The Optimum One-Dimensional Magnetohydrodynamic Slider Bearing

1970 ◽  
Vol 92 (3) ◽  
pp. 530-534 ◽  
Author(s):  
J. B. Shukla
Keyword(s):  
Magnetic Fields ◽  
Film Thickness ◽  
Calculus Of Variations ◽  
Bearing Surface ◽  
Slider Bearing ◽  
Bounded Control ◽  
Control Variables ◽  
Type Function ◽  
One Dimensional ◽  
Step Type

In this paper, the techniques of the calculus of variations are used to study the general problems in magnetohydrodynamic lubrication when two or more control variables are bounded. The load integral, in the case of a magnetohydrodynamic slider bearing, is maximized with film thickness and conductivity functions as bounded control variables. It is shown that if the conductivity of a bearing surface is of step-type function then uniformly applied magnetic fields are more advantageous.

Generalized Reynolds Equation for Micropolar Lubricants and its Application to Optimum One-Dimensional Slider Bearings: Effects of Solid-Particle Additives in Solution

1975 ◽  
Vol 17 (5) ◽  
pp. 280-284 ◽  
Author(s):  
J. B. Shukla ◽  
M. Isa
Keyword(s):  
Calculus Of Variations ◽  
Solid Particle ◽  
Micropolar Fluid ◽  
Reynolds Equation ◽  
Slider Bearing ◽  
One Dimensional ◽  
Slider Bearings ◽  
Generalized Reynolds Equation

The effects of solid-particle additives in the lubricant are considered by characterizing this suspension as a micropolar fluid. The generalized Reynolds equation for this case has been derived and the optimum one-dimensional slider bearing is studied by using the techniques of the calculus of variations.

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.

The Optimum Slider Bearing in Terms of Friction

1972 ◽  
Vol 94 (3) ◽  
pp. 275-279 ◽  
Author(s):  
S. M. Rohde
Keyword(s):  
Variational Method ◽  
Film Thickness ◽  
Variational Formulation ◽  
Slider Bearing ◽  
One Dimensional ◽  
First Case ◽  
Minimum Film Thickness ◽  
The Coefficient Of Friction ◽  
Constant Viscosity ◽  
Film Profile

The film profile which minimizes the coefficient of friction and the film profile which minimizes the total friction force for a given load for a one-dimensional slider bearing are determined using a variational method. The lubricant is assumed to be incompressible and of constant viscosity. The flow is assumed to be laminar, and the optimization in the first case is based upon an assumed minimum film thickness. It is shown by the use of the nonlocal variational formulation that these profiles do yield a minimum among all admissible profiles.

A Variational Principle Applied to the Plane Slider Bearing

1965 ◽  
Vol 87 (4) ◽  
pp. 1081-1082
Author(s):  
Clarence J. Maday
Keyword(s):  
Variational Principle ◽  
Film Thickness ◽  
Power Loss ◽  
Minimum Principle ◽  
Slider Bearing ◽  
One Dimensional ◽  
Dimensional Plane ◽  
Minimum Film Thickness ◽  
The One ◽  
Made In

A minimum principle from hydrodynamics is applied to the one-dimensional plane slider bearing which is provided with a self-seeking pivot mechanism. An analysis was made in which a certain integral was minimized subject to the constraint that the load, speed, and viscosity were held fixed. This analysis showed that this corresponded exactly to that combination of minimum film thickness and inclination which would minimize the power loss subject to the above-mentioned constraint. It was also found that, in order to satisfy the minimum principle, there exists a definite numerical ratio between the slider inclination and the nondimensional minimum film thickness. This, in turn, fixed the pivot location relative to the length of the slider.

The Optimum Rayleigh Bearing in Terms of Load Capacity or Friction for Non-Newtonian Lubricants

1985 ◽  
Vol 107 (1) ◽  
pp. 59-67 ◽  
Author(s):  
P. Bourgin ◽  
B. Gay
Keyword(s):  
Film Thickness ◽  
Load Capacity ◽  
Maximum Load ◽  
Slider Bearing ◽  
Generalized Newtonian Fluid ◽  
Newtonian Viscosity ◽  
One Dimensional ◽  
Minimum Film Thickness ◽  
The One ◽  
Numerical Program

Pontryagin’s Maximum Principle is used to show that the configuration of the one-dimensional slider bearing which carries the maximum load for a specified minimum film thickness, is a modified Rayleigh bearing. The lubricant may be any Generalized Newtonian Fluid. Having selected two optimization criteria (1: maximum load capacity for a given minimum film thickness—2: minimum friction force for a specified load), a numerical program allows one to determine the optimal step bearing associated with the lubricant non-Newtonian viscosity. Several examples are worked out, showing that significant gains are expected, in comparison with the results obtained for the classical (Newtonian) Rayleigh bearing.

scholarly journals Quantitative Experimental and Theoretical Investigations on the Interaction of Shock Waves with Magnetic Fields

1969 ◽  
Vol 24 (10) ◽  
pp. 1449-1457
Author(s):  
H. Klingenberg ◽  
F. Sardei ◽  
W. Zimmermann
Keyword(s):  
Magnetic Fields ◽  
Shock Waves ◽  
Physical Model ◽  
Spectroscopic Method ◽  
Experimental Results ◽  
Dimensional Theory ◽  
One Dimensional ◽  
Theoretical Investigations ◽  
Theoretical Predictions ◽  
Interaction Of Shock Waves

Abstract In continuation of the work on interaction between shock waves and magnetic fields 1,2 the experiments reported here measured the atomic and electron densities in the interaction region by means of an interferometric and a spectroscopic method. The transient atomic density was also calculated using a one-dimensional theory based on the work of Johnson3 , but modified to give an improved physical model. The experimental results were compared with the theoretical predictions.

scholarly journals Hamiltonian Approach to Magnetic Fields with Toroidal Surfaces

1985 ◽  
Vol 40 (10) ◽  
pp. 959-967
Author(s):  
A. Salat
Keyword(s):  
Magnetic Field ◽  
Hamiltonian System ◽  
Magnetic Fields ◽  
Constant Pressure ◽  
Time Dependent ◽  
Hamiltonian Approach ◽  
One Dimensional ◽  
Mhd Equilibrium ◽  
Magnetic Surfaces ◽  
Rotational Transform

The equivalence of magnetic field line equations to a one-dimensional time-dependent Hamiltonian system is used to construct magnetic fields with arbitrary toroidal magnetic surfaces I = const. For this purpose Hamiltonians H which together with their invariants satisfy periodicity constraints have to be known. The choice of H fixes the rotational transform η(I). Arbitrary axisymmetric fields, and nonaxisymmetric fields with constant η(I) are considered in detail.Configurations with coinciding magnetic and current density surfaces are obtained. The approach used is not well suited, however, to satisfying the additional MHD equilibrium condition of constant pressure on magnetic surfaces.

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