An Interior Collocation Method for Static and Dynamic Analysis of Finite Length Gas Journal Bearings

1988 ◽  
Vol 110 (3) ◽  
pp. 456-461 ◽  
Author(s):  
N. Arakere ◽  
H. D. Nelson
Keyword(s):  
Differential Equation ◽  
Finite Difference ◽  
Dynamic Analysis ◽  
Collocation Method ◽  
Finite Length ◽  
Journal Bearings ◽  
Computational Domain ◽  
Static And Dynamic Analysis ◽  
Order Of Magnitude ◽  
Governing Differential Equation

The general Reynolds equation for self-acting, finite length gas-lubricated journal bearings is solved using an interior collocation method. The method assumes an approximate solution to the governing differential equation in the form of a series of trial functions, wn, with n unknown coefficients. The coefficients are evaluated by substituting the assumed solution in the governing differential equation, and forcing the residual (error) at n collocation points in the computational domain to be zero. The effectiveness of the collocation method is demonstrated by using the technique for the static and dynamic analysis of a journal supported by a plane gas bearing. The results from the collocation method agree very well with those obtained from a finite difference technique. Periodic orbit plots for a single journal in a finite bearing are presented for various journal unbalances and speeds. The collocation method is shown to be an order of magnitude computationally faster than the finite difference method. The method can be extended to other bearing types such as slider, hydrostatic and tilting pad bearings.

Collocation Method for Finite Length Squeeze Film Dampers With Variable Clearance

1988 ◽  
Vol 110 (4) ◽  
pp. 685-692 ◽  
Author(s):  
N. K. Arakere ◽  
H. D. Nelson
Keyword(s):  
Finite Difference ◽  
Collocation Method ◽  
Finite Length ◽  
Squeeze Film ◽  
Eccentricity Ratio ◽  
Static And Dynamic Analysis ◽  
Damping Coefficients ◽  
Highly Nonlinear ◽  
Secant Stiffness ◽  
Stiffness And Damping

The Collocation method is used to study the characteristics of a finite length Squeeze Film Damper (SFD) with variable clearance. The secant stiffness and damping coefficients of the SFD for centered circular orbits is compured by solving Reynolds equation for a finite bearing, using the collocation method. A significant disadvantage with a conventional SFD is the highly nonlinear variation of secant stiffness and damping coefficients with eccentricity ratio. It is shown in this paper that by suitably varying the parameters α and λ, which control the nature of clearance variation, the secant stiffness and damping coefficient variation with eccentricity; ratio can be altered to better suit specific design needs. Nonlinear transient analysis of a single journal in a finite SFD is carried out, and, periodic journal center orbits due to unbalance response for uncentered finite SFD’s are also obtained. The centered circular orbit response for different unbalance values and speeds is obtained. Results from the Collocation method show excellent agreement with finite difference results. The Collocation method is shown to be an efficient and viable technique that is an order of magnitude computationally faster than the finite difference method, for static and dynamic analysis of finite length SFD’s.

Effect of Lubricant Compressibility on Hydrodynamic Behavior of Finite Length Journal Bearings Running under Heavy Load Conditions

2015 ◽  
Vol 32 (1) ◽  
pp. 101-111 ◽  
Author(s):  
M. Besanjideh ◽  
S. A. Gandjalikhan Nassab
Keyword(s):  
Finite Length ◽  
Complex Geometry ◽  
Three Dimensional ◽  
Mass Conservation ◽  
Variable Density ◽  
Journal Bearings ◽  
Computational Domain ◽  
Heavy Load ◽  
Compressibility Effect ◽  
Hydrodynamic Behaviour

ABSTRACTThe journal bearings which are designed for heavy-duty operations could experience different lubricant density due to high bearing loads. In the present work, hydrodynamic behaviour of finite length journal bearings under laminar and isoviscous flow with variable density are investigated. For this purpose, three-dimensional continuity and momentum equations along with a proper density-pressure relation are solved numerically, using CFD technique. Also, an appropriate cavitation model based on mass conservation is involved in the computation. Because of complex geometry of journal bearing, a conformal mapping is employed to generate an orthogonal grid and the governing equations are transformed in the computational domain. Since the degree of oil compressibility can be depended to the type of lubricant, typical mineral and synthetic oils treatments are modelled, separately. Results indicate that the oil compressibility effect leads to increasing load carrying capacity such that this increase is slightly more for the synthetic oil.

scholarly journals The Application of the Collocation Method Using Hermite Cubic Splines to Nonlinear Transient One-Dimensional Heat Conduction Problems

1975 ◽  
Vol 97 (4) ◽  
pp. 562-569 ◽  
Author(s):  
T. C. Chawla ◽  
G. Leaf ◽  
W. L. Chen ◽  
M. A. Grolmes
Keyword(s):  
Differential Equation ◽  
Heat Conduction ◽  
Finite Difference ◽  
Differential Equations ◽  
Collocation Method ◽  
Gaussian Quadrature ◽  
Time Step ◽  
One Dimensional ◽  
Hermite Splines ◽  
Nonlinear Transient

A collocation method for the solution of one-dimensional parabolic partial differential equations using Hermite splines as approximating functions and Gaussian quadrature points as collocation points is described. The method consists of expanding dependent variables in terms of piece-wise cubic Hermite splines in the space variable at each time step. The unknown coefficients in the expansion are obtained at every time step by requiring that the resultant differential equation be satisfied at a number of points (in particular at the Gaussian quadrature points) in the field equal to the number of unknown coefficients. This collocation procedure reduces the partial differential equation to a system of ordinary differential equations which is solved as an initial value problem using the steady-state solution as the initial condition. The method thus developed is applied to a two-region nonlinear transient heat conduction problem and compared with a finite-difference method. It is demonstrated that because of high-order accuracy only a small number of equations are needed to produce desirable accuracy. The method has the desirable characteristic of an analytical method in that it produces point values as against nodal values in the finite-difference scheme.

scholarly journals On an improved computational solution for the 3D HCIR PDE in finance

2019 ◽  
Vol 27 (3) ◽  
pp. 207-230 ◽  
Author(s):  
Fazlollah Soleymani ◽  
Ali Akgül ◽  
Esra Karatas Akgül
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Finite Difference ◽  
Three Dimensional ◽  
Time Dependent ◽  
Computational Domain ◽  
Order Of Accuracy ◽  
Spatial Variables ◽  
Stability Bound ◽  
Computational Solution

AbstractThe aim of this work is to tackle the three–dimensional (3D) Heston– Cox–Ingersoll–Ross (HCIR) time–dependent partial differential equation (PDE) computationally by employing a non–uniform discretization and gathering the finite difference (FD) weighting coe cients into differentiation matrices. In fact, a non–uniform discretization of the 3D computational domain is employed to achieve the second–order of accuracy for all the spatial variables. It is contributed that under what conditions the proposed procedure is stable. This stability bound is novel in literature for solving this model. Several financial experiments are worked out along with computation of the hedging quantities Delta and Gamma.

Static and Dynamic Properties of Partial Journal Bearings

1963 ◽  
Vol 85 (2) ◽  
pp. 247-255 ◽  
Author(s):  
Paul C. Warner
Keyword(s):  
Differential Equation ◽  
Journal Bearing ◽  
Dynamic Properties ◽  
Journal Bearings ◽  
Wide Range ◽  
Governing Differential Equation

The liquid lubricated partial journal bearing is analyzed in an approximate yet accurate manner in order to obtain its static and dynamic properties. The solution of the governing differential equation is analytical rather than numerical, permitting inexpensive computation of results over a very wide range of the parameters involved.

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