The one-dimensional gas-lubricated slider bearing

1980 ◽  
Vol 22 (1) ◽  
pp. 75-97
Author(s):  
J. J. Shepherd
Keyword(s):  
Singular Perturbation ◽  
Pressure Distribution ◽  
Singular Perturbation Theory ◽  
Slider Bearing ◽  
Perturbation Problem ◽  
Jump Discontinuities ◽  
Uniqueness Results ◽  
The One ◽  
Bearing Number ◽  
Film Profile

AbstractUnder the appropriate physical hypotheses, the problem of determining the pressure distribution in a gas-filled slider bearing becomes a singular perturbation problem as Λ, the bearing number, tends to infinity. This paper extends the results of an earlier one by the author to consider the case where the film profile has jump discontinuities in slope at points interior to the bearing. Application of the methods of general singular perturbation theory establishes the appropriate existence-uniqueness results for this problem, and a means is devised by which uniformly valid asymptotic approximations to the pressure distribution may be obtained for large values of Λ.

An Approximate Solution for the Infinitely Long, Gas Lubricated Slider Bearing

1965 ◽  
Vol 87 (4) ◽  
pp. 1085-1086
Author(s):  
H. J. Sneck
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Pressure Distribution ◽  
Carrying Capacity ◽  
Load Carrying Capacity ◽  
Slider Bearing ◽  
Load Carrying ◽  
Numerical Design ◽  
The One ◽  
Design Calculations

The only exact solution for the infinitely long, gas-lubricated slider bearing is the one obtained by Harrison [1] for the plane wedge isothermal film. The resultant formulas for the pressure distribution and load-carrying capacity are complicated and therefore quite cumbersome in numerical design calculations. In the analysis to follow, a simplified, approximate solution is developed which can be applied to any infinitely long slider geometry.

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 Analysis of a Finite Gas Slider Bearing of Narrow Geometry

1983 ◽  
Vol 105 (3) ◽  
pp. 491-495 ◽  
Author(s):  
J. J. Shepherd ◽  
R. C. DiPrima
Keyword(s):  
Steady State ◽  
Asymptotic Analysis ◽  
Pressure Distribution ◽  
Asymptotic Expansions ◽  
Matched Asymptotic Expansions ◽  
Slider Bearing ◽  
Load Bearing ◽  
Bearing Number ◽  
Isothermal Gas ◽  
State Pressure

The method of matched asymptotic expansions is used to analyze the steady state pressure distribution and load bearing properties of a finite rectangular isothermal gas slider bearing when ε, the ratio of transverse to longitudinal dimensions of the bearing, is small and the bearing number Λ is moderate. General expressions for the pressure and load are obtained. Specific results are given for bearings with shallow crowning. The effects of the bearing number becoming large and the interaction between the two effects ε→0 and Λ→∞ are discussed.

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

scholarly journals Renormalization group and fractional calculus methods in a complex world: A review

2021 ◽  
Vol 24 (1) ◽  
pp. 5-53
Author(s):  
Lihong Guo ◽  
YangQuan Chen ◽  
Shaoyun Shi ◽  
Bruce J. West
Keyword(s):  
Perturbation Theory ◽  
Renormalization Group ◽  
Fractional Calculus ◽  
Singular Perturbation ◽  
World View ◽  
Singular Perturbation Theory ◽  
Asymptotic Behavior Of Solutions ◽  
Quantum Field ◽  
Self Similarity ◽  
Way Of Thinking

Abstract The concept of the renormalization group (RG) emerged from the renormalization of quantum field variables, which is typically used to deal with the issue of divergences to infinity in quantum field theory. Meanwhile, in the study of phase transitions and critical phenomena, it was found that the self–similarity of systems near critical points can be described using RG methods. Furthermore, since self–similarity is often a defining feature of a complex system, the RG method is also devoted to characterizing complexity. In addition, the RG approach has also proven to be a useful tool to analyze the asymptotic behavior of solutions in the singular perturbation theory. In this review paper, we discuss the origin, development, and application of the RG method in a variety of fields from the physical, social and life sciences, in singular perturbation theory, and reveal the need to connect the RG and the fractional calculus (FC). The FC is another basic mathematical approach for describing complexity. RG and FC entail a potentially new world view, which we present as a way of thinking that differs from the classical Newtonian view. In this new framework, we discuss the essential properties of complex systems from different points of view, as well as, presenting recommendations for future research based on this new way of thinking.

Thermohydrodynamic Behavior of a Slider Pocket Bearing

2005 ◽  
Vol 128 (2) ◽  
pp. 312-318 ◽  
Author(s):  
Mihai B. Dobrica ◽  
Michel Fillon
Keyword(s):  
Pressure Distribution ◽  
Reverse Flow ◽  
Three Dimensional ◽  
Convective Boundary ◽  
Inertia Effects ◽  
Good Convergence ◽  
Bearing Design ◽  
The One ◽  
Volume Method ◽  
Generalized Reynolds Equation

Pocket-pads or steps are often used in journal bearing design, allowing improvement of the latter’s dynamic behavior. Similar “discontinuous” geometries are used in designing thrust bearing pads. A literature review shows that, to date, only isoviscous and adiabatic studies of such geometries have been performed. The present paper addresses this gap, proposing a complete thermohydrodynamic (THD) steady model, adapted to three-dimensional (3D) discontinuous geometries. The model is applied to the well-known geometry of a slider pocket bearing, operating with an incompressible viscous lubricant. A model based on the generalized Reynolds equation, with concentrated inertia effects, is used to determine the 2D pressure distribution. On this basis, a 3D field of velocities is constructed which, in turn, allows the resolution of the 3D energy equation. Using a variable-size grid improves the accuracy in the discontinuity region, allowing an evaluation of the magnitude of error induced by Reynolds assumptions. The equations are solved using the finite volume method. This ensures good convergence even when a significant reverse flow is present. Heat evacuation through the pad is taken into account by solving the Laplace equation with convective boundary conditions that are realistic. The runner’s temperature, assumed constant, is determined by imposing a zero value for the global heat flux balance. The constructed model gives the pressure distribution and velocity fields in the fluid, as well as the temperature distribution across the fluid and solid pad. Results show important transversal temperature gradients in the fluid, especially in the areas of minimal film thickness. This further justifies the use of a complete THD model such as the one employed.

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