Asymptotic Methods for a General Finite Width Gas Slider Bearing

1978 ◽  
Vol 100 (2) ◽  
pp. 254-260 ◽  
Author(s):  
J. A. Schmitt ◽  
R. C. DiPrima
Keyword(s):  
Film Thickness ◽  
Asymptotic Expression ◽  
Asymptotic Methods ◽  
Finite Width ◽  
Slider Bearing ◽  
Sliding Direction ◽  
Special Cases ◽  
Load Carrying ◽  
The Individual ◽  
Bearing Number

The method of matched asymptotic expansions is used to develop an asymptotic expression for the load-carrying capacity of a finite width gas slider bearing for large bearing numbers and for film thicknesses varying both in the sliding and transverse directions. The individual terms in the formula for the load are independent of the bearing number and are related to the interior portion, the side edge boundary layers, and the trailing edge boundary layer of the bearing. Only the terms associated with the side leakage phenomena must be computed numerically. Two special cases are discussed: (i) the film thickness varying only in the sliding direction, and (ii) the film thickness having linear or parabolic variation in the sliding direction and parabolic variation in the transverse direction.

Asymptotic Methods for an Infinite Slider Bearing With a Discontinuity in Film Slope

1976 ◽  
Vol 98 (3) ◽  
pp. 446-452 ◽  
Author(s):  
J. A. Schmitt ◽  
R. C. DiPrima
Keyword(s):  
Boundary Layer ◽  
Film Thickness ◽  
Asymptotic Expansions ◽  
Asymptotic Expression ◽  
Asymptotic Methods ◽  
Matched Asymptotic Expansions ◽  
Slider Bearing ◽  
Trailing Edge ◽  
Slider Bearings ◽  
Special Cases

The method of matched asymptotic expansions is used to develop an asymptotic expression for the pressure for large bearing numbers for the case of an infinite slider bearing with a general film thickness that has a discontinuous slope at a point. It is shown that, in addition to the boundary layer of the pressure at the trailing edge, there is also a boundary layer in the derivative of the pressure at the point of discontinuity. The corresponding load formula is also derived. The special cases of the taper-flat and taper-taper slider bearings are discussed.

The Optimum One-Dimensional Hydrodynamic Gas Rayleigh Step Bearing

1970 ◽  
Vol 92 (3) ◽  
pp. 504-508 ◽  
Author(s):  
G. M. Wylie ◽  
C. J. Maday
Keyword(s):  
Load Capacity ◽  
Slider Bearing ◽  
Optimum Configuration ◽  
Inherent Feature ◽  
Load Carrying ◽  
Lower Load ◽  
Data Points ◽  
Rayleigh Gas ◽  
Bearing Number ◽  
High Degree

The optimum Rayleigh gas slider bearing is determined for a range of bearing numbers. Numerical methods are used to calculate step location, step pressure, and load capacity for given values of step height ratio, bearing number, and flow parameter. These methods are used to determine as many data points as desired so that it is possible to obtain the optimum configuration dimensions to a very high degree of accuracy. An inherent feature of this analytical experiment is the acquisition of data pertaining to the near-optimum bearings and such data are presented for bearings with load capacities ranging down to seven-tenths of the load associated with the optimum Rayleigh bearing. At low bearing numbers it is found that the optimum Rayleigh bearing has only slightly lower load-carrying capability than the optimum gas slider bearing. For bearing numbers of 50, 100, and 500 the optimum Rayleigh slider bearings were, respectively, 5.8, 8.3, and 15.3 percent lower in load-carrying capability than the corresponding optimum bearings.

The One-Dimensional Optimum Hydrodynamic Gas Slider Bearing

1968 ◽  
Vol 90 (1) ◽  
pp. 281-284 ◽  
Author(s):  
C. J. Maday
Keyword(s):  
Film Thickness ◽  
Load Capacity ◽  
Maximum Load ◽  
Slider Bearing ◽  
Compression Process ◽  
One Dimensional ◽  
One Step ◽  
The Difference ◽  
The One ◽  
Bearing Number

Bounded variable methods of the calculus of variations are used to determine the optimum or maximum load capacity hydrodynamic one-dimensional gas slider bearing. A lower bound is placed on the minimum film thickness in order to keep the load finite, and also to satisfy the boundary conditions. Using the Weierstrass-Erdmann corner conditions and the Weierstrass E-function it is found that the optimum gas slider bearing is stepped with a convergent leading section and a uniform thickness trailing section. The step location and the leading section film thickness depend upon the bearing number and compression process considered. It is also shown that the bearing contains one and only one step. The difference in the load capacity and maximum film pressure between the isothermal and adiabatic cases increases with increasing bearing number.

Asymptotic Methods for an Infinitely Long Step Slider Squeeze Bearing

1973 ◽  
Vol 95 (2) ◽  
pp. 208-215 ◽  
Author(s):  
R. C. DiPrima
Keyword(s):  
Steady State ◽  
Singular Perturbation ◽  
Film Thickness ◽  
Asymptotic Methods ◽  
Perturbation Techniques ◽  
Bearing Number

The step slider squeeze bearing is analyzed by a combination of singular perturbation techniques and numerical procedures. It is assumed that the bearing number associated with the trailing section is large and that the amplitude of the squeezing motion is small. There is no restriction on the film clearance or the jump in the film thickness at the step, so the bearing number associated with the leading section of the bearing need not be large. Results for the steady-state load are given for several geometries. Stability characteristics of different geometries are discussed.

Measurement of Load-Carrying Capacity of Thin Lubricating Films

2012 ◽  
Vol 134 (4) ◽  
Author(s):  
Xia Li ◽  
Feng Guo ◽  
Shuyan Yang ◽  
P. L. Wong
Keyword(s):  
Film Thickness ◽  
Carrying Capacity ◽  
Constant Load ◽  
Experimental Results ◽  
Hydrodynamic Theory ◽  
Load Carrying Capacity ◽  
Slider Bearing ◽  
Lubricating Film ◽  
Single Function ◽  
Load Carrying

This paper presents an experimental procedure to evaluate the load-carrying capacity of a fixed-incline slider bearing (dimensionless load W versus convergence ratio K) using a slider-on-disk lubricating film test rig. In general, the applied load is the dependent variable and is directly measured for different convergence ratios such that the relation of the load-carrying capacity W and the convergence ratio K can be obtained. The load and slider inclination are fixed in the present approach, and the film thickness is measured at different speeds. As the dimensionless load can be a function of speed and film thickness, the variation of load-carrying capacity with respect to speed can be obtained even under a constant load and a fixed incline. It is shown that the measured load-carrying capacity is lower than that predicted by the classical hydrodynamic theory. Nevertheless, the experimental results acquire the same trend in the variation of dimensionless loads with convergence ratios. The theory holds that the load-carrying capacity is a single function of the convergence ratio. However, the experimental results show that the dimensionless load-carrying capacity is affected by the inclination angle of the slider, load, and the properties of lubricating oils.

Higher Order Approximations in the Asymptotic Solution of the Reynolds Equation for Slider Bearings at High Bearing Numbers

1969 ◽  
Vol 91 (1) ◽  
pp. 45-51 ◽  
Author(s):  
R. C. DiPrima
Keyword(s):  
Carrying Capacity ◽  
Reynolds Equation ◽  
Journal Bearing ◽  
Higher Order ◽  
Load Carrying Capacity ◽  
Slider Bearing ◽  
Load Carrying ◽  
Higher Order Terms ◽  
Parabolic Shape ◽  
Bearing Number

The methods of matched asymptotic expansions are used in a systematic manner to obtain the load-carrying capacity of an infinitely long slider bearing correct through terms 0 (1/Λ) where Λ is the bearing number. The expression for the load is extremely simple. It is shown that the error is 0 (1/Λ2), and the procedure for obtaining higher order terms is discussed. Results are given for the case of a converging film thickness with a parabolic shape and for a partial arc journal bearing.

Perturbation Solution of the 1-D Reynolds Equation With Slip Boundary Conditions

1978 ◽  
Vol 100 (1) ◽  
pp. 70-73 ◽  
Author(s):  
Aron Sereny ◽  
Vittorio Castelli
Keyword(s):  
Boundary Conditions ◽  
Numerical Solution ◽  
Carrying Capacity ◽  
Reynolds Equation ◽  
Load Carrying Capacity ◽  
Slip Boundary Conditions ◽  
Slider Bearing ◽  
Slip Boundary ◽  
Load Carrying ◽  
Bearing Number

The method of matched asymptotic expansion is applied to obtain the pressure distribution and the load carrying capacity for an infinitely long slider bearing, operating under high-speed, low-height, with slip boundary conditions. The pressure distribution is easily applicable as the starting solution for the iterative numerical solution of Reynolds equation. Two examples given show extremely good correlation between this expansion and the numerical solution. It is shown that, for a tapered slider bearing with a bearing number above 100, the reduction in load because of slip is minimal and that, for a parabolic slider, there exists a certain unique bearing number for which the load carrying capacity is independent of the parabolic crown of the slider. It is shown that for a wide slider bearing with large bearing number, the effect of slip is on the order of 1/A.

scholarly journals Study on the Load-Carrying Capacity of Finite-Width Slider Bearing with Wavy Surface

2013 ◽  
Vol 29 (1) ◽  
pp. 13-18 ◽  
Author(s):  
Jung-Hun Shin ◽  
Gi-Chun Lee ◽  
Jong-Won Park ◽  
Bo-Sik Kang ◽  
Kyung Woong Kim
Keyword(s):  
Carrying Capacity ◽  
Wavy Surface ◽  
Finite Width ◽  
Load Carrying Capacity ◽  
Slider Bearing ◽  
Load Carrying

Asymptotic Methods for an Infinitely Long Slider Squeeze-Film Bearing

1968 ◽  
Vol 90 (1) ◽  
pp. 173-183 ◽  
Author(s):  
R. C. DiPrima
Keyword(s):  
Thrust Bearing ◽  
Relative Rate ◽  
Asymptotic Methods ◽  
Squeeze Film ◽  
Singular Perturbation Theory ◽  
Slider Bearing ◽  
Approach Infinity ◽  
Bearing Number ◽  
Squeeze Number ◽  
Gas Bearing

The application of the techniques of singular perturbation theory (boundary layer theory) to several problems in gas bearing lubrication is discussed. The leading terms in asymptotic expansions for the pressure are obtained for the cases: A slider bearing with large bearing number, a squeeze-film thrust bearing with large squeeze number, and a combined slider squeeze-film bearing with large bearing number and/or large squeeze number. For the latter problem it is necessary to distinguish several cases depending upon the relative rate at which the bearing number and squeeze number approach infinity

Optimal Consumption and Insurance: A Continuous-time Markov Chain Approach

2008 ◽  
Vol 38 (01) ◽  
pp. 231-257 ◽  
Author(s):  
Holger Kraft ◽  
Mogens Steffensen
Keyword(s):  
Continuous Time ◽  
Optimal Solution ◽  
Decision Problems ◽  
Optimal Consumption ◽  
Continuous Time Markov Chain ◽  
Financial Decision Making ◽  
Markov Chain Approach ◽  
Special Cases ◽  
Financial Decision ◽  
The Individual

Personal financial decision making plays an important role in modern finance. Decision problems about consumption and insurance are in this article modelled in a continuous-time multi-state Markovian framework. The optimal solution is derived and studied. The model, the problem, and its solution are exemplified by two special cases: In one model the individual takes optimal positions against the risk of dying; in another model the individual takes optimal positions against the risk of losing income as a consequence of disability or unemployment.

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