The Maximum Principle Approach to the Optimum One-Dimensional Journal Bearing

1970 ◽  
Vol 92 (3) ◽  
pp. 482-487 ◽  
Author(s):  
C. J. Maday
Keyword(s):  
Maximum Principle ◽  
Film Thickness ◽  
Journal Bearing ◽  
Maximum Load ◽  
Load Direction ◽  
One Dimensional ◽  
The Maximum Principle ◽  
Minimum Film Thickness ◽  
Constant Viscosity ◽  
The One

Pontryagin’s Maximum Principle is used to determine the journal bearing which supports the maximum load for a given minimum film thickness and a specified load direction. The one-dimensional configuration which uses a constant-viscosity, incompressible lubricant is considered. Comparison shows that the optimum bearing carries a load about 13.5 percent greater than the maximum carried by the usual full-Sommerfeld bearing and about 121 percent greater than that carried by the half-Sommerfeld unit. The problem is formulated subject to the constraints of a fixed load direction and a specified minimum film thickness while the only boundary condition imposed is that the pressure must vanish at the inlet and at the outlet. The actual extent of the bearing is determined in the optimization process and it is shown that this extent is 360 deg. Further, the bearing is stepped with only two regions of different but constant film thickness.

A Demonstrably Optimum One Dimensional Journal Bearing

1972 ◽  
Vol 94 (2) ◽  
pp. 188-192 ◽  
Author(s):  
S. M. Rohde
Keyword(s):  
Film Thickness ◽  
Carrying Capacity ◽  
Journal Bearing ◽  
Infinite Length ◽  
Load Carrying Capacity ◽  
One Dimensional ◽  
Load Carrying ◽  
Minimum Film Thickness ◽  
Constant Viscosity ◽  
Nonlocal Formulation

By the use of a new variational technique, the bearing profile which maximizes the load carrying capacity of an infinite length journal bearing is obtained. The lubricant is assumed to be incompressible and of constant viscosity. The flow is assumed to be laminar and the optimization is based upon a minimum film thickness. The solution obtained is a concentric step bearing with a film thickness ratio of 1.812 and a ridge to pad ratio of 0.328. It is mathematically shown by the use of the “nonlocal” formulation that this step profile does yield a maximum among all profiles sufficiently “close.”

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.

Dynamic Analysis of Tilting-Pad Journal Bearing—Influence of Pad Deformations

1994 ◽  
Vol 116 (3) ◽  
pp. 621-627 ◽  
Author(s):  
H. Desbordes ◽  
M. Fillon ◽  
C. Chan Hew Wai ◽  
J. Frene
Keyword(s):  
Film Thickness ◽  
Pressure Field ◽  
Journal Bearing ◽  
Maximum Pressure ◽  
Tilting Pad ◽  
Minimum Film Thickness ◽  
Dimensional Finite Element ◽  
The One ◽  
Tilting Pad Journal Bearing ◽  
Tilting Pad Journal Bearings

A theoretical nonlinear analysis of tilting-pad journal bearings is presented for small and large unbalance loads under isothermal conditions. The radial displacements of internal pad surface due to pressure field are determined by a two-dimensional finite element method in order to define the actual film thickness. The influence of pad deformations on the journal orbit, on the minimum film thickness and on the maximum pressure is studied. The effects of pad displacements are to decrease the minimum film thickness and to increase the maximum pressure. The orbit amplitude is also increased by 20 percent for the large unbalance load compared to the one obtained for rigid pad.

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.

scholarly journals The persistence of logconcavity for positive solutions of the one dimensional heat equation

1990 ◽  
Vol 48 (2) ◽  
pp. 246-263 ◽  
Author(s):  
G. Keady
Keyword(s):  
Maximum Principle ◽  
Heat Equation ◽  
Positive Solutions ◽  
Space Variable ◽  
Time Variable ◽  
One Dimensional ◽  
The Maximum Principle ◽  
Proof Techniques ◽  
The One

AbstractConsider positive solutions of the one dimensional heat equation. The space variable x lies in (–a, a): the time variable t in (0,∞). When the solution u satisfies (i) u (±a, t) = 0, and (ii) u(·, 0) is logconcave, we give a new proof based on the Maximum Principle, that, for any fixed t > 0, u(·, t) remains logconcave. The same proof techniques are used to establish several new results related to this, including results concerning joint concavity in (x, t) similar to those considered in Kennington [15].

Stress-driven diffusion in a drying liquid paint layer

1998 ◽  
Vol 9 (5) ◽  
pp. 447-461 ◽  
Author(s):  
B. W. VAN DE FLIERT ◽  
R. VAN DER HOUT
Keyword(s):  
Maximum Principle ◽  
Diffusion Process ◽  
Viscoelastic Material ◽  
Polymer Network ◽  
Paint Layer ◽  
Dimensional Model ◽  
One Dimensional ◽  
The Maximum Principle ◽  
The One ◽  
Liquid Paint

A model is presented for the diffusion-driven drying of a polymeric solution such as liquid paint. Included is a stress build-up and relaxation in the polymer network of the viscoelastic material, which influences the diffusion process. The behaviour of the (one-dimensional) model is analysed by means of the maximum principle and illustrated with numerical calculations.

scholarly journals Discrete Maximum Principle in One-Dimensional Heat Equation

2016 ◽  
Vol 2 ◽  
pp. 5
Author(s):  
Durga Jang K.C. ◽  
Ganesh Bahadur Basnet
Keyword(s):  
Maximum Principle ◽  
Heat Equation ◽  
Discrete Maximum Principle ◽  
Finite Difference Schemes ◽  
Discrete Version ◽  
One Dimensional ◽  
The Maximum Principle ◽  
Linear Partial Differential Equations ◽  
The One ◽  
College Of Engineering

<p>The maximum principle plays key role in the theory and application of a wide class of real linear partial differential equations. In this paper, we introduce ‘Maximum principle and its discrete version’ for the study of second-order parabolic equations, especially for the one-dimensional heat equation. We also give a short introduction of formation of grid as well as finite difference schemes and a short prove of the ‘Discrete Maximum principle’ by using different schemes of heat equation.</p><p><strong>Journal of Advanced College of Engineering and Management</strong>, Vol. 2, 2016, Page: 5-10</p>

The Foundation of the Sommerfeld Transformation

2002 ◽  
Vol 124 (3) ◽  
pp. 645-646 ◽  
Author(s):  
Clarence J. Maday
Keyword(s):  
Solar System ◽  
Film Thickness ◽  
Journal Bearing ◽  
Closed Orbit ◽  
True Anomaly ◽  
Central Angle ◽  
Eccentric Anomaly ◽  
One Dimensional ◽  
Analytic Singularity ◽  
The One

The analysis of the one-dimensional journal bearing leads to an interesting integral that is continuous but has an analytic singularity involving the inverse tangent at π/2. This difficulty was resolved by a clever and non-intuitive transformation attributed to Sommerfeld. In this technical brief we show that the transformation has its origin in the geometry of the ellipse and Kepler’s equation that is based upon his observations of the planets in the Solar system. The derivation of the transformation is a problem or exercise in Sommerfeld’s monograph, Mechanics. The transformation is the relation between the two angles that characterize the ellipse, the closed orbit of a body in a central inverse square force field. The angle measured about the focus is the true anomaly (angle) and the angle measured about the center is the eccentric anomaly (angle). We establish the analogy between the orbital radius in terms of the eccentric anomaly and the film thickness of the journal bearing in terms of its central angle.

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