Effects of Foil Bending Rigidity on Spacing Height Characteristics of Hydrostatic Porous Foil Bearings for Web Handling Processes

1997 ◽  
Vol 119 (3) ◽  
pp. 422-427 ◽  
Author(s):  
H. Hashimoto
Keyword(s):  
Reynolds Equation ◽  
Numerical Solutions ◽  
Computation Time ◽  
Finite Width ◽  
Bending Rigidity ◽  
Weighted Residual Method ◽  
Web Handling ◽  
Foil Bearings ◽  
Wide Range ◽  
Foil Bending

In this paper, the effects of foil bending rigidity on the spacing height characteristics of hydrostatic foil bearings with a hollow porous shaft for web handling processes are analyzed by the finite width bearing theory. In the analysis, in order to save computation time and to improve the convergence of solutions, the two-dimensional modified Reynolds equation considering the added flow through porous shaft is reduced to an ordinary differential equation based on the weighted residual method. The reduced Reynolds equation and elastic equation for the foil are discretized by the finite difference method and solved numerically by the iterative technique. The numerical solutions for the pressure and film thickness distributions between foil and shaft are obtained for a wide range of bearing width-to-diameter ratio under various combinations of foil bending rigidity and foil wrap angle, and the spacing height characteristics of the foil bearings are examined theoretically.

Theoretical Analysis of Externally Pressurized Porous Foil Bearings—Part I: In the Case of Smooth Surface Porous Shaft

1995 ◽  
Vol 117 (1) ◽  
pp. 103-111 ◽  
Author(s):  
H. Hashimoto
Keyword(s):  
Theoretical Analysis ◽  
Film Thickness ◽  
Reynolds Equation ◽  
Numerical Solutions ◽  
Computation Time ◽  
Hybrid Performance ◽  
Foil Bearings ◽  
Wide Range ◽  
Wrap Angle ◽  
Element Technique

In this paper, the theoretical analysis for the hybrid performance of externally pressurized foil bearings with a hollow porous shaft is described. In the analysis, in order to save computation time and to improve the convergency of solutions, the two-dimensional modified Reynolds equation considering the added flow through hollow porous shaft is reduced to the ordinary differential equation by expanding the film pressure to the Fourier series with respect to the axial coordinate. The reduced Reynolds equation and the equilibrium equation for the perfectly flexible foil are solved iteratively by the finite element technique. The numerical solutions for the pressure and film thickness distributions between the foil and the shaft are obtained for a wide range of bearing width-to-diameter ratio under the various combinations of dimensionless supply pressure, dimensionless permeability of porous shaft and dimensionless wrap angle of foil, and the hybrid performance of foil bearings are examined theoretically. It is found from the numerical results that the bearing width has the significant effect on the pressure and film thickness distributions between the foil and the shaft.

Air Film Thickness Estimation in Web Handling Processes

1999 ◽  
Vol 121 (1) ◽  
pp. 50-55 ◽  
Author(s):  
Hiromu Hashimoto
Keyword(s):  
Film Thickness ◽  
Reynolds Equation ◽  
Experimental Results ◽  
Finite Width ◽  
Web Handling ◽  
Foil Bearing ◽  
Guide Roller ◽  
The Finite Difference Method ◽  
Air Film ◽  
The Web

In this paper, in order to estimate an air film thickness between moving web and guide roller (web spacing height), an air film thickness formula was derived based on the finite width compressible foil bearing theory. In the derivation of the air film thickness formula, the two-dimensional Reynolds equation and foil equilibrium equation were discretized by the finite difference method and solved iteratively to obtain the pressure and air film thickness distributions for various parameters. Based on the numerical results, the simplified convenience formula for the estimation of air film thickness between web and guide roller was obtained. On the other hand, the air film thickness between web and guide roller was measured by an optical sensor, and the experimental results were compared with the calculated results. Moreover, the variation of air film thickness between two layers in web winding processes was analyzed by making use of the air film thickness formula derived above. From the theoretical and experimental results obtained, the effects of air film thickness on the web transporting systems were clarified.

Theoretical and Experimental Investigations Into Spacing Characteristics Between Roller and Three Types of Webs With Different Permeabilities

2004 ◽  
Author(s):  
H. Hashimoto ◽  
M. Okajima
Keyword(s):  
Film Thickness ◽  
Reynolds Equation ◽  
Numerical Solutions ◽  
Experimental Investigations ◽  
First Order ◽  
Wide Range ◽  
Entrained Air ◽  
Air Diffusion ◽  
Air Film ◽  
The Web

A new theoretical model for estimating the entrained air film thickness between a web and roller is presented for both impermeable and permeable webs. A simple closed-form formula for estimating the air film thickness, which considers the effects of air leakage from the web edges and air diffusion due to the permeability of web, was obtained based on a large number of simultaneous numerical solutions of the compressible Reynolds equation and the web equilibrium equation. The variation of air film thickness with roller velocity is measured for three typical webs: PET (polyethylene terephthalate), coated paper, and newsprint. The effects of web permeability, web width and web tension on the air film thickness are examined theoretically and experimentally for a wide range of roller velocity. Reasonable agreement is seen both quantatively and qualitatively between the predicted and measured results. The validity of the formula for the first order estimation of web-roller interface problems is verified experimentally.

Theoretical and Experimental Investigations Into Spacing Characteristics Between Roller and Three Types of Webs With Different Permeabilities

2005 ◽  
Vol 128 (2) ◽  
pp. 267-274 ◽  
Author(s):  
H. Hashimoto ◽  
M. Okajima
Keyword(s):  
Film Thickness ◽  
Reynolds Equation ◽  
Numerical Solutions ◽  
Experimental Investigations ◽  
Curve Fit ◽  
Wide Range ◽  
Entrained Air ◽  
Air Diffusion ◽  
Air Film ◽  
The Web

A new theoretical model for estimating the entrained air film thickness between a web and roller is presented for both impermeable and permeable webs. A simple curve fit formula for estimating the air film thickness, which considers the effects of air leakage from the web edges and air diffusion due to the permeability of web, was obtained based on a large number of simultaneous numerical solutions of the compressible Reynolds equation and the web equilibrium equation. The variation of air film thickness with roller velocity is measured for three typical webs: polyethylene terephthalate, coated paper, and newsprint. The effects of web permeability, web width, and web tension on the air film thickness are examined theoretically and experimentally for a wide range of roller velocity. Reasonable agreement is seen both quantitatively and qualitatively between the predicted and measured results. The validity of the formula for the first-order estimation of web-roller interface problems is verified experimentally.

Effects of Power—Law, Non-Newtonian Lubricants on Load Capacity and Friction for Plane Slider Bearings

1986 ◽  
Vol 108 (1) ◽  
pp. 86-91 ◽  
Author(s):  
R. H. Buckholz
Keyword(s):  
Power Law ◽  
Reynolds Equation ◽  
Numerical Solutions ◽  
Lagrange Equation ◽  
Load Capacity ◽  
Approximate Solutions ◽  
Finite Width ◽  
Aspect Ratios ◽  
Modified Reynolds Equation ◽  
Power Law Index

The lubrication of a conventional, finite width plane bearing, using a power-law, non-Newtonian lubricant, is studied. The basic assumptions in this analysis are: thin fluid-film, no thermal effects, and a modified Reynolds’ equation for small bearing aspect ratios. Results from this study include bearing pressure, load, and friction formulas. Similar results for the not-so-small bearing aspect ratios are found via an Euler-Lagrange equation. This Euler-Lagrange equation is derived from the optimization integral for the modified Reynolds’ equation. Approximate solutions to the modified Reynolds’ equation and to the Euler-Lagrange equation are contrasted with numerical solutions for the modified Reynolds equation. Bearing aspect ratios in the range 0.1 to 0.6, clearance ratios in the range 1.2 to 4.0, and non-Newtonian power-law index in the range 0.4 to 1.0 are considered.

The Effect of Journal Bearing Misalignment on Load and Cavitation for Non-Newtonian Lubricants

1986 ◽  
Vol 108 (4) ◽  
pp. 645-654 ◽  
Author(s):  
R. H. Buckholz ◽  
J. F. Lin
Keyword(s):  
Reynolds Equation ◽  
Numerical Solutions ◽  
Journal Bearings ◽  
Surface Boundary ◽  
Film Rupture ◽  
Aspect Ratios ◽  
Free Surface Boundary Condition ◽  
Load Carrying ◽  
Boundary Location ◽  
Surface Boundary Condition

An analysis for hydrodynamic, non-Newtonian lubrication of misaligned journal bearings is given. The hydrodynamic load-carrying capacity for partial arc journal bearings lubricated by power-law, non-Newtonian fluids is calculated for small valves of the bearing aspect ratios. These results are compared with: numerical solutions to the non-Newtonian modified Reynolds equation, with Ocvirk’s experimental results for misaligned bearings, and with other numerical simulations. The cavitation (i.e., film rupture) boundary location is calculated using the Reynolds’ free-surface, boundary condition.

Theoretical Condition for the Cxy=Cyx Relation in Fluid Film Journal Bearings

1989 ◽  
Vol 111 (3) ◽  
pp. 426-429 ◽  
Author(s):  
T. Kato ◽  
Y. Hori
Keyword(s):  
Reynolds Equation ◽  
Journal Bearing ◽  
Journal Bearings ◽  
Fluid Film ◽  
Finite Width ◽  
Damping Coefficients ◽  
Dynamic Coefficients ◽  
Cross Terms ◽  
Linear Damping ◽  
Theoretical Condition

A computer program for calculating dynamic coefficients of journal bearings is necessary in designing fluid film journal bearings and an accuracy of the program is sometimes checked by the relation that the cross terms of linear damping coefficients of journal bearings are equal to each other, namely “Cxy = Cyx”. However, the condition for this relation has not been clear. This paper shows that the relation “Cxy = Cyx” holds in any type of finite width journal bearing when these are calculated under the following condition: (I) The governing Reynolds equation is linear in pressure or regarded as linear in numerical calculations; (II) Film thickness is given by h = c (1 + κcosθ); and (III) Boundary condition is homogeneous such as p=0 or dp/dn=0, where n denotes a normal to the boundary.

Data-Driven Analysis of Engine Mission Severity Using Non-Dimensional Groups

2021 ◽  
Author(s):  
Tim Brandes ◽  
Stefano Scarso ◽  
Christian Koch ◽  
Stephan Staudacher
Keyword(s):  
Support Vector Machine ◽  
Computation Time ◽  
Principal Component ◽  
General Term ◽  
Support Vector ◽  
Physical Parameters ◽  
Machine Model ◽  
Precision Error ◽  
Operational Conditions ◽  
Wide Range

Abstract A numerical experiment of intentionally reduced complexity is used to demonstrate a method to classify flight missions in terms of the operational severity experienced by the engines. In this proof of concept, the general term of severity is limited to the erosion of the core flow compressor blade and vane leading edges. A Monte Carlo simulation of varying operational conditions generates a required database of 10000 flight missions. Each flight is sampled at a rate of 1 Hz. Eleven measurable or synthesizable physical parameters are deemed to be relevant for the problem. They are reduced to seven universal non-dimensional groups which are averaged for each flight. The application of principal component analysis allows a further reduction to three principal components. They are used to run a support-vector machine model in order to classify the flights. A linear kernel function is chosen for the support-vector machine due to its low computation time compared to other functions. The robustness of the classification approach against measurement precision error is evaluated. In addition, a minimum number of flights required for training and a sensible number of severity classes are documented. Furthermore, the importance to train the algorithms on a sufficiently wide range of operations is presented.

Static Determinacy in the Theory of Finite Width Foil Bearings

1974 ◽  
Vol 41 (1) ◽  
pp. 51-54 ◽  
Author(s):  
W. E. Langlois
Keyword(s):  
Force Balance ◽  
Finite Width ◽  
Foil Bearings ◽  
Lubrication Equation ◽  
Foil Bearing ◽  
Edge Conditions ◽  
Single Force ◽  
Static Determinacy ◽  
Force Balance Equation ◽  
Determinate Problem

The assumption of “perfect flexibility” is shown to be self-consistent in an important class of finite-width foil bearing problems. When the membrane equations are written in the “stretched coordinates” of foil bearing theory, the usual edge conditions on the tape result in a statically determinate problem. The tape dynamics couples to the Reynolds lubrication equation through a single force-balance equation which does not entail the elastic strain.

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