The Rotor Dynamic Coefficients of Eccentric Mechanical Face Seals

1996 ◽  
Vol 118 (1) ◽  
pp. 215-224 ◽  
Author(s):  
J. Wileman ◽  
I. Green
Keyword(s):  
Degrees Of Freedom ◽  
Reynolds Equation ◽  
Equations Of Motion ◽  
Shear Stresses ◽  
Dynamic Coefficients ◽  
Angular Deflection ◽  
Face Seals ◽  
Radial Forces ◽  
Rotor Dynamic ◽  
Additional Coupling

The Reynolds equation is extended to include the effects of radial deflection in a seal with two flexibly mounted rotors. The resulting pressures are used to obtain the forces and moments introduced in the axial and angular modes by the inclusion of eccentricity in the analysis. The rotor dynamic coefficients relating the forces and moments in these modes to the axial and angular deflection are shown to be the same as those presented in the literature for the concentric case. Additional coefficients are obtained to express the dependence of these forces and moments upon the radial deflections and velocities. The axial force is shown to be decoupled from both the angular and radial modes, but the angular and radial modes are coupled to one another by the dependence of the tilting moments upon the radial deflections. The shear stresses acting upon the element faces are derived and used to obtain the radial forces acting upon the rotors. These forces are used to obtain rotor dynamic coefficients for the two radial degrees of freedom of each rotor. The additional rotor dynamic coefficients can be used to obtain the additional equations of motion necessary to include the radial degrees of freedom in the dynamic analysis. These coefficients introduce additional coupling between the angular and radial degrees of freedom, but the axial degrees of freedom remain decoupled.

Dynamics of a motorized spindle supported on water-lubricated bearings

2016 ◽  
Vol 231 (3) ◽  
pp. 459-472 ◽  
Author(s):  
Huihui Feng ◽  
Shuyun Jiang
Keyword(s):  
Reynolds Equation ◽  
Dynamic Stiffness ◽  
Journal Bearings ◽  
Motorized Spindle ◽  
Dynamic Coefficients ◽  
Numerical Predictions ◽  
Tilting Angle ◽  
Dynamic Performances ◽  
Experimental Values ◽  
Rotor Dynamic

The purpose of this paper is to investigate the dynamic performances of a motorized spindle supported on water-lubricated bearings. A modified transfer matrix method considering both of the translational and tilting dynamic coefficients of the bearings is established. The turbulent Reynolds equation is adopted and numerically solved by the perturbation method and the finite difference method, and the dynamic characteristics of the water-lubricated journal bearings are obtained; the effects of the eccentricity ratio, tilting angle, and the rotational speed on the dynamic coefficients of the water-lubricated journal bearings are analyzed. The critical speed, the dynamic stiffness of spindle nose, and unbalance response of the motorized spindle are investigated. Finally, a comparative study of rotor dynamic behaviors between the 32- and the eight-coefficient bearing models is conducted. The numerical predictions obtained by the 32-coefficient bearing models correlate well with the experimental values available in the literature.

Stability Analysis of a Whirling Rigid Rotor Supported by FDBs Considering Five Degrees of Freedom of a General Rotor-Bearing System

2014 ◽  
Author(s):  
M. H. Lee ◽  
J. H. Lee ◽  
G. H. Jang
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Time Varying ◽  
Algebraic Equations ◽  
Dynamic Coefficients ◽  
Mass Unbalance ◽  
Whirling Motion ◽  
Rotor Bearing ◽  
The Stability ◽  
Bearing System

A rotor supported by fluid dynamic bearings (FDBs) has a whirling motion by centrifugal force due to the mass unbalance or by the flexibility of shaft. This whirling motion also generates periodic time-varying oil-film reaction and dynamic coefficients even in case of the stationary grooved FDBs. This paper proposes a method to determine the stability of a whirling rotor supported by stationary grooved FDBs considering five degrees of freedom of a general rotor-bearing system. Dynamic coefficients are calculated by using the finite element method and the perturbation method, and they are represented as periodic harmonic functions by considering whirling motion. Because of the periodic time-varying dynamic coefficients, the equations of motion of the rotor supported by FDBs can be represented as a parametrically excited system. The solution of the equations of motion can be assumed as the Fourier series so that the equations of motion can be rewritten as simultaneous algebraic equations with respect to the Fourier coefficients. Hill’s infinite determinant is calculated by using these algebraic equations in order to determine the stability. The stability of the FDBs decreases with the increase of rotational speed. The stability of the FDBs increases with the increase of whirl radius, because the average and variation of Cxx increase faster than those of Kxx. The proposed method is verified by solving the equations of motion by using the forth Runge-Kutta method to determine the convergence and divergence of whirl radius.

Experimental Analysis of Dynamic Characteristics of a Hybrid Aerostatic Bearing

2011 ◽  
Author(s):  
Laurent Rudloff ◽  
Mihai Arghir ◽  
Olivier Bonneau ◽  
Sebastien Guingo ◽  
Guillaume Chemla ◽  
...  
Keyword(s):  
Dynamic Characteristics ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Experimental Information ◽  
Rigid Rotor ◽  
Aerostatic Bearing ◽  
Test Rig ◽  
Impulse Turbine ◽  
Dynamic Coefficients ◽  
The Impact

The dynamic characteristics of a hybrid aerostatic bearing are experimentally investigated on a test rig consisting of a rigid rotor driven by an impulse turbine located at its midlength. The rotor is horizontally mounted and is supported by two identical aerostatic bearings at its ends. Both the impulse turbine and the aerostatic hybrid bearings are fed with air. The actually available resources enable to attain feeding pressures up to 5 bar in the bearings and rotation speeds up to 60 krpm. Under these conditions the dynamic load on the rotor is much larger than the static load engendered by its weight. Dynamic loads consist either of impacts provided by a hammer or of added unbalance masses. The test rig can measure the bearing feeding pressures, the rotation speed, the impact force, the displacements of the two bearings and the bearing housing accelerations. This experimental information together with the equations of motion of the rotor enables the identification of the dynamic coefficients of the bearings. A second identification procedure using the same impact hammer is enabled by the fact force transducers are mounted between the bearing housing and its support. The dynamic coefficients of the bearings can then be obtained from the equation of motion of its housing. Unbalance response provide a convenient way for verifying the accuracy of the identified dynamic coefficients. Therefore these coefficients are injected in the equations of motion of a four degrees of freedom rigid rotor and the theoretical results are compared with values measured on the test rig. Comparisons show that predictions are acceptable but become less accurate at high rotation speeds where large dynamic forces are needed for exciting the corresponding synchronous frequencies.

scholarly journals Fluid Film Dynamic Coefficients in Mechanical Face Seals

1983 ◽  
Vol 105 (2) ◽  
pp. 297-302 ◽  
Author(s):  
I. Green ◽  
I. Etsion
Keyword(s):  
Small Perturbation ◽  
Equilibrium Position ◽  
Degrees Of Freedom ◽  
Fluid Film ◽  
Seal Ring ◽  
Damping Coefficients ◽  
Dynamic Coefficients ◽  
Face Seals ◽  
Stiffness And Damping ◽  
Analytical Expressions

The stiffness and damping coefficients of the fluid film in mechanical face seals are calculated for the three major degrees of freedom of the primary seal ring. The calculation is based on small perturbation of the ring from its equilibrium position. Analytical expressions are presented for the various coefficients and a comparison is made with results of accurate but more complex analyses to establish the range of applicability.

Dynamics of Air-Lubricated Slider Bearings for Noncontact Magnetic Recording

1971 ◽  
Vol 93 (2) ◽  
pp. 272-278 ◽  
Author(s):  
T. Tang
Keyword(s):  
High Speed ◽  
Degrees Of Freedom ◽  
Reynolds Equation ◽  
Equations Of Motion ◽  
Two Dimensions ◽  
Direct Access ◽  
Slider Bearing ◽  
Storage Devices ◽  
Disk Storage ◽  
Slider Bearings

One of the key technologies which led to the success of modern magnetic disk storage devices is the development of self acting gas lubricated slider bearings for positioning a magnetic head precisely over a high speed rotating recording disk. This paper covers a dynamic simulation of such an air bearing system used in direct access disk storage devices. In the simulation model, the Reynolds equation, which describes the dynamics of the lubricating air film, is solved by finite difference techniques in two dimensions and time for compressible, isothermal flow. The equations of motion of the slider bearing are solved simultaneously with the Reynolds equation for three degrees of freedom. Applications of the simulation are demonstrated, and experimental measurements to verify the theory are presented and discussed.

Experimental Analysis of the Dynamic Characteristics of a Hybrid Aerostatic Bearing

2012 ◽  
Vol 134 (8) ◽  
Author(s):  
Laurent Rudloff ◽  
Mihai Arghir ◽  
Olivier Bonneau ◽  
Sébastien Guingo ◽  
Guillaume Chemla ◽  
...  
Keyword(s):  
Dynamic Characteristics ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Measuring Instruments ◽  
Rigid Rotor ◽  
Aerostatic Bearing ◽  
Test Rig ◽  
Impulse Turbine ◽  
Dynamic Coefficients ◽  
The Impact

The dynamic characteristics of a hybrid aerostatic bearing are experimentally investigated on a test rig consisting of a rigid rotor driven by an impulse turbine. The rotor is horizontally mounted and is supported by two identical aerostatic bearings. Both the impulse turbine and the aerostatic hybrid bearings are fed with air. The feeding pressures in the bearings can be as high as 7 bars and rotation speeds can reach 60 krpm so the dynamic load on the rotor is much larger than the static load engendered by its weight. Excitations are applied either via an impact hammer or via unbalancing masses. The measuring instruments record the bearing feeding pressures, the rotation speed, the impact force, the displacements of the two bearings, and the bearing housing accelerations. The experimental data together with the equations of motion of the rotor enables the identification of the dynamic coefficients of the bearings. A second identification procedure using the same impact hammer is also possible as force transducers are mounted between the bearing housing and its support. The dynamic coefficients of the bearings can then be obtained from the equation of motion of its housing. Unbalance response provide a convenient way for verifying the accuracy of the identified dynamic coefficients. Therefore these coefficients are injected in the equations of motion of a four degrees of freedom rigid rotor and the theoretical results are compared with values measured on the test rig. Comparisons show that predictions are acceptable but become less accurate at high rotation speeds where large dynamic forces are needed for exciting the corresponding synchronous frequencies.

scholarly journals The Rotor Dynamic Coefficients of Coned-Face Mechanical Seals With Inward or Outward Flow

1987 ◽  
Vol 109 (1) ◽  
pp. 129-135 ◽  
Author(s):  
I. Green
Keyword(s):  
Fluid Film ◽  
Mechanical Seals ◽  
Dynamic Coefficients ◽  
Face Seals ◽  
Stiffness And Damping ◽  
Rotor Dynamic

The linearized fluid film dynamic coefficients, i.e., stiffness and damping, of flexibly-mounted rotor noncontacting mechanical face seals are found. The coefficients are derived from a previous study where the flexibly mounted element was the stator. The two cases of inward and outward flows, both having converging gaps in the direction of flow, are analyzed for the two mounting configurations, and it is found that the later case possesses higher angular stiffness.

scholarly journals Curvilinear Squeeze Film Bearing Lubricated with a Dehaven Fluid or with Similar Fluids

2017 ◽  
Vol 22 (3) ◽  
pp. 697-715
Author(s):  
A. Walicka ◽  
P. Jurczak ◽  
J. Falicki
Keyword(s):  
Reynolds Equation ◽  
Equations Of Motion ◽  
Unified Model ◽  
Squeeze Film ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Shear Stresses ◽  
Strain Rates ◽  
Load Carrying Capacity ◽  
Load Carrying

AbstractIn the paper, the model of a DeHaven fluid and some other models of non-Newtonian fluids, in which the shear strain rates are known functions of the powers of shear stresses, are considered. It was demonstrated that these models for small values of material constants can be presented in a form similar to the form of a DeHaven fluid. This common form, called a unified model of the DeHaven fluid, was used to consider a curvilinear squeeze film bearing. The equations of motion of the unified model, given in a specific coordinate system are used to derive the Reynolds equation. The solution to the Reynolds equation is obtained by a method of successive approximations. As a result one obtains formulae expressing the pressure distribution and load-carrying capacity. The numerical examples of flows of the unified DeHaven fluid in gaps of two simple squeeze film bearings are presented.

The motion of a lunar orbiter

1966 ◽  
Vol 25 ◽  
pp. 373
Author(s):  
Y. Kozai
Keyword(s):  
Degrees Of Freedom ◽  
Dimensional Space ◽  
Artificial Satellite ◽  
Equations Of Motion ◽  
Triaxial Ellipsoid ◽  
The Moon ◽  
Secular Motion ◽  
The Earth ◽  
Close Satellite ◽  
The Mean

The motion of an artificial satellite around the Moon is much more complicated than that around the Earth, since the shape of the Moon is a triaxial ellipsoid and the effect of the Earth on the motion is very important even for a very close satellite.The differential equations of motion of the satellite are written in canonical form of three degrees of freedom with time depending Hamiltonian. By eliminating short-periodic terms depending on the mean longitude of the satellite and by assuming that the Earth is moving on the lunar equator, however, the equations are reduced to those of two degrees of freedom with an energy integral.Since the mean motion of the Earth around the Moon is more rapid than the secular motion of the argument of pericentre of the satellite by a factor of one order, the terms depending on the longitude of the Earth can be eliminated, and the degree of freedom is reduced to one.Then the motion can be discussed by drawing equi-energy curves in two-dimensional space. According to these figures satellites with high inclination have large possibilities of falling down to the lunar surface even if the initial eccentricities are very small.The principal properties of the motion are not changed even if plausible values ofJ3andJ4of the Moon are included.This paper has been published in Publ. astr. Soc.Japan15, 301, 1963.

The Influence of Loading Position in A Priori High Stress Detection using Mode Superposition

2020 ◽  
Vol 1 (1) ◽  
pp. 93-102
Author(s):  
Carsten Strzalka ◽  
◽  
Manfred Zehn ◽  
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
A Priori ◽  
High Stress ◽  
Von Mises Stress ◽  
Detailed Knowledge ◽  
Variable Loading ◽  
Numerical Effort ◽  
Von Mises

For the analysis of structural components, the finite element method (FEM) has become the most widely applied tool for numerical stress- and subsequent durability analyses. In industrial application advanced FE-models result in high numbers of degrees of freedom, making dynamic analyses time-consuming and expensive. As detailed finite element models are necessary for accurate stress results, the resulting data and connected numerical effort from dynamic stress analysis can be high. For the reduction of that effort, sophisticated methods have been developed to limit numerical calculations and processing of data to only small fractions of the global model. Therefore, detailed knowledge of the position of a component’s highly stressed areas is of great advantage for any present or subsequent analysis steps. In this paper an efficient method for the a priori detection of highly stressed areas of force-excited components is presented, based on modal stress superposition. As the component’s dynamic response and corresponding stress is always a function of its excitation, special attention is paid to the influence of the loading position. Based on the frequency domain solution of the modally decoupled equations of motion, a coefficient for a priori weighted superposition of modal von Mises stress fields is developed and validated on a simply supported cantilever beam structure with variable loading positions. The proposed approach is then applied to a simplified industrial model of a twist beam rear axle.

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