Analytical Solution to the Hydrodynamic Lubrication of Fan-Shaped Thrust Step Bearings

2010 ◽  
Vol 132 (2) ◽  
Author(s):  
Shuangbiao Liu ◽  
W. Wayne Chen ◽  
Diann Y. Hua
Keyword(s):  
Analytical Solution ◽  
Flow Rate ◽  
Analytical Solutions ◽  
Hydrodynamic Lubrication ◽  
Load Capacity ◽  
Finite Width ◽  
Inner Radius ◽  
Cylindrical Coordinate ◽  
Torque Loss ◽  
Lubrication Models

Step bearings are frequently used in industries for better load capacities. Analytical solutions to the Rayleigh step bearing and a rectangular slider with a finite width are available in literature, but none for a fan-shaped thrust step bearing. This study starts with a known solution to the Laplace equation in a cylindrical coordinate system, which is in the form of an infinite summation. A set of analytical solutions to pressure, load capacity, flow rate, and torque loss is derived in this paper for hydrodynamic lubrication problems encountered in the fan-shaped step bearing. These analytical solutions are compared with those for the rectangular slider and the Rayleigh step bearing to reveal relationships among them. When the inner radius becomes smaller, the load capacity increases, almost linearly in a certain region. The effects of inner radius, step height, and step location on pressure distribution and load capacity are studied in general and under a specific set of bearing geometry as an example. The presented solutions can be useful for designers to maximize bearing performance as well as for researchers to benchmark numerical lubrication models.

Hydrodynamic Pressure of Thrust Step Bearings

2009 ◽  
Author(s):  
Shuangbiao Liu ◽  
W. Wayne Chen ◽  
Diann Y. Hua
Keyword(s):  
Analytical Solution ◽  
Coordinate System ◽  
Laplace Equation ◽  
Analytical Solutions ◽  
Hydrodynamic Lubrication ◽  
Load Capacity ◽  
Hydrodynamic Pressure ◽  
Finite Width ◽  
Cylindrical Coordinate ◽  
Lubrication Models

Step bearings are frequently used in industry for better load capacity. Analytical solutions to the Rayleigh step bearing and a rectangular slider with a finite width are available in literature, but none for a fan-shaped thrust step bearing. This study starts with a known solution to the Laplace equation in a cylindrical coordinate system, which is in the form of infinite summation. An analytical solution to pressure is derived in this paper for hydrodynamic lubrication problems encountered in the fan-shaped step bearing. The presented solutions can be useful for designers to maximize bearing performance as well as for researchers to benchmark numerical lubrication models.

Steady State Performance of a Hydrodynamic Journal Bearing With a Pseudoplastic Lubricant

1979 ◽  
Vol 101 (4) ◽  
pp. 497-502 ◽  
Author(s):  
C. Rajalingham ◽  
B. S. Prabhu ◽  
B. V. A. Rao
Keyword(s):  
Steady State ◽  
Flow Rate ◽  
Journal Bearing ◽  
Hydrodynamic Lubrication ◽  
Load Capacity ◽  
Slenderness Ratio ◽  
Correlation Effect ◽  
Polynomial Type ◽  
Successive Over Relaxation ◽  
Steady State Performance

Starting from the general shear stress-shear strain relation of the odd cubic polynomial type considering correlation effect for pseudoplastic (shear thinning) lubricant, a modified form of Reynolds’ equation (nonlinear) is derived under conventional hydrodynamic lubrication approximations. The finite difference technique with successive over relaxation is used adopting the solution of the associated linear equation as a first approximation to obtain the pressure distribution of a finite cylindrical journal bearing incorporating Reynolds’ boundary conditions. Steady state performance characteristics such as load capacity, attitude, friction and flow rate are obtained for various values of nonlinear parameter and bearing slenderness ratio. The results presented in nondimensional form are compared with established results for Newtonian lubricants. The investigation shows a significant effect of correlation on the bearing load capacity and flow rate.

Zur Lösung der BETHE-GOLDSTONE-Gleichung bei nicht-verschwindendem Gesamtimpuls I

1963 ◽  
Vol 18 (4) ◽  
pp. 531-538
Author(s):  
Dallas T. Hayes
Keyword(s):  
Approximate Solution ◽  
Analytical Solution ◽  
Nuclear Matter ◽  
Analytical Solutions ◽  
Projection Operator ◽  
Center Of Mass ◽  
Separable Potential ◽  
Localized Solutions ◽  
Non Local ◽  
Square Well

Localized solutions of the BETHE—GOLDSTONE equation for two nucleons in nuclear matter are examined as a function of the center-of-mass momentum (c. m. m.) of the two nucleons. The equation depends upon the c. m. m. as parameter due to the dependence upon the c. m. m. of the projection operator appearing in the equation. An analytical solution of the equation is obtained for a non-local but separable potential, whereby a numerical solution is also obtained. An approximate solution for small c. m. m. is calculated for a square-well potential. In the range of the approximation the two analytical solutions agree exactly.

Performance of Hydrostatic Spherical Gas Gyro Bearings

1976 ◽  
Vol 98 (1) ◽  
pp. 111-116 ◽  
Author(s):  
A. Gu ◽  
L. Cziglenyi
Keyword(s):  
Numerical Solution ◽  
Flow Rate ◽  
Gas Flow ◽  
Load Capacity ◽  
Performance Data ◽  
Drag Torque ◽  
Fixed Bearing ◽  
Gas Flow Rate

Analysis and method of numerical solution for evaluating the performance of hydrostatic spherical gas gyro bearings at any gimbal angle and at any eccentricity have been developed. Performance data on load capacity, gas flow rate, drag torque, and error torque over some ranges of gimbal angle and eccentricity are presented. Comparison has been made between the equatorially vented and nonvented bearings of fixed bearing angles.

Hydrodynamic Lubrication of Pocket Bearings and Effect of Contouring the Inner and Outer Regions, and Optimum Design Data

1989 ◽  
Author(s):  
C. Bagci ◽  
C. J. McClure ◽  
S. K. Rajavenkateswaran
Keyword(s):  
Optimum Design ◽  
Hydrodynamic Lubrication ◽  
Load Capacity ◽  
Load Flow ◽  
Temperature Dependent Viscosity ◽  
Design Data ◽  
Planar Surfaces ◽  
Friction Factors ◽  
The Coefficient Of Friction ◽  
Dependent Viscosity

Abstract The article investigates pocket bearings with contoured profiles of exponential forms on both surfaces inside and outside of the step boundary forming hydro-dynamic action surfaces, and develops optimum design data yielding efficient slider bearings with small pockets with higher load capacities than conventional pocket bearings. In the case of a pocket bearings, in addition to the Reynolds equation used for the regions inside and outside the pocket, the continuity equation along the pocket boundary is satisfied to form the complete model of the bearing. The optimum design data includes dimensionless load-, flow-, temperature rise-, power loss-, stiffness-, and the coefficient of friction factors. Incompressible lubricant with temperature dependent viscosity is considered. Detailed study of conventional pocket bearings with planar surfaces is included. Some optimum exponential pocket bearings yield up to 561 percent increase in load capacity as compared to the conventional tapered bearings.

Two-dimensional buoyant plumes in a uniform co-flow

2021 ◽  
Vol 932 ◽  
Author(s):  
Gary R. Hunt ◽  
Jamie P. Webb
Keyword(s):  
Theoretical Investigation ◽  
Analytical Solutions ◽  
Richardson Number ◽  
Relative Magnitude ◽  
Dynamical Behaviour ◽  
Finite Width ◽  
Two Dimensional ◽  
Buoyant Plumes ◽  
The Subject ◽  
Flow Case

The behaviour of turbulent, buoyant, planar plumes is fundamentally coupled to the environment within which they develop. The effect of a background stratification directly influences a plumes buoyancy and has been the subject of numerous studies. Conversely, the effect of an ambient co-flow, which directly influences the vertical momentum of a plume, has not previously been the subject of theoretical investigation. The governing conservation equations for the case of a uniform co-flow are derived and the local dynamical behaviour of the plume is shown to be characterised by the scaled source Richardson number and the relative magnitude of the co-flow and plume source velocities. For forced, pure and lazy plume release conditions the co-flow acts to narrow the plume and reduce both the dilution and the asymptotic Richardson number relative to the classic zero co-flow case. Analytical solutions are developed for pure plumes from line sources, and for highly forced and highly lazy releases from sources of finite width in a weak co-flow. Contrary to releases in quiescent surroundings, our solutions show that all classes of release can exhibit plume contraction and the associated necking. For entraining plumes, a dynamical invariance spatially only occurs for pure and forced releases and we derive the co-flow strengths that lead to this invariance.

scholarly journals Dynamic Analytical Solution of a Piezoelectric Stack Utilized in an Actuator and a Generator

2018 ◽  
Vol 8 (10) ◽  
pp. 1779 ◽  
Author(s):  
Xinnan Liu ◽  
Jianjun Wang ◽  
Weijie Li
Keyword(s):  
Analytical Solution ◽  
Dynamic Characteristics ◽  
Analytical Solutions ◽  
External Load ◽  
Piezoelectric Actuators ◽  
Displacement Method ◽  
Proposed Model ◽  
Special Cases ◽  
Piezoelectric Stacks ◽  
Comprehensive Manner

This paper presents the dynamic analytical solution of a piezoelectric stack utilized in an actuator and a generator based on the linear piezo-elasticity theory. The solutions for two different kinds of piezoelectric stacks under external load were obtained using the displacement method. The effects of load frequency and load amplitude on the dynamic characteristics of the stacks were discussed. The analytical solutions were validated using the available experimental results in special cases. The proposed model is able not only to predict the output properties of the devices, but also to reflect the inner electrical and mechanical components, which is helpful for designing piezoelectric actuators and generators in a comprehensive manner.

Modeling and experimental research on engaging characteristics of wet clutch

2019 ◽  
Vol 71 (1) ◽  
pp. 94-101 ◽  
Author(s):  
Yanzhong Wang ◽  
Yuan Li ◽  
Yang Liu ◽  
Wei Zhang
Keyword(s):  
Contact Surface ◽  
Hydrodynamic Lubrication ◽  
Load Capacity ◽  
Lubricating Oil ◽  
Centrifugal Effect ◽  
Element Analysis ◽  
Content Type ◽  
Clutch Engagement ◽  
Wet Clutch ◽  
Engagement Process

PurposeTo gain in-depth understandings of engaging characteristics, the purpose of this paper is to improve the model of wet clutches to predict the transmitted torque during the engagement process.Design/methodology/approachThe model of wet clutch during the engagement process took main factors into account, such as the centrifugal effect of lubricant, permeability of friction material, slippage factor of lubricant on contact surface and roughness of contact surface. Reynolds’ equation was derived to describe the hydrodynamic lubrication characteristics of lubricant film between the friction plate and the separated plate, and an elastic-plastic model of the rough surfaces contact based on the finite element analysis was used to indicate the loading force and friction torque of the contact surface.FindingsThe dynamic characteristics of wet clutch engagement time, relative speed, hydrodynamic lubrication of lubricating oil, rough surface contact load capacity and transfer torque can be obtained by the wet clutch engagement model. And the influence of the groove shape and depth on the engaging characteristics is also analyzed.Originality/valueThe mathematical model of the wet clutch during the engagement process can be used to predict the engaging characteristics of the wet clutch which could be useful to the design of the wet clutch.

Behavior of Thermal Effects and Velocity-Slip on Performance of Externally Pressurized Porous Incompressible Gas Thrust Bearing

1979 ◽  
Vol 46 (2) ◽  
pp. 465-468 ◽  
Author(s):  
V. K. Kapur ◽  
J. S. Yadav
Keyword(s):  
Mass Flow Rate ◽  
Mass Flow ◽  
Flow Rate ◽  
Thermal Effects ◽  
Thrust Bearing ◽  
Load Capacity ◽  
Velocity Slip ◽  
Experimental Results ◽  
Slip Condition ◽  
Static Stiffness

In the present analysis, the interactions of thermal effects and velocity slip on the performance of externally pressurized porous incompressible gas thrust bearing have been studied. Numerical results for load capacity, mass flow rate, and static stiffness have been obtained and their behavior is illustrated in figures. The results for slip as well as no-slip condition have also been compared with the experimental results of Gargiulo and Gilmour [7].

scholarly journals CONCERNING THE EXTRACTION OF A SLACK SOIL LAYER FROM COMPRESSIBLE THICKNESS OF FOUNDATION STRATUM OF FINITE WIDTH FOUNDATIONS

2019 ◽  
Vol 7 (4) ◽  
pp. 49-56
Author(s):  
Zaven Ter-Martirosyan ◽  
Armen Ter-martirosyan ◽  
Valery DEMYANENKO
Keyword(s):  
Analytical Solutions ◽  
Soft Soil ◽  
Soil Layer ◽  
Horizontal Displacement ◽  
Creep Model ◽  
Shear Stresses ◽  
Finite Width ◽  
Relative Depth ◽  
Weak Layer ◽  
Horizontal Displacements

The paper provides a quantitative assessment of the deflected mode of foundation stratum of finite width foundation, in the compressible thickness of which there is a slack clay soil layer. A number of criteria for assessing the possibility or impossibility of extruding a slack layer depending on its strength and rheological properties, as well as the relative thickness of the layer to its length (h/l) and the relative depth of the layer (h/d) have been given. Closed analytical solutions are given to determine the rate of Foundation precipitation depending on the rate of extrusion of the weak layer, including taking into account the damped and undamped creep. The analytical solutions in the article are supported by the graphical part made with the help of the Mathcad program. Plots of changes in shear stresses in the layer along the x axis at different distances from the axis and at different values 0, contours of horizontal displacement velocities in the weak layer at different distances from the x axis, plots of horizontal displacement velocities in the middle of the weak layer and plots of horizontal displacement velocities in the weak layer at different distances from the x axis are given. As a calculation model for describing the creep of a slack layer, rheological ones of the soil using power and hyperbolic functions and their modifications have been considered. In addition, most modern rheological models that take into account soil hardening during creep have been considered. Based on these models, the problem is solved by means analytical and numerical methods using the Mathcad PC and the PLAXIS PC according to the Soft Soil Creep model. The graphical part shows the isofields of horizontal displacements for 300 days and 600 days and the corresponding contours of horizontal displacements.

Sign in / Sign up

Export Citation Format

Share Document