ◽  
Ren-Ming Hwang ◽  
Shrh-Shyong Shyr
Keyword(s):  
Reynolds Equation ◽  
Inertial Force ◽  
Adiabatic Process ◽  
Rigid Rotor ◽  
Double Row ◽  
Stability Threshold ◽  
Spindle Rotation ◽  
The Stability ◽  
Air Film ◽  
Bearing System

This paper has studied the influence of the restriction effect on the stability of a rigid rotor in rotation supported by double-row, orifice compensated aerostatic bearings. The air which is assumed to be perfect gas, passes through orifice restrictor into the bearing clearance undergoing the adiabatic process is governed by Reynolds equation including the coupled effects of wedge due to spindle rotation and squeezed film due to journal oscillation. The Ph-method is used to analyze Reynolds equation and which is then solved by the finite difference method and numerical integration to yield static and dynamic characteristics of air film. The motion equation of the rotor-bearing system is obtained by using the perturbation method and the eigensolution method is used to determine the stability threshold and critical whirl ratio. The variations of stability threshold of both critical inertial force and critical whirl ratio with restriction parameters are analyzed for various whirl ratios, speed of journal rotation and eccentricity ratios.

The Nonlinear Vibration of a Vehicle and Passenger System

2001 ◽  
Author(s):  
Ke Yu ◽  
Albert C. J. Luo ◽  
Yuancheng He
Keyword(s):  
Nonlinear Vibration ◽  
Nonlinear Equations ◽  
Equations Of Motion ◽  
Dynamic Responses ◽  
Pavement Surface ◽  
Torsional Spring ◽  
Nonlinear Equations Of Motion

Abstract The vibration of passengers in a vehicle traveling on a rough pavement surface is investigated. The nonlinear equations of motion for a vehicle and passenger system with impacts are derived, and the corresponding equilibrium and stability are investigated. The dynamic responses for the vehicle and passenger system with and without impacts are simulated numerically. This investigation shows that the strong torsional spring is required in order to reduce the vibration amplitudes of passengers and to avoid impacts between the vehicle and passenger.

Linear Stability Analysis of a Waveboard Multibody Model With a Minimal Set of Equations

2021 ◽  
Author(s):  
A. G. Agúndez ◽  
D. García-Vallejo ◽  
E. Freire ◽  
A. M. Mikkola
Keyword(s):  
Stability Analysis ◽  
Multibody System ◽  
Equations Of Motion ◽  
Nonholonomic Constraints ◽  
Design Parameters ◽  
Multibody Model ◽  
Aspect Ratios ◽  
Holonomic And Nonholonomic Constraints ◽  
The Stability ◽  
Nonlinear Equations Of Motion

Abstract In this paper, the stability of a waveboard, the skateboard consisting in two articulated platforms, coupled by a torsion bar and supported of two caster wheels, is analysed. The waveboard presents an interesting propelling mechanism, since the rider can achieve a forward motion by means of an oscillatory lateral motion of the platforms. The system is described using a multibody model with holonomic and nonholonomic constraints. To perform the stability analysis, the nonlinear equations of motion are linearized with respect to the forward upright motion with constant speed. The linearization is carried out resorting to a novel numerical linearization procedure, recently validated with a well-acknowledged bicycle benchmark, which allows the maximum possible reduction of the linearized equations of motion of multibody systems with holonomic and nonholonomic constraints. The approach allows the expression of the Jacobian matrix in terms of the main design parameters of the multibody system under study. This paper illustrates the use of this linearization approach with a complex multibody system as the waveboard. Furthermore, a sensitivity analysis of the eigenvalues considering different scenarios is performed, and the influence of the forward speed, the casters’ inclination angle and the tori aspect ratios of the toroidal wheels on the stability of the system is analysed.

scholarly journals Instability Threshold of a Rigid Rotor-Tilting Pad Journal Bearings System

1997 ◽  
Vol 3 (3) ◽  
pp. 199-213 ◽  
Author(s):  
Stefano Pagano ◽  
Ernesto Rocca ◽  
Michele Russo ◽  
Riccardo Russo
Keyword(s):  
Differential Equations ◽  
Numerical Integration ◽  
Equations Of Motion ◽  
Journal Bearings ◽  
Stable Operation ◽  
Rigid Rotor ◽  
Tilting Pad ◽  
The Stability ◽  
Tilting Pad Journal Bearings ◽  
Rotor Axis

The stability of a rigid rotor supported on radial tilting pad journal bearings is analysed. This study has been tackled both for small unbalance values by linearising the equations of motion, and also in the case where, because of the high unbalance value, the rotor axis describes orbits with an amplitude such that the system's non-linearity cannot be ignored. In both cases the system's stable operation maps have been obtained and verified through numerical integration of the differential equations of motion.

Nonlinear vibration and stability analysis of an electrically actuated piezoelectric nanobeam considering surface effects and intermolecular interactions

2015 ◽  
Vol 23 (12) ◽  
pp. 1873-1889 ◽  
Author(s):  
S Mehrdad Pourkiaee ◽  
Siamak E Khadem ◽  
Majid Shahgholi
Keyword(s):  
Stability Analysis ◽  
Nonlinear Vibration ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Van Der Waals ◽  
Surface Effects ◽  
Van Der Waals Forces ◽  
System Response ◽  
Nonlinear Equations Of Motion ◽  
Piezoelectric Nanobeam

This paper investigates the nonlinear vibration and stability analysis of a doubly clamped piezoelectric nanobeam, as a nano resonator actuated by a combined alternating current and direct current loadings, including surface effects and intermolecular van der Waals forces. The governing equation of motion is obtained using the extended Hamilton principle. The multiple scales method is used to solve nonlinear equations of motion. The influence of van der Waals forces, piezoelectric voltages and surface effects are investigated on the static equilibria, pull-in voltages and dynamic primary resonances of the nano resonator. It is shown that for accurate and exact investigation of the system response, it is necessary to consider the surface effects. To validate the analytical results, numerical simulation is performed. It is seen that the perturbation results are in accordance with numerical results.

Parametric Stability of a Two-Degree-of-Freedom Machine System: Part I—Equations of Motion and Stability

1987 ◽  
Vol 109 (2) ◽  
pp. 210-215 ◽  
Author(s):  
R. I. Zadoks ◽  
A. Midha
Keyword(s):  
Equations Of Motion ◽  
Monodromy Matrix ◽  
Degree Of Freedom ◽  
Computationally Efficient ◽  
Machine System ◽  
Stable Operating ◽  
System Part ◽  
The Stability ◽  
Nonlinear Equations Of Motion ◽  
Two Degree Of Freedom

An important question facing a designer is whether a certain machine system will have a stable operating condition. To date, the investigations which deal with this question have been scarce. This study treats an elastic two-degree-of-freedom system with position-dependent inertia and external forcing. In Part I, the nonlinear equations of motion are derived and linearized about the system’s steady-state rigid-body response. The stability of the linearized equations is examined using Floquet theory, and a computationally efficient method for approximating the monodromy matrix is presented. A specific example is proposed and the results are presented in Part II of this paper.

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