Critical Speed of Hyperbolic Disks and Eccentricity Effects on The Stability of The Rotor Bearing System

The critical speed of hyperbolic disks for the selected model is calculated in three different cases, anisotropic material and thermal case, isotropic material and thermal case finally isotropic material and thermal with generated heat case. The second case is found to be the best because the critical speed occurs above maximum speed values. For recognition of the eccentricity (e) effects on the stability, the original bearings of the selected model were tilting pads bearing replaced by adjustable hydrodynamic pads bearing. The results proved that stability criteria (Sc), critical mass ( cM ), whirling frequency ratio (


Introduction
The rotordynamics of turbomachinery encompasses the structural analysis of rotors (shafts and disks) and the design of fluid film bearings that determine the best dynamic performance given the required operating conditions.This best performance is denoted by wellcharacterized natural frequencies and critical speeds with amplitudes of synchronous dynamic response within required standards and demonstrated absence of subsynchronous vibration instabilities, [1].Lewis et al,(1987), [2] have developed a spilt bearing with a new configuration, this bearing surfaces consists of a five pad tilting pad fluid bearings with one pad having a mean for being moved radially.This radial motion causes changes in the proportionality constants, the stiffness and the damping capabilities of the bearing which in turn changes the critical speeds of the rotor system Wang and Shin, (1990), [3] improved the stability of rotor bearing system by optimization technique used to find the optimum diameters of shaft elements so that the optimized rotor can sustain maximum fluid leakage excitation.Chivens and Nelson (2000), [4] presented an analytical investigation into the influence of disk flexibility on the transverse bending natural frequencies and critical speeds of a rotating shaft-disk system.The partial differential equations governing the system motion and the associated exact Eng. & Technology, Vol.25, No.8, 2007 Critical Speed of Hyperbolic Disks and Eccentricity Effects on The Stability of The Rotor Bearing System 1000 solution form had been developed.Numerical solutions are presented covering a wide range of nondimensional parameters and general conclusions are drawn.Krodkiewski and Sun (2002),[5] presented the modeling and analysis of a rotorbearing system with a new type of active oil bearing, the active bearing is supplied with a flexible sleeve whose deformation can be changed during operation of the rotor.The flexible sleeve is also a part of a hydraulic damper whose parameters can be controlled during operation as well, the self exciting vibration can be eliminated or greatly reduced during operation by properly controlled deformation of the flexible sleeve and optimal choice of the hydraulic damper parameters.In this paper the topic has been divided into two parts, the first is confined the evaluation of the critical speeds of hyperbolic disk by thermoelastic relations in three operating cases, while the second contains the replacement of the two tilting bearings type used in the selected model with two hydrodynamic adjustable pads bearings and also study the effects of the eccentricity on the stability criteria at high speeds, and the effect of thermohydrodynamic (THD) analysis on the stability of rotor-bearing system.

Theoretical Analysis of The Model
It takes similar characteristics of centrifugal compressor, it's consisting of three disks (hyperbolic profile) clamped on a shaft which is supported by two tilting pads bearings driven by steam turbine using in "southern refineries company", all dimensions in origin specifications as shown in figure (1).

Mechanical Specification (i)-Type of compressors RB43B;
Serial No. 33F3047 (ii)-shaft properties : -total length: 1411 mm -shaft material; all properties as shown in table (1).-maximum diameter:142 mm -minimum diameter:108 mm at bearing and 63 mm (at shaft end).(iii)-Two tilting pads bearing (right and left).(iv)-Numbers and type of disks; three hyperbolic profile variable thickness disk, some whose dimensions: -inner and outer diameters respectively :139mm -420mm -disk thickness: 100 mm (at inner diameter) -disk material; all properties as shown in table (1

Thickness and Inhomogeneity
Disks of hyperbolic axial thickness with profile shown in figure (2) and described by [6].
where: 2h =outer thickness at radius (a) β= constant in variable thickness expression The governing equations for thermoelastic analysis of an inhomogeneous, rotating disk with arbitrary varying thickness, small deformations and rotational symmetry are assumed.If the thickness of the disk is small compared to its radius, the axial stress is negligible and the disk is then in a state of plane stress.The material is assumed to be isotropic but inhomogeneous along the radial direction.That is the Young's modulus E=E (r), the density of the disk material ρ = ρ(r), the coefficient of linear expansion α = α (r), and the Poisson's ratio ) (r υ υ = .The temperature field is steady, axisymmetric but inhomogeneous, that is, T = T ( r) where T is the temperature.
From equilibrium consideration of an element as shown in figure (2) The exact elastic displacement in radial direction u(r) can be written as: The critical speed (ω c ) depends on thermo-mechanical anisotropy of the disk material, size of the disk and temperature at center of the disk, so that the selection of material and size of the disk become important for design consideration.Equation ( 9) can be written for disk material which is isotropic material: Poisson's ratio respectively.if we consider such a thermal situation that rate of generation of heat is fixed amount "Q" per unit volume, the critical speed: where: K*=thermal conductivity where: =

Study The Rotor Bearing System
Parametric Stability Analysis Method of analysis based on Liapunov's direct method is used for studying the behavior of the nonlinear system of differential equations governing the motion of a rotor-bearing system in the neighborhood of its equilibrium point.Among the results reached in Liapunov ' s direct method is the demonstration of the roles played by, [13].
Applying Sylvester's Theorem to test the positive definiteness of ( ik U ), its successive principal minor determinants must be positive definite, thus yielding the following sufficient conditions for system whirl stability.

Discussion of Results
The results of this paper are illustrated below.Fig. ( 4) show the relationship between temperature rise (suction and discharge temperatures) and increasing value of critical speed ( ω c ) for three cases, also shows the critical speed (ω c ) as a variable quantity.The second case is found to be the best case because the critical speed occurs at over the range of the high rotational speed disk N=12000 r.p.m.

(centrifugal compressor driven by steam turbine).
It is quite easy to observe the dependence of critical speed  (ωc ) on the thermo-mechanical anisotropy of the disk material (size of the disk and temperature at center of the disk).From the values of the critical speed (ωc ) given in equations (9,10 and 11) it is noted that the critical speed  (ωc ) is independent of the variation of disk thickness that is the critical speed  (ωc ) will remain the same for all hyperbolic profiles.From Fig.( 5), the rotor critical mass ( c M ) increases as the eccentricity (e) increases to reach system stability.The rotor critical mass ( c M ) must be negative this means the curve no.3 is more stable especially at high values of eccentricity (e). of eccentricity ratio (εο).The system becomes more stable at the negative value of this ratio as shown in curve (3) at rotational speed equal 12000 r. p .m.. Whirl begins with the rotor operating relatively close to center of the bearing.When the journal is close to the center of the bearing (the eccentricity ratio is small) the bearing stiffness is much lower than the shaft stiffness.When the journal is located relatively close to the bearing wall (the eccentricity ratio is near 0.9) the bearing stiffness is typically much higher than rotor shaft stiffness.One can improved the stability of rotor bearing system by selecting softer surface material of bearings.
It can be seen from Fig.( 7) the thermohydrodynamic (THD) analysis effects on the threshold speed at all eccentricity ratios (ε).The best value of threshold speed (Ω) at ε=0.4 is in the stable region.This figure shows the increase of temperature rise parameter( ψ ) with increase of eccentricity ratio (when the maximum rotational speed N=12000 r.p.m the value of ε=0.66 at ψ=0.2 but when ε=0.696 at ψ=0.26).
Table (2) shows the relationship between the stability criteria with change of eccentricity (e).It also shows the stability criteria increases with the increase of eccentricity, but the values of stability criteria must be larger than ( 2α ).