On the Dynamics of Gas Lubricated Triboelements

2007 ◽  
Author(s):  
Itzhak Green
Keyword(s):  
Gas Turbines ◽  
High Speed ◽  
Reynolds Equation ◽  
Equations Of Motion ◽  
Series Representation ◽  
Prony Series Representation ◽  
Stiffness And Damping ◽  
Orbit Codes ◽  
Mathematical Justification ◽  
First Time

Gas lubricated triboelements are ubiquitous in many applications, from gas turbines seals, high speed dental drills, cryogenic refrigerators, oil-free bearings for turbo chargers, to read-write self acting heads in magnetic recording. In all these application the film must be maintained sufficiently precise especially in the presence of disturbances. Hence, it is insufficient to analyze such triboelements quasi statically, where a complete dynamic analysis of the system is needed. Numerous techniques have been developed over the past five decades, some are purely numerical, and others are semi-analytical. Complete analytical methods are practically non-existent because of the non-linear nature of the Reynolds equation. This work discusses a few of the common techniques, explores their intricate, and highlights limitations and flaws in implementation. Particularly: (1) in the so called “orbit codes” the equations of motion and the Reynolds equation must be solved simultaneously (and not “side-by-side”), (2) a common representation of linearized stiffness and damping is shown to be produce unrealistic results even in the simplest application, and (3) for the first time a mathematical justification is given for the use of Prony series representation of the hereditary stiffness in the so called “step-jump” method.

scholarly journals High-Speed Rotor Analytical Dynamics on Flexible Foundation Subjected to Internal and External Excitation

2016 ◽  
Vol 46 (4) ◽  
pp. 3-18
Author(s):  
Venelin S. Jivkov ◽  
Evtim V. Zahariev
Keyword(s):  
Gas Turbines ◽  
High Speed ◽  
Asymptotic Methods ◽  
Numerical Procedure ◽  
Rigid Bodies ◽  
Dynamics Simulation ◽  
Geometrical Approach ◽  
Rotating Machine ◽  
Exciting Frequency ◽  
Stiffness And Damping

Abstract The paper presents a geometrical approach to dynamics simulation of a rigid and flexible system, compiled of high speed rotating machine with eccentricity and considerable inertia and mass. The machine is mounted on a vertical flexible pillar with considerable height. The stiffness and damping of the column, as well as, of the rotor bearings and the shaft are taken into account. Non-stationary vibrations and transitional processes are analyzed. The major frequency and modal mode of the flexible column are used for analytical reduction of its mass, stiffness and damping properties. The rotor and the foundation are modelled as rigid bodies, while the flexibility of the bearings is estimated by experiments and the requirements of the manufacturer. The transition effects as a result of limited power are analyzed by asymptotic methods of averaging. Analytical expressions for the amplitudes and unstable vibrations throughout resonance are derived by quasi-static approach increasing and decreasing of the exciting frequency. Analytical functions give the possibility to analyze the influence of the design parameter of many structure applications as wind power generators, gas turbines, turbo-generators, and etc. A numerical procedure is applied to verify the effectiveness and precision of the simulation process. Nonlinear and transitional effects are analyzed and compared to the analytical results. External excitations, as wave propagation and earthquakes, are discussed. Finite elements in relative and absolute coordinates are applied to model the flexible column and the high speed rotating machine. Generalized Newton - Euler dynamics equations are used to derive the precise dynamics equations. Examples of simulation of the system vibrations and nonstationary behaviour are presented.

Dynamic Analysis of High-Speed Hybrid Thrust Bearings

2013 ◽  
Vol 365-366 ◽  
pp. 304-308
Author(s):  
Lei Wang
Keyword(s):  
Dynamic Analysis ◽  
High Speed ◽  
Reynolds Equation ◽  
Thrust Bearing ◽  
Dynamic Performance ◽  
Orifice Diameter ◽  
Thrust Bearings ◽  
Supply Pressure ◽  
Bearing Stiffness ◽  
Stiffness And Damping

An analysis is conducted and solutions are provided for the dynamic performance of high speed hybrid thrust bearing. By adopting bulk flow theory, the turbulent Reynolds equation is solved numerically with the different orifice diameter and supply pressure. The results show that increasing supply pressure can significantly improve the bearing stiffness and damping, while the orifice diameters make a different effect on the bearing stiffness and damping.

Turbulent Flow, Flexure-Pivot Hybrid Bearings for Cryogenic Applications

1996 ◽  
Vol 118 (1) ◽  
pp. 190-200 ◽  
Author(s):  
Luis San Andres
Keyword(s):  
High Speed ◽  
Energy Transport ◽  
Load Capacity ◽  
Equations Of Motion ◽  
Liquid Oxygen ◽  
Hydrodynamic Bearings ◽  
Numerical Predictions ◽  
Tilting Pad ◽  
Direct Stiffness ◽  
Stiffness And Damping

The thermal analysis of flexure-pivot tilting-pad hybrid (combination hydrostatic-hydrodynamic) bearings for cryogenic turbopumps is presented. The advantages of this type of bearing for high speed operation are discussed. Turbulent bulk-flow, variable properties, momentum and energy transport equations of motion govern the flow in the bearing pads. Zeroth-order equations for the flow field at a journal equilibrium position render the bearing flow rate, load capacity, drag torque, and temperature rise. First-order equations for perturbed flow fields due to small amplitude journal motions provide rotordynamic force coefficients. A method to determine the tilting-pad moment coefficients from the force displacement coefficients is outlined. Numerical predictions correlate well with experimental measurements for tilting-pad hydrodynamic bearings. The design of a liquid oxygen, flexure-pad hybrid bearing shows a reduced whirl frequency ratio and without loss in load capacity or reduction in direct stiffness and damping coefficients.

Analysis of Hybrid (Hydrodynamic/Hydrostatic) Journal Bearing

2013 ◽  
Vol 650 ◽  
pp. 385-390 ◽  
Author(s):  
Vijay Kumar Dwivedi ◽  
Satish Chand ◽  
K.N. Pandey
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Gas Turbines ◽  
High Speed ◽  
Reynolds Equation ◽  
Journal Bearing ◽  
Load Capacity ◽  
Natural Boundary Condition ◽  
Dimensional Solution

The Hybrid (hydrodynamic/ hydrostatic) journal bearing system has found wide spread application in high speed rotating machines such as compressors, gas turbines, steam turbines, etc. The present studies include solution of Reynolds equation for hydrodynamic journal bearing with infinitely long approximation (ILA), infinitely short bearing approximation (ISA) and finite journal bearing approximation. Further Finite Journal bearing approximation considers two dimensional solution of Reynolds equation with natural boundary condition, which cannot be solved by analytical method. So, here the solutions for finite journal bearing have been done with finite difference method (a MATLAB® code is prepared for finite difference method) to get bearing performance parameters such as load capacity, Sommerfeld no., etc.

Dynamic Analysis of a High-Speed Rotor-Ball Bearing System Under Elastohydrodynamic Lubrication

2014 ◽  
Vol 136 (6) ◽  
Author(s):  
Yu-Yan Zhang ◽  
Xiao-Li Wang ◽  
Xiao-Qing Zhang ◽  
Xiao-Liang Yan
Keyword(s):  
High Speed ◽  
Ball Bearing ◽  
Chaotic Behavior ◽  
Equations Of Motion ◽  
Power Spectra ◽  
Elastohydrodynamic Lubrication ◽  
Lubricated Contacts ◽  
Rolling Elements ◽  
Stiffness And Damping ◽  
Bearing System

The nonlinear dynamic behaviors of a high-speed rotor-ball bearing system under elastohydrodynamic lubrication (EHL) are investigated. First, the numerical curve fittings for stiffness and damping coefficients of lubricated contacts between rolling elements and races are undertaken, and then the fitted formulae are introduced to the equations of motion of the rotor-ball bearing system to investigate its nonlinear characteristics. Furthermore, the time responses, power spectra, phase trajectories, orbit plots, and bifurcation diagrams for cases of ignoring and considering the lubrication condition in bearings are inspected and compared. The results indicate that, when lubrication is taken into account, the amplitudes of vibration displacements and velocities of the rotor system increase, and the appearance of different regions of periodic, quasi-periodic, and chaotic behavior is strongly dependent on the speed and load.

Lubricant Inertia in Water Lubricated Bearings

2017 ◽  
Author(s):  
Xin Deng ◽  
Cori Watson ◽  
Brian Weaver ◽  
Houston Wood ◽  
Roger Fittro
Keyword(s):  
High Speed ◽  
Reynolds Equation ◽  
Current Status ◽  
Shear Stresses ◽  
Turbulent Regime ◽  
Fluid Inertia ◽  
Film Pressure ◽  
Inertia Effects ◽  
Inertia Model ◽  
Stiffness And Damping

Oil-lubricated bearings are widely used in high speed rotating machines such as those used in the aerospace and automotive industries. However, with some applications including underwater machinery and environmentally friendly applications, water lubricated bearings have become increasingly used. Due to the different fluid properties between oil and water — namely viscosity — the use of water increases the Reynolds numbers drastically and, therefore, makes water-lubricated bearings prone to turbulence and fluid inertia effects. In other words, the linear approximation of the fluid film reaction forces due to the stiffness and damping parameters — as suggested in the traditional Reynolds equation — is not adequate and should be amended to include lubricant added mass. This is because water-lubricated bearings exhibit large lubricant inertia forces on the order of viscous forces. Additionally, stiffness and damping coefficients should be calculated with the turbulence effects included. The aim of this study was to investigate the methodology of modifying the traditional Reynolds equation to include lubricant inertia effects. This paper reviews the current status of research in the lubricant inertia of bearings and explores the development of methodologies to modify the Reynolds equation to include lubricant inertia in bearings. The Reynolds equation is a partial differential equation governing the pressure distribution of thin viscous fluid films in lubrication theory. The thin film hypothesis is used to directly relate the bearing film thickness to the lubricant film pressure. Adding lubricant inertia to the Reynolds equation is vital to improving the accuracy of the bearing model and more specifically its film pressure which is essential to predicting load carrying capabilities. The film pressure relates the gradient of the velocity tensor through the Reynolds equation, and resulting shear stresses then allow the turbulent momentum equations to be written in terms of an eddy-viscosity value. An extended Reynolds equation should be developed which takes into account turbulence and both convective and temporal inertia. The most complete form of the temporal inertia effect model should be developed and applied to the turbulent regime, consisting of both primary and secondary temporal inertia terms. The convective inertia model follows Constantinescu’s approach. This analysis develops a lubricant inertia model applicable to water-lubricated bearings. The results of this study could aid in improving future designs and models of water-lubricated bearings.

Pressure Boundary Layer in a Transient or Steady Gas Film and Mass Interaction

1970 ◽  
Vol 37 (4) ◽  
pp. 945-953 ◽  
Author(s):  
F. C. Hsing ◽  
H. S. Cheng
Keyword(s):  
Boundary Layer ◽  
Steady State ◽  
High Speed ◽  
Numerical Scheme ◽  
Reynolds Equation ◽  
Equations Of Motion ◽  
Pressure Profile ◽  
Tilting Pad ◽  
Static Solutions ◽  
Gas Bearing

This paper presents a numerical scheme capable of yielding accurate pressure profile for the transient and steady hydrodynamic gas film generated by high-speed relative motion of two nonparallel surfaces. The numerical difficulties associated with high compressibility numbers for the gas film Reynolds equation were overcome by employing a set of systematically generated irregular grid spacings based on a coordinate transformation. By coupling the fluid-film solution with the equations of motion of a tilting pad, the dynamics of the mass film interaction were treated. Results are presented for both steady-state and dynamical solutions. Static solutions for a 120-deg partial-arc gas bearing have been used for comparison.

Limiting Stiffness and Damping Coefficients of Foil Bearing

2005 ◽  
Author(s):  
Jerzy T. Sawicki ◽  
T. V. V. L. N. Rao
Keyword(s):  
High Speed ◽  
Reynolds Equation ◽  
Necessary Conditions ◽  
Load Capacity ◽  
Damping Coefficients ◽  
Foil Bearing ◽  
Infinitesimal Perturbation ◽  
Stiffness And Damping ◽  
Limiting Values ◽  
Gas Journal Bearing

The limiting values of load capacity, stiffness and damping coefficients for a foil bearing are presented. The necessary conditions for high bearing numbers (journal operating at high speed) are obtained by simplifying the compressible Reynolds equation. Linearized stiffness and damping coefficients are obtained using infinitesimal perturbation method. Results of load capacity, stiffness and damping coefficients, for foil bearing are compared with those obtained for a rigid gas journal bearing. The limiting values of dynamic characteristics for a foil bearing are constant for all eccentricity ratios.

Dynamics of Air-Lubricated Slider Bearings for Noncontact Magnetic Recording

1971 ◽  
Vol 93 (2) ◽  
pp. 272-278 ◽  
Author(s):  
T. Tang
Keyword(s):  
High Speed ◽  
Degrees Of Freedom ◽  
Reynolds Equation ◽  
Equations Of Motion ◽  
Two Dimensions ◽  
Direct Access ◽  
Slider Bearing ◽  
Storage Devices ◽  
Disk Storage ◽  
Slider Bearings

One of the key technologies which led to the success of modern magnetic disk storage devices is the development of self acting gas lubricated slider bearings for positioning a magnetic head precisely over a high speed rotating recording disk. This paper covers a dynamic simulation of such an air bearing system used in direct access disk storage devices. In the simulation model, the Reynolds equation, which describes the dynamics of the lubricating air film, is solved by finite difference techniques in two dimensions and time for compressible, isothermal flow. The equations of motion of the slider bearing are solved simultaneously with the Reynolds equation for three degrees of freedom. Applications of the simulation are demonstrated, and experimental measurements to verify the theory are presented and discussed.

Ball bearing turbocharger vibration management: application on high speed balancer

2020 ◽  
Vol 21 (6) ◽  
pp. 619
Author(s):  
Kostandin Gjika ◽  
Antoine Costeux ◽  
Gerry LaRue ◽  
John Wilson
Keyword(s):  
Dynamic Behavior ◽  
High Speed ◽  
Internal Combustion Engines ◽  
Ball Bearing ◽  
Squeeze Film ◽  
Design And Technology ◽  
Squeeze Film Damper ◽  
Damping Coefficients ◽  
Bearing Loads ◽  
Stiffness And Damping

Today's modern internal combustion engines are increasingly focused on downsizing, high fuel efficiency and low emissions, which requires appropriate design and technology of turbocharger bearing systems. Automotive turbochargers operate faster and with strong engine excitation; vibration management is becoming a challenge and manufacturers are increasingly focusing on the design of low vibration and high-performance balancing technology. This paper discusses the synchronous vibration management of the ball bearing cartridge turbocharger on high-speed balancer and it is a continuation of papers [1–3]. In a first step, the synchronous rotordynamics behavior is identified. A prediction code is developed to calculate the static and dynamic performance of “ball bearing cartridge-squeeze film damper”. The dynamic behavior of balls is modeled by a spring with stiffness calculated from Tedric Harris formulas and the damping is considered null. The squeeze film damper model is derived from the Osborne Reynolds equation for incompressible and synchronous fluid loading; the stiffness and damping coefficients are calculated assuming that the bearing is infinitely short, and the oil film pressure is modeled as a cavitated π film model. The stiffness and damping coefficients are integrated on a rotordynamics code and the bearing loads are calculated by converging with the bearing eccentricity ratio. In a second step, a finite element structural dynamics model is built for the system “turbocharger housing-high speed balancer fixture” and validated by experimental frequency response functions. In the last step, the rotating dynamic bearing loads on the squeeze film damper are coupled with transfer functions and the vibration on the housings is predicted. The vibration response under single and multi-plane unbalances correlates very well with test data from turbocharger unbalance masters. The prediction model allows a thorough understanding of ball bearing turbocharger vibration on a high speed balancer, thus optimizing the dynamic behavior of the “turbocharger-high speed balancer” structural system for better rotordynamics performance identification and selection of the appropriate balancing process at the development stage of the turbocharger.

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