A Simultaneous Numerical Solution for the Lubrication and Dynamic Stability of Noncontacting Gas Face Seals

2000 ◽  
Vol 123 (2) ◽  
pp. 388-394 ◽  
Author(s):  
Itzhak Green ◽  
Roger M. Barnsby
Keyword(s):  
Numerical Solution ◽  
Equations Of Motion ◽  
Critical Value ◽  
Pressure Drops ◽  
Mode 2 ◽  
Face Seals ◽  
Natural Response ◽  
The Stability ◽  
Film Lubrication ◽  
Fluid Film Lubrication

A numerical solution is presented for the dynamic analysis of gas lubricated noncontacting mechanical face seals having a single grounded flexibly mounted stator. Seal dynamics is solved in axial and angular modes of motion. Both the Reynolds equation and the equations of motion are arranged into a single state space form, allowing the fluid film lubrication and the dynamics to be solved simultaneously. The resulting set of equations is solved using a high-order multistep ordinary differential equation solver, yielding a complete simulation for the seal dynamic behavior. Examples of seal motion are given in detailed transient responses. The stability threshold is investigated to gauge the influence of seal parameters such as inertia, speed, coning, and the direction of sealed pressure drops. The results show two modes of instability: (1) When the inertia effect is larger than a critical value, the natural response of the seal grows monotonically in a half-frequency-whirl mode. (2) When the seal coning is less than some critical value in an outside pressurized seal, the minimum film thickness diminishes because of hydrostatic instability, and face contact occurs. Conversely, an inside pressurized seal is shown to be hydrostatically stable and have a superior dynamic response at any coning.

scholarly journals About the Accuracy and the Grid Convergence of the Numerical Solution of the Energy Equation in Fluid Film Lubrication

2018 ◽  
Author(s):  
Silun Zhang ◽  
Mohamed-Amine Hassini ◽  
Mihai Arghir
Keyword(s):  
Numerical Solution ◽  
Reynolds Equation ◽  
Energy Equation ◽  
Computation Time ◽  
Fluid Film ◽  
Hydrodynamic Analysis ◽  
One Dimensional ◽  
Grid Convergence ◽  
Film Lubrication ◽  
Fluid Film Lubrication

The present work is focused on the numerical solution of the complete energy equation used in fluid film lubrication. The work was motivated by the fact the complete energy equation has no analytic solution that could be used for validations. Its accuracy and computation time are related to the employed numerical method and to the grid resolution. The natural discretization method (NDM) applied on different grids is systematically compared with the spectral method (the Lobatto Point Colocation Method or LPCM) with different polynomial degrees. A one dimensional inclined slider is used for the numerical tests and the energy equation is artificially decoupled from Reynolds. This approach enables to focus all the attention on the numerical solution of the energy equation. The results show that the LPCM is one or two orders of magnitudes more efficient than the NDM in terms of computation time. The energy equation is then coupled with Reynolds equation in a thermo-hydrodynamic analysis of the same 1D slider; the numerical results confirm again the efficiency of the LPCM. A thermo-hydrodynamic analysis of a two-lobe journal bearing is then presented as a practical application.

On the Mechanics of Periodic Motion and Control of Tethered Satellite System in Elliptical Orbit

2019 ◽  
Vol 11 (06) ◽  
pp. 1950057
Author(s):  
Z. J. Pang ◽  
Z. Zhao ◽  
Q. T. Wang ◽  
Z. H. Du
Keyword(s):  
Periodic Solutions ◽  
Equations Of Motion ◽  
Satellite System ◽  
Critical Value ◽  
Stable Region ◽  
Periodic Motions ◽  
Orbital Eccentricity ◽  
Tethered Satellite ◽  
Tethered Satellite System ◽  
The Stability

This study investigates the periodic motions of an in-plane tethered satellite system in elliptical orbits. The equations of motion of the system are derived, and periodic solutions are obtained by perturbation method. Then, the stability properties of the periodic solutions are studied. Analysis results show that the periodic solutions become unstable when orbital eccentricity is larger than a critical value. Two classical control schemes are used to convert the unstable periodic motions to stable ones. Stability analyses of periodic solutions of the two controlled systems show that the two control methods can improve the critical value of orbital eccentricity. Numerical simulations of the controlled system are carried out to demonstrate the validity of the stable region.

scholarly journals Accuracy and Grid Convergence of the Numerical Solution of the Energy Equation in Fluid Film Lubrication: Application to the 1D Slider

Lubricants ◽  
2018 ◽  
Vol 6 (4) ◽  
pp. 95
Author(s):  
Silun Zhang ◽  
Mohamed-Amine Hassini ◽  
Mihai Arghir
Keyword(s):  
Numerical Solution ◽  
Reynolds Equation ◽  
Energy Equation ◽  
Computation Time ◽  
Fluid Film ◽  
Hydrodynamic Analysis ◽  
One Dimensional ◽  
Grid Convergence ◽  
Film Lubrication ◽  
Fluid Film Lubrication

The present work is focused on the numerical solution of the complete energy equation used in fluid film lubrication. The work was motivated by the fact that the complete energy equation has no analytical solution that can be used for validations. Its accuracy and computation time are related to the employed numerical method and to the grid resolution. The natural discretization method (NDM) applied on different grids is systematically compared with the spectral method (the Lobatto Point Colocation Method or LPCM) with different polynomial degrees. A one dimensional inclined slider is used for the numerical tests, and the energy equation is artificially decoupled from the Reynolds equation. This approach enables us to focus all the attention on the numerical solution of the energy equation. The results show that the LPCM is one or two orders of magnitude more efficient than the NDM in terms of computation time. The energy equation is then coupled with the Reynolds equation in a thermo-hydrodynamic analysis of the same 1D slider; the numerical results confirm again the efficiency of the LPCM. A thermo-hydrodynamic analysis of a two-lobe journal bearing is then presented as a practical application.

Stability of an electron beam in a two-frequency wiggler with a self-generated field

2010 ◽  
Vol 77 (2) ◽  
pp. 257-263 ◽  
Author(s):  
SOON-KWON NAM ◽  
KI-BUM KIM
Keyword(s):  
Fourier Transformation ◽  
Equations Of Motion ◽  
Critical Value ◽  
The Self ◽  
Fast Fourier Transformation ◽  
Electron Motion ◽  
Surface Of Section ◽  
Poincaré Surface Of Section ◽  
Wiggler Magnetic Field ◽  
The Stability

AbstractWe investigate the relativistic electron motions in a two-frequency wiggler magnetic field with self-generated fields. The equations of motion are derived from the Hamiltonian which include the self-generated field, and we find the steady-state orbit from the equations of motion. The stability of electron motion in a two-frequency wiggler is examined by the numerical simulation. We analyze the a dynamical systems using the fast Fourier transformation and the Poincarè surface of section to find the critical value which have the periodical electron motion and to optimize the two-frequency wiggler.

Semi-Analytical Dynamic Analysis of Spiral-Grooved Mechanical Gas Face Seals

2003 ◽  
Vol 125 (2) ◽  
pp. 403-413 ◽  
Author(s):  
Brad A. Miller ◽  
Itzhak Green
Keyword(s):  
Constitutive Model ◽  
Dynamic Analysis ◽  
Closed Form ◽  
Prony Series ◽  
Closed Form Solutions ◽  
Face Seals ◽  
Mechanical Face Seal ◽  
Natural Response ◽  
Nonlinear Numerical Simulation ◽  
The Stability

A novel semi-analytical formulation is presented for the linearized dynamic analysis of spiral-grooved mechanical gas face seals. The linearized rotordynamic properties of the gas film are numerically computed and then represented analytically by a constitutive model consisting of a cosine modified Prony series. The cosine modification enables the Prony series to characterize the gas film properties of face seals in applications with large compressibility numbers. The gas film correspondence principle is then employed to couple the constitutive model to the dynamics of the mechanical face seal. Closed-form solutions are presented for the transient natural response to initial velocity conditions, the steady-state response to rotor runout and initial stator misalignment, the transmissibility ratios, and the stability threshold. Results from the closed-form solutions are all within a few percent of the results from a full nonlinear numerical simulation.

A Mass-Conserving Complementarity Formulation to Study Fluid Film Lubrication in the Presence of Cavitation for Non-Newtonian and Compressible Fluids

2012 ◽  
Author(s):  
Luca Bertocchi ◽  
Matteo Giacopini ◽  
Antonio Strozzi ◽  
Mark T. Fowell ◽  
Daniele Dini
Keyword(s):  
Reynolds Equation ◽  
Incompressible Fluids ◽  
Compressible Fluids ◽  
Contact Pressure Distribution ◽  
One Dimensional ◽  
The Stability ◽  
Mathematical Derivation ◽  
Dimensional Domain ◽  
Film Lubrication ◽  
Fluid Film Lubrication

A mass-conserving formulation of the Reynolds equation has been recently developed using the concept of complementarity [1]. The mathematical derivation of the Linear Complementarity Problem (LCP) implemented in the solver favoured in [1] overcomes the drawbacks previously associated with the use of such complementarity formulations for the solution of cavitation problems in which reformation of the liquid film occurs. In the present paper, the proposed methodology, already successfully applied to solve textured bearing and squeeze problems in the presence of cavitation in a one dimensional domain and for incompressible fluids [1], has been extend to a two dimensional domain and the fluid compressibility has been included in the formulation. The evolution of the cavitated region and the contact pressure distribution are studied for a number of different configurations. Some of the results obtained with the proposed scheme are critically analysed and compared with the predictions obtained using alternative formulations (including full CFD calculations). The stability of the proposed algorithm and its flexibility in terms of the implementation of different compressibility laws is highlighted.

A Parametric Analysis of the Transient Forced Response of Noncontacting Coned-Face Gas Seals

2001 ◽  
Vol 124 (1) ◽  
pp. 151-157 ◽  
Author(s):  
Itzhak Green ◽  
Roger M. Barnsby
Keyword(s):  
Equations Of Motion ◽  
Forced Response ◽  
Coupled Analysis ◽  
Face Seal ◽  
Reference Case ◽  
Stable Mode ◽  
Pressure Drops ◽  
Stability Issue ◽  
Gas Seals ◽  
The Stability

A properly designed mechanical face seal must satisfy two requirements: (1) the seal must be stable, and (2) the seal forced response must be such that the stator tracks the misaligned rotor with the smallest clearance possible, with the smallest relative tilt, and with the largest minimum film thickness. The stability issue was investigated in a previous paper. Here a numerical solution is presented for the transient response of a noncontacting gas lubricated face seal that is subjected to stator and rotor forcing misalignments. The seal dynamic response is obtained in axial and angular modes of motion in a coupled analysis where the Reynolds equation and the equations of motion are solved simultaneously. The steady-state response is first identified for a reference case. Subsequently a parametric study is performed to gauge the influence of the various seal effects, such as speeds, inner to outer radii ratios, face coning heights, pressure drops, support stiffness and damping, and forcing misalignments. The transient responses to static stator misalignment and rotor runout are given, showing that properly designed coned face seals can operate in a stable mode with the stator tracking dynamically a misaligned rotor.

scholarly journals Stability of nonaxisymmetric ferrofluid flow in rotating cylinders with magnetic field

2005 ◽  
Vol 2005 (23) ◽  
pp. 3727-3737 ◽  
Author(s):  
Jitender Singh ◽  
Renu Bajaj
Keyword(s):  
Magnetic Field ◽  
Applied Magnetic Field ◽  
Critical Value ◽  
Annular Space ◽  
Rotating Cylinders ◽  
The Stability

Effect of an axially applied magnetic field on the stability of a ferrofluid flow in an annular space between two coaxially rotating cylinders with nonaxisymmetric disturbances has been investigated numerically. The critical value of the ratioΩ∗of angular speeds of the two cylinders, at the onset of the first nonaxisymmetric mode of disturbance, has been observed to be affected by the applied magnetic field.

On the Parametric Excitation of Pendulum-Type Vibration Absorber

1961 ◽  
Vol 28 (3) ◽  
pp. 330-334 ◽  
Author(s):  
Eugene Sevin
Keyword(s):  
Energy Transfer ◽  
Numerical Solution ◽  
Parametric Excitation ◽  
Equations Of Motion ◽  
Vibration Absorber ◽  
Single Cycle ◽  
Linearized System ◽  
Nonlinear Equations Of Motion ◽  
Beam Oscillation ◽  
Energy Interchange

The free motion of an undamped pendulum-type vibration absorber is studied on the basis of approximate nonlinear equations of motion. It is shown that this type of mechanical system exhibits the phenomenon of auto parametric excitation; a type of “instability” which cannot be accounted for on the basis of the linearized system. Complete energy transfer between modes is shown to occur when the beam frequency is twice the simple pendulum frequency. On the basis of a numerical solution, approximately 150 cycles of the beam oscillation take place during a single cycle of energy interchange.

Paper 3: Recent Developments in Fluid Film Lubrication Theory

1967 ◽  
Vol 182 (1) ◽  
pp. 36-50 ◽  
Author(s):  
T. Lloyd ◽  
H. McCallion
Keyword(s):  
Finite Difference ◽  
High Speed ◽  
Lubrication Theory ◽  
Fluid Film ◽  
Special Topic ◽  
Finite Difference Schemes ◽  
Great Activity ◽  
Wide Range ◽  
Film Lubrication ◽  
Fluid Film Lubrication

Developments in high-speed electronic computers have greatly influenced the progress in fluid film lubrication over the past ten years. Static and dynamic oil film parameters have been computed for a wide range of finite geometries, for hydrostatic and hydrodynamic bearings lubricated by compressible and incompressible lubricants. These are either sufficient in themselves or else act as a yardstick against which approximate formulas may be tested. Much use has been made of iterative finite difference schemes, which are particularly well suited to digital computers, and these methods are now more fully understood. Other methods of solution include direct inversion of finite difference matrices and solution by expression of the pressure by some infinite series, a finite number of terms of which give adequate representation. Besides the increase in design data available, there has been substantial progress through a re-examination of the effects of modifying some of the assumptions inherent in most of the available solutions of the Reynolds equation. These include the assumption of constant lubricant viscosity, of rigid surfaces and of laminar flow. Major progress has been witnessed in two fields. The interaction of the lubricant film with elastic boundaries has been shown to be of prime importance in highly loaded contacts such as gears. This has led to the development of the special topic of elastohydrodynamic lubrication theory. The applicability of gas bearings in such growing industries as computers, space vehicles and nuclear reactors has resulted in great activity and progress in this field.

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