The Narrow Groove Theory of Spiral Grooved Gas Bearings: Development and Application of a Generalized Formulation for Numerical Solution

1972 ◽  
Vol 94 (1) ◽  
pp. 86-92 ◽  
Author(s):  
A. J. Smalley
Keyword(s):  
Numerical Solution ◽  
Small Perturbation ◽  
Governing Equation ◽  
Spiral Groove ◽  
Gas Bearings ◽  
Rotationally Symmetric ◽  
Spherical Bearing ◽  
Symmetric Geometry

Application of the narrow groove theory to spiral groove bearings of any rotationally symmetric geometry is described. An effective method for numerical solution of the governing equation to yield radially centered pressures, and small perturbation stiffness performance, is developed. The power of the approach is demonstrated by application to two problems of widely different geometries, one involving a spool bearing, and one involving a spherical bearing.

scholarly journals Solutions of the Force-Free Duffing-van der Pol Oscillator Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Najeeb Alam Khan ◽  
Muhammad Jamil ◽  
Syed Anwar Ali ◽  
Nadeem Alam Khan
Keyword(s):  
Approximate Solution ◽  
Numerical Solution ◽  
Approximate Method ◽  
Governing Equation ◽  
Laplace Transformation ◽  
Large Time ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Alternative Framework ◽  
Runge Kutta

A new approximate method for solving the nonlinear Duffing-van der pol oscillator equation is proposed. The proposed scheme depends only on the two components of homotopy series, the Laplace transformation and, the Padé approximants. The proposed method introduces an alternative framework designed to overcome the difficulty of capturing the behavior of the solution and give a good approximation to the solution for a large time. The Runge-Kutta algorithm was used to solve the governing equation via numerical solution. Finally, to demonstrate the validity of the proposed method, the response of the oscillator, which was obtained from approximate solution, has been shown graphically and compared with that of numerical solution.

From dawn to dusk

2018 ◽  
Vol 102 (554) ◽  
pp. 246-256
Author(s):  
John D. Mahony
Keyword(s):  
Point Source ◽  
Numerical Solution ◽  
Governing Equation ◽  
Winter Solstice ◽  
Tangency Condition ◽  
Problem Corner

It used to be the case in some jurisdictions that a rhythmic knell of the Angelus bell would mark the onset of dawn, noon and the passing of a day at dusk. Between these times there is daylight whose duration varies from place to place and from day to day and which can be predicted either exactly or approximately. An illuminating problem concerning the number of daylight hours at a winter solstice in London was posed recently and answered in the Problem Corner of The Mathematical Gazette [1]. It was shown, for example, that a calculation of daylight hours rested strictly upon the numerical solution to a transcendental trigonometric equation. Related references to earlier works in the Gazette involving a point source Sun were also given.The purpose of this note is multifold. First, it is to point out that the above-cited equation might be viewed also as a “Sun-ray-to-Earth tangency condition”. Such a condition was developed earlier by the author in a publication that is now defunct [2], and so for completeness the steps necessary to establish the condition will be produced again here. Second, it will be evident from the manner of its derivation that the governing equation is valid at all orbit points, not just at a given solstice.

A Numerical Solution for the Design of Externally Pressurized Porous Gas Bearings: Thrust Bearings

1968 ◽  
Vol 90 (4) ◽  
pp. 810-817 ◽  
Author(s):  
E. P. Gargiulo ◽  
P. W. Gilmour
Keyword(s):  
Porous Media ◽  
Numerical Solution ◽  
Experimental Verification ◽  
Anisotropic Permeability ◽  
Gas Bearings ◽  
Thrust Bearings ◽  
Porous Bearing ◽  
Bearing Load ◽  
Typical Design

A numerical solution for the flow in a porous bearing is presented which includes provision for compressibility of the lubricant and anisotropic permeability of the porous media. Typical design curves for the bearing load and flow are included with experimental verification.

scholarly journals Two-dimensional Analysis of Stiffness and Damping Factor of Spiral Groove Spherical Bearing

1982 ◽  
Vol 25 (202) ◽  
pp. 663-670 ◽  
Author(s):  
Yutaka MIYAKE ◽  
Nobuyoshi KAWABATA ◽  
Akira TOMINAGA ◽  
Susumu MURATA
Keyword(s):  
Dimensional Analysis ◽  
Damping Factor ◽  
Two Dimensional ◽  
Spiral Groove ◽  
Spherical Bearing ◽  
Stiffness And Damping

On the Dynamic Stability of the Spiral-Grooved Gas-Lubricated Thrust Bearing

1987 ◽  
Vol 109 (1) ◽  
pp. 183-188 ◽  
Author(s):  
V. N. Constantinescu ◽  
S. Galetuse
Keyword(s):  
Dynamic Stability ◽  
Small Perturbation ◽  
Thrust Bearing ◽  
Critical Mass ◽  
Practical Interest ◽  
Unconditionally Stable ◽  
Gas Bearings ◽  
Air Hammer

The dynamic stability of a blocked center inward pumping spiral grooved thrust bearing is investigated. For this purpose two methods are considered comparatively, namely a standard small perturbation one, and an extension of the method used previously to determine air-hammer phenomena in externally pressurized gas bearings. The first method gives a more detailed description of situations in which the film is stable or unstable, while the second one gives a limit for the speed (or compressibility parameter Λc) up to which the bearing is unconditionally stable. The second method is simpler and of practical interest since at higher speeds the critical mass furnished by the first method is of little practical interest (being too small).

The Numerical Solution of Linear Elliptic Equations

1968 ◽  
Vol 90 (4) ◽  
pp. 773-776 ◽  
Author(s):  
R. Coleman
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Boundary Conditions ◽  
Numerical Solution ◽  
Difference Method ◽  
Elliptic Equations ◽  
Linear Boundary ◽  
Numerical Techniques ◽  
Spiral Groove ◽  
Gas Bearing

A noniterative finite difference method is given for the solutions of linear elliptic equations with linear boundary conditions. This method is applicable to a variety of gas bearing problems. Comparisons are made with other numerical techniques, and an application to the spiral groove thrust plate is given.

New perspectives in the use of the Ffowcs Williams–Hawkings equation for aeroacoustic analysis of rotating blades

2007 ◽  
Vol 570 ◽  
pp. 79-127 ◽  
Author(s):  
S. IANNIELLO
Keyword(s):  
Numerical Solution ◽  
Governing Equation ◽  
Rotational Velocity ◽  
Nonlinear Problems ◽  
Integration Scheme ◽  
Noise Prediction ◽  
Solution Approach ◽  
Rotating Blades ◽  
Surface Integration ◽  
Pressure Peak

The Ffowcs Williams–Hawkings equation represents a standard approach in the prediction of noise from rotating blades. It is widely used for linear aeroacoustic problems concerning helicopter rotors and aircraft propellers and over the last few years, through the use of the so called porous (or permeable) surface formulation, has replaced the Kirchhoff approach in the numerical solution of nonlinear problems. Nevertheless, because of numerical difficulties in evaluating the contribution from supersonic sources, most of the computing tools are still unable to treat the critical velocities at which the shock delocalization occurs. At those conditions, the attention is usually limited to the comparison between the noise prediction and the experimental data in the narrow time region where the pressure peak value is located, but there has been little attention paid to the singular behaviour of the governing equation at supersonic speeds. The aim of this paper is to couple the advantages of the porous formulation to an emission surface integration scheme in order to show if and how the singularities affect the noise prediction and to demonstrate a practical way to remove them. Such an analysis enables an investigation of some interesting and somewhat hidden features of the numerical solution of the governing equation and suggests a new solution approach to predicting the noise of a rotor at any rotational velocity.

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