Limitations of the Short Bearing Approximation in Dynamically Loaded Narrow Hydrodynamic Bearings

1993 ◽  
Vol 115 (3) ◽  
pp. 544-549 ◽  
Author(s):  
M. A. Rezvani ◽  
E. J. Hahn
Keyword(s):  
Numerical Solutions ◽  
Parametric Design ◽  
Pressure Profile ◽  
Squeeze Film ◽  
Aspect Ratios ◽  
Transient Solutions ◽  
Dynamically Loaded ◽  
Axial Profile ◽  
Dynamic Loading Conditions ◽  
Improved Accuracy

Transient solutions are still widely used for evaluating the vibrational behavior of rotor bearing systems containing dynamically loaded journal bearings with large unbalance, or noncircular orbit type squeeze film dampers, such as dampers without centralizing springs. For parametric design studies, such transient analyses need rapid means for evaluating the motion dependent fluid film forces and for narrow bearings or dampers (aspect ratios less than 0.5) the short bearing approximation (SBA) to the Reynolds equation has generally been assumed. Comparisons with exact numerical solutions under conditions of static loading and pure squeezing show that the SBA pressure profile predictions are significantly in error for aspect ratios as low as 0.25 at eccentricities around 0.9, whereas the optimal parabolic axial profile approximation (MSBA), while retaining all the rapid calculation features of the SBA, is accurate to within 1 percent under the same conditions and to within 3 percent for aspect ratios around 1.0. Using the MSBA as a yardstick under transient solution conditions, the SBA, while satisfactory for aspect ratios of 0.05, was found to be inadequate in predicting transient and steady state orbits and transmitted forces at aspect ratios of 0.5. At these aspect ratios, jump speeds and instability threshold speeds were also found to be erroneously predicted, with speed overestimates of 30 percent possible for practical unbalance situations. In view of the vastly improved accuracy obtainable by the MSBA, its use is to be preferred to the SBA under dynamic loading conditions for aspect ratios around 0.5, and probably around 0.25 or lower.

Electrohydrodynamic stability: Taylor–Melcher theory for a liquid bridge suspended in a dielectric gas

2002 ◽  
Vol 452 ◽  
pp. 163-187 ◽  
Author(s):  
C. L. BURCHAM ◽  
D. A. SAVILLE
Keyword(s):  
Surface Tension ◽  
Dielectric Constant ◽  
Electric Fields ◽  
Liquid Bridge ◽  
Numerical Solutions ◽  
Specific Conductivity ◽  
Axisymmetric Deformation ◽  
Aspect Ratios ◽  
The Stability ◽  
Theory And Experiment

A liquid bridge is a column of liquid, pinned at each end. Here we analyse the stability of a bridge pinned between planar electrodes held at different potentials and surrounded by a non-conducting, dielectric gas. In the absence of electric fields, surface tension destabilizes bridges with aspect ratios (length/diameter) greater than π. Here we describe how electrical forces counteract surface tension, using a linearized model. When the liquid is treated as an Ohmic conductor, the specific conductivity level is irrelevant and only the dielectric properties of the bridge and the surrounding gas are involved. Fourier series and a biharmonic, biorthogonal set of Papkovich–Fadle functions are used to formulate an eigenvalue problem. Numerical solutions disclose that the most unstable axisymmetric deformation is antisymmetric with respect to the bridge’s midplane. It is shown that whilst a bridge whose length exceeds its circumference may be unstable, a sufficiently strong axial field provides stability if the dielectric constant of the bridge exceeds that of the surrounding fluid. Conversely, a field destabilizes a bridge whose dielectric constant is lower than that of its surroundings, even when its aspect ratio is less than π. Bridge behaviour is sensitive to the presence of conduction along the surface and much higher fields are required for stability when surface transport is present. The theoretical results are compared with experimental work (Burcham & Saville 2000) that demonstrated how a field stabilizes an otherwise unstable configuration. According to the experiments, the bridge undergoes two asymmetric transitions (cylinder-to-amphora and pinch-off) as the field is reduced. Agreement between theory and experiment for the field strength at the pinch-off transition is excellent, but less so for the change from cylinder to amphora. Using surface conductivity as an adjustable parameter brings theory and experiment into agreement.

The Effect of Journal Bearing Misalignment on Load and Cavitation for Non-Newtonian Lubricants

1986 ◽  
Vol 108 (4) ◽  
pp. 645-654 ◽  
Author(s):  
R. H. Buckholz ◽  
J. F. Lin
Keyword(s):  
Reynolds Equation ◽  
Numerical Solutions ◽  
Journal Bearings ◽  
Surface Boundary ◽  
Film Rupture ◽  
Aspect Ratios ◽  
Free Surface Boundary Condition ◽  
Load Carrying ◽  
Boundary Location ◽  
Surface Boundary Condition

An analysis for hydrodynamic, non-Newtonian lubrication of misaligned journal bearings is given. The hydrodynamic load-carrying capacity for partial arc journal bearings lubricated by power-law, non-Newtonian fluids is calculated for small valves of the bearing aspect ratios. These results are compared with: numerical solutions to the non-Newtonian modified Reynolds equation, with Ocvirk’s experimental results for misaligned bearings, and with other numerical simulations. The cavitation (i.e., film rupture) boundary location is calculated using the Reynolds’ free-surface, boundary condition.

Thermal Elastohydrodynamic Lubrication Analysis for Point Contact Transmission

2009 ◽  
Author(s):  
Hai-zhou Huang ◽  
Xi-chuan Niu ◽  
Xiao-yang Yuan
Keyword(s):  
Film Thickness ◽  
Numerical Solutions ◽  
Point Contact ◽  
Elastohydrodynamic Lubrication ◽  
Surface Velocity ◽  
Squeeze Film ◽  
Elastic Displacement ◽  
Circular Arc ◽  
Tooth Width ◽  
Contact Transmission

To investigate the thermal EHL (elastohydrodynamic lubrication) in point contact transmission, a model considering the two-dimensional surface velocity of tooth face and the running-in is proposed. The numerical solutions for pressure, temperature and film thickness distribution in the contact zone are obtained by solving equations including the Reynolds, Energy and the elastic displacement with variable dimension meshing method. The model was used to study the point contact transmission of the circular arc gear in a windlass. The main results show that it is pure rolling along the direction of tooth width, and the rolling speed plays a leading role in improving the lubricating performance and transmission efficiency of circular arc gear. The squeeze film effect makes the pressure peak tend to be gentle and the film thickness increase slightly.

Numerical investigation of supercritical Taylor-vortex flow for a wide gap

1984 ◽  
Vol 138 ◽  
pp. 21-52 ◽  
Author(s):  
H. Fasel ◽  
O. Booz
Keyword(s):  
Flow Field ◽  
Vortex Flow ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Taylor Number ◽  
Taylor Vortex ◽  
Plane Boundary ◽  
Aspect Ratios ◽  
Taylor Vortex Flow ◽  
Wide Gap

For a wide gap (R1/R2= 0.5) and large aspect ratiosL/d, axisymmetric Taylor-vortex flow has been observed in experiments up to very high supercritical Taylor (or Reynolds) numbers. This axisymmetric Taylor-vortex flow was investigated numerically by solving the Navier–Stokes equations using a very accurate (fourth-order in space) implicit finite-difference method. The high-order accuracy of the numerical method, in combination with large numbers of grid points used in the calculations, yielded accurate and reliable results for large supercritical Taylor numbers of up to 100Tac(or 10Rec). Prior to this study numerical solutions were reported up to only 16Tac. The emphasis of the present paper is placed upon displaying and elaborating the details of the flow field for large supercritical Taylor numbers. The flow field undergoes drastic changes as the Taylor number is increased from just supercritical to 100Tac. Spectral analysis (with respect toz) of the flow variables indicates that the number of harmonics contributing substantially to the total solution increases sharply when the Taylor number is raised. The number of relevant harmonics is already unexpectedly high at moderate supercriticalTa. For larger Taylor numbers, the evolution of a jetlike or shocklike flow structure can be observed. In the axial plane, boundary layers develop along the inner and outer cylinder walls while the flow in the core region of the Taylor cells behaves in an increasingly inviscid manner.

On the Oscillatory Instability of Closed-Loop Thermosyphons

1985 ◽  
Vol 107 (4) ◽  
pp. 826-832 ◽  
Author(s):  
K. Chen
Keyword(s):  
Turbulent Flows ◽  
Closed Loop ◽  
Numerical Solutions ◽  
Angular Frequency ◽  
Oscillatory Instability ◽  
Neutral Stability ◽  
Horizontal Segment ◽  
Aspect Ratios ◽  
The Stability ◽  
Laminar And Turbulent Flows

The stability of natural convection flows in single-phase closed-loop thermosyphons is investigated. The thermosyphons considered in the present analysis are fluid-filled tubes bent into rectangular shapes. The fluid is heated over the lower horizontal segment and cooled over the upper horizontal segment. Analytical and numerical solutions are presented for a range of loop aspect ratios and radii for both laminar and turbulent flows. It is found that the steady-state results for thermosyphons with different aspect ratios and radii can be expressed in terms of a single dimensionless parameter. When this parameter is less than a critical value, the flow is always stable. Above this critical point, oscillatory instability exists for a narrow range of a friction parameter. The calculated neutral stability conditions show that the flow is least stable when the aspect ratio of the loop approaches unity. The frequency of the convection-induced oscillation is slightly higher than the angular frequency of a fluid particle traveling along the loop.

SEMI-ANALYTICAL SOLUTIONS FOR THE BRUSSELATOR REACTION–DIFFUSION MODEL

2017 ◽  
Vol 59 (2) ◽  
pp. 167-182 ◽  
Author(s):  
H. Y. ALFIFI
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Partial Differential ◽  
Analytical Results ◽  
Transient Solutions ◽  
The Impact

Semi-analytical solutions are derived for the Brusselator system in one- and two-dimensional domains. The Galerkin method is processed to approximate the governing partial differential equations via a system of ordinary differential equations. Both steady-state concentrations and transient solutions are obtained. Semi-analytical results for the stability of the model are presented for the identified critical parameter value at which a Hopf bifurcation occurs. The impact of the diffusion coefficients on the system is also considered. The results show that diffusion acts to stabilize the systems better than the equivalent nondiffusive systems with the increasing critical value of the Hopf bifurcation. Comparison between the semi-analytical and numerical solutions shows an excellent agreement with the steady-state transient solutions and the parameter values at which the Hopf bifurcations occur. Examples of stable and unstable limit cycles are given, and Hopf bifurcation points are shown to confirm the results previously calculated in the Hopf bifurcation map. The usefulness and accuracy of the semi-analytical results are confirmed by comparison with the numerical solutions of partial differential equations.

Inertial effect on gas squeeze film for large radius disc excited by standing waves with complex modal shapes

2019 ◽  
Vol 33 (24) ◽  
pp. 1950282 ◽  
Author(s):  
Yi Qiang Fan ◽  
M. Miyatake ◽  
S. Kawada ◽  
Bin Wei ◽  
S. Yoshimoto
Keyword(s):  
Reynolds Equation ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Standing Waves ◽  
Inertial Effect ◽  
Squeeze Film ◽  
Navier Stokes ◽  
Large Radius ◽  
Navier Stokes Equations ◽  
Sinusoidal Functions

In order to investigate the gas inertial effect on bearing capacity of acoustic levitation on condition of complex exciting shapes, a new kind of numerical model including inertial effect in cylindrical coordinates was proposed. The inertial terms in Navier–Stokes equations are packaged to derive modified Reynolds equations. The amplitudes of standing waves were tested by distance probe in experiment and film thickness equation were reconstructed by sum of the sinusoidal functions. The theoretical and experimental results implied that the inertial effect is strongly related to the exciting modal shapes. It is concluded that the proposal of modified Reynolds equation can provide more optimized numerical solutions to solve the problems about the deviation between theoretical and experimental data.

scholarly journals Design and numerical simulation for the development of an expandable paediatric heart valve

2020 ◽  
pp. 039139882097750
Author(s):  
Monica M Kerr ◽  
Terence Gourlay
Keyword(s):  
Numerical Simulation ◽  
Aortic Valve ◽  
Immersed Boundary Method ◽  
Inverse Relationship ◽  
Parametric Design ◽  
Somatic Growth ◽  
Parameter Design ◽  
Immersed Boundary ◽  
Aspect Ratios ◽  
Base Radius

Current paediatric valve replacement options cannot compensate for somatic growth, leading to an obstruction of flow as the child outgrows the prosthesis. This often necessitates an increase in revision surgeries, leading to legacy issues into adulthood. An expandable valve concept was modelled with an inverse relationship between annulus size and height, to retain the leaflet geometry without requiring additional intervention. Parametric design modelling was used to define certain valve parameter aspect ratios in relation to the base radius, Rb, including commissural radius, Rc, valve height, H and coaptation height, x. Fluid-structure simulations were subsequently carried out using the Immersed Boundary method to radially compress down the fully expanded aortic valve whilst subjecting it to diastolic and systolic loading cycles. Leaflet radial displacements were analysed to determine if valve performance is likely to be compromised following compression. Work is ongoing to optimise valvular parameter design for the paediatric patient cohort.

Vibration analysis of a rotor supported on magnetorheological squeeze film damper with short bearing approximation: A contrast between short and long bearing approximations

2015 ◽  
Vol 23 (11) ◽  
pp. 1792-1808 ◽  
Author(s):  
Mostafa Irannejad ◽  
Abdolreza Ohadi
Keyword(s):  
Dynamic Behavior ◽  
Electrical Current ◽  
Magnetorheological Fluid ◽  
Squeeze Film ◽  
Fluid Viscosity ◽  
Squeeze Film Damper ◽  
Fluid Forces ◽  
Rotating Systems ◽  
Aspect Ratios ◽  
Variable Damping

Squeeze film dampers are widely used to reduce the vibration of rotating systems. Using magnetorheological fluid in these dampers can lead to a variable-damping damper called Magnetorheological Squeeze Film Damper (MRSFD). Magnetorheological fluid viscosity alter under different values of magnetic field. The previous research have widely used long bearing approximation to derive the equations governing the hydrodynamic behavior of MRSFDs. In this paper, the behavior of MRSFDs has been studied using short bearing approximation. Next, the effects of MRSFDs on the dynamic behavior of a flexible rotor have been studied, using finite element method (FEM). Synchronous whirl motion has not been imposed on the system behavior, as an external assumption. Damper pressure distribution and forces, dynamic trajectories, eccentricity and the frequency response of the rotor are tools used to analyze the dynamic behavior of MRSFDs and rotor system. As the results show, it seems to be more precise to use short bearing approximation to analyze dampers with aspect ratios lower than a limit (especially L/D < 1). Furthermore, by controlling electrical current one can control the dynamic behavior of a rotor, to avoid failure and damage. Finally, the whirl motion of the rotor was observed to remain synchronous, even when fluid forces are present.

Nonlinear Response of Short Squeeze Film Dampers

1980 ◽  
Vol 102 (1) ◽  
pp. 51-58 ◽  
Author(s):  
D. L. Taylor ◽  
B. R. K. Kumar
Keyword(s):  
Frequency Response ◽  
Phase Angle ◽  
Transient Response ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Polar Coordinates ◽  
Squeeze Film ◽  
Response Curves ◽  
Squeeze Film Damper ◽  
Fluid Forces

This paper considers the methodology of numerical integration for prediction of dynamic response of squeeze film damper systems. A planar rotor carried in a squeeze film damper with linear centering spring is considered. Governing differential equations are expressed in polar coordinates, and fluid forces are obtained from the Ocvirk short bearing integrals. The rotating unbalance response is presented as a function of speed, unbalance, and a bearing parameter. Runge Kutta integration techniques are used to obtain numerical solutions for transient response and frequency response. The 2π film approximation results in almost linear frequency response curves. However, the π film response is very nonlinear, demonstrating the well known multiple valued response and associated hardening jump/drop phenomenon. The π film transient response is analyzed within the speed range of bistable operation to determine the effects of initial conditions, the domains of convergence, and the relative strengths of stability of each solution. The transient response is found to be most sensitive to initial values of phase angle and phase angle velocity. Initial eccentricity and eccentric velocity are much less important. In general, of the two steady state solutions, the one with lower eccentricity appears to be more stable, with a larger domain of convergence. Examples show how premature termination of the integration can lead to erroneous conclusions.

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