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Nonlinear Dynamic Analysis of a Rigid Rotor Supported

by a Three-Pad Hydrostatic Squeeze Film Dampers

A. Bouzidane a & M. Thomas b

a Research Laboratory of Industrial Technologies Department of Science and Technology , Ibn

Khaldun's University of Tiaret BP 78 , Tiaret , 14000 , Algeria

b Ecole Supérieur de Technologie 1100 Notre Dame Ouest , Montréal , H1C 1K3 , Québec ,

Canada

Accepted author version posted online: 28 Mar 2013.Published online: 14 Jun 2013.

To cite this article: A. Bouzidane & M. Thomas (2013): Nonlinear Dynamic Analysis of a Rigid Rotor Supported by a Three-Pad

Hydrostatic Squeeze Film Dampers, Tribology Transactions, 56:5, 717-727

To link to this article: http://dx.doi.org/10.1080/10402004.2013.788238

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Tribology Transactions, 56: 717-727, 2013

Copyright C

Society of Tribologists and Lubrication Engineers

ISSN: 1040-2004 print / 1547-397X online

DOI: 10.1080/10402004.2013.788238

Nonlinear Dynamic Analysis of a Rigid Rotor Supported

by a Three-Pad Hydrostatic Squeeze Film Dampers

A. BOUZIDANE1and M. THOMAS2

1Research Laboratory of Industrial Technologies

Department of Science and Technology

Ibn Khaldun’s University of Tiaret

BP 78, Tiaret 14000, Algeria

2Ecole Sup´

erieur de Technologie

1100 Notre Dame Ouest

Montr´

eal, H1C 1K3

Qu´

ebec, Canada

The aim of this article is to study the nonlinear dynamic be-

havior of rigid rotors, supported by a new hydrostatic squeeze

ﬁlm damper (HSFD). A nonlinear model of an HSFD is pre-

sented. The results are compared with those obtained from a

linear approach that is only valid for small vibrations around

the equilibrium position. Compared to the four-pad HSFD, the

advantage of using a three-pad HSFD consists of reducing the

cost due to the need for a feeding system and the volumetric

ﬂow rate used in hydrostatic lubrication. In this study, the ef-

fects of the pad dimensions ratios, capillary diameters, and ro-

tational speed on the ﬂow rate, unbalance responses, and trans-

mitted forces are investigated using a nonlinear method and the

results are analyzed and discussed. The results obtained show

that this type of HSFD provides hydrostatic designers with a

new bearing conﬁguration suitable to control rotor vibrations

and bearing transmitted forces for high speeds.

KEY WORDS

Hydrostatic Bearing Flat Pad; Squeeze Film Lubrication; Ro-

tor Bearing Dynamics; Nonlinear Method

INTRODUCTION

Hydrostatic squeeze ﬁlm dampers (HSFDs) composed of pads

bearing are often used to guide vertical or horizontal shafts (with

very lightly loads). This journal bearing type has better dynamic

characteristics than other antiwhirl conﬁgurations; that is, good

suppression of whirl, good damping at critical speeds, overall

good performance, wide range of design parameters, and mod-

erate cost (Allaire (1)). HSFDs can be employed in applications

involving heavy loads, high radial stiffness, zero or low speeds,

and low eccentricity ratios. These types of journal bearings are

of moderate cost, provide excellent low friction characteristics,

Manuscript received December 22, 2012

Manuscript accepted March 15, 2013

Review led by Luis San Andres

and have an extremely long life. They are superior to conven-

tional ball bearings and sleeve bearings in many applications.

HSFDs mounted on rolling-element bearings can be used in high-

speed gas turbine engines and power turbines to attenuate the un-

balance responses and bearing transmitted forces. Squeeze ﬁlm

dampers (SFDs) generate their damping force capability in reac-

tion to dynamic journal motions, squeezing a thin ﬁlm of lubri-

cant in the clearance between a stationary housing and a whirling

journal (San Andres and De Santiago (2)).

Many researchers have studied the effects of HSFDs in or-

der to use it them as a device to actively control rotors. Burrows,

et al. (3) investigated the possibility of controlling the pressure in

an SFD as a means of controlling rotating machinery. Mu, et al.

(4) suggested an active SFD with a movable conical damper ring.

San Andres (5) developed an approximate solution for the pres-

sure ﬁeld and dynamic force coefﬁcients in turbulent ﬂow, in a

symmetric hydrostatic bearing with its journal centered within the

bearing clearance. The model includes the effects of recess vol-

ume liquid compressibility and introduces the model for a HJB

with end seals. The results of its investigation show that HJBs

with end seals have increased damping, better dynamic stability

characteristics than conventional HJBs. Braun, et al. (6),(7) per-

formed an extensive analysis of the variation in lubricant viscosity

with pressure and temperature and also analyzed the ﬂow pattern

in the recesses. Hathout, et al. (8) summarized the modeling and

control of hybrid SFDs for active vibration control of rotors ex-

hibiting multiple modes. Sawicki, et al. (9) investigated the effects

of dynamic eccentricity ratio on the dynamic characteristic of a

four-pocket, oil-fed, oriﬁce-compensation hydrostatic bearing in-

cluding the hybrid effects of journal rotation. Adams and Zahloul

(10) studied the vibration of rotors by controlling the pressure in

hydrostatic four-pad SFDs. They showed that stiffness is quite

controllable with supply pressure and damping is nearly insen-

sitive to supply pressure changes using a linear method. Using

a similar system, Bouzidane and Thomas (11) investigated the

effects of ﬁlm thickness, recess pressure, and geometric conﬁg-

uration on the equivalent stiffness and damping of a four-pad

717

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718 A. BOUZIDANE AND M. THOMAS

NOMENCLATURE

A=Bearing pad length (m)

˜

A=Dimensionless vibratory amplitude

A/B =Bearing pad dimensions ratio

a/A =b/B =Dimension ratio

B=Bearing pad width (m)

Cpi =Damping in the yidirection of the ith hydrostatic

bearing pad (N.s/m)

dc=Capillary diameter (m)

e=Eccentricity (m)

Fpi =Force hydrostatic of the ith hydrostatic bearing pad

FT=Transmitted force (N)

Fx,Fy=Represent the hydrostatic forces in the Xand Y

directions (N)

h0=Film thickness at the center position of the

hydrostatic squeeze ﬁlm damper (m)

hi=Film thickness of the ith hydrostatic bearing pad (m)

hi=Squeeze velocity of the ith hydrostatic bearing

pad (m/s)

Kpi =Stiffness in the yidirection of the ith hydrostatic

bearing pad (N/m)

lc=Capillary length (m)

m=Mass of the rotor (kg)

Pri =Recess pressure of the ith

hydrostatic bearing pad (Pa)

Pr0,Pa0=Recess pressure at the center position of the

hydrostatic squeeze ﬁlm damper (Pa)

Ps=Supply pressure (Pa)

Qoi,Qoxi ,

Qoyi,Qozi =Flow rate requirement of the ith hydrostatic

bearing pad (m3/s) ﬂow rate requirement

in the x,y,andzdirections, respectively,

of the ith hydrostatic bearing pad (m3/s)

Qri =Flow through an oriﬁce or capillary of the ith

hydrostatic bearing pad (m3/s)

QT=Total ﬂow rate requirement (m3/s)

S=Cross-section area (m2)

Sb=Area of hydrostatic bearing pad (m2)

Sr=Area of hydrostatic bearing recess (m2)

uxi,u

zi =Flow velocities in the xand ydirections,

respectively, of the ith hydrostatic bearing

pad (m/s)

(xi,zi,yi)=Coordinate system used in the Reynolds equation

(x,y)=Coordinate system used to describe the

rotor motion

β0=Pr0/Ps=Ratio of recess pressure over supply pressure

at the center position of the hydrostatic

squeeze ﬁlm damper

βi=Pressure ratio of the ith hydrostatic bearing pad

ε=e/h0=Unbalance eccentricity

μ=Viscosity (Pa.s)

ω=Excitation frequency (rad/s)

HJB. Their results reveals that because of its higher stiffness,

good damping and zero cross-coupling terms, the four-pad HJB

has better dynamic characteristics and stability than the hybrid

journal bearing. They found that an optimal equivalent stiffness

of a four-pad HJB is obtained for a pressure ratio at the centered

position ßo close to 0.67. Bouzidane, et al. (12) investigated the

effect of pressure ratio, supply pressure, viscosity, and rotational

speed on the unbalance response and transmitted force of a rigid

rotor supported by a four-pad HSFD. Shen, et al. (13) presented

a new model to calculate the ﬂuid-ﬁlm forces under the Reynolds

boundary condition in order to study the nonlinear dynamics be-

havior of a rigid rotor in the elliptical bearing support. Their nu-

merical results showed that the balanced rotor undergoes a su-

percritical Hopf bifurcation as the rotor speed increases. A novel

numerical method to compute Floquet multipliers was presented

to predict the nonlinear response of rotor with an elastically sup-

ported SFD reported in the literature (Qin, et al. (14)). This

method can begin integration from any point near a stable tra-

jectory and avoid the numerical oscillation in the ﬁrst periods of

integration. Chang-Jian, et al. (15) theoretically investigated the

nonlinear dynamic behavior of a hybrid SFD-mounted rigid rotor

lubricated with coupled stress ﬂuid. The numerical results showed

that due to the nonlinear factors of the oil ﬁlm force, the trajec-

tory of the rotor demonstrated a complex dynamic with rotational

speed ratio. The effect of load orientation on the stability of a

three-lobe pressure dam bearing was studied (Rattan, et al. (16)).

The published literature on the performance of HSFDs com-

posed of pad bearings is mainly limited to four-pad HSFDs and

no information is available on the performance of three-pad

HSFDs. It should also be noted that there is no information ex-

isting in current published research on actual application of these

types of journal bearings in rotating machines. The objective of

this work was to study the effects of hydrostatic bearing dimen-

sions ratios, capillary diameters, and rotational speed on the ﬂow

rate, unbalance responses, and transmitted forces of a rigid rotor

supported by three-pad HSFDs fed by capillary restrictors. Non-

linear and linear models for an HSFD are presented. The results

are discussed and compared with those obtained from a linear

approach that is only valid for small vibrations around the equi-

librium position. The main advantages of three-pad HSFDs over

multipad HSFDs are the reduced cost due to the need for a feed-

ing system (pumps, ﬁlters, tanks, etc.), the volumetric ﬂow rate,

and control of the rotor vibrations and bearing transmitted forces

at high speeds.

MATHEMATICAL MODELING

Two kinds of conﬁgurations for segregated multipad hydro-

static bearings may be considered from among the simplest ge-

ometries: cylindrical and rectangular. The cylindrical pad conﬁg-

uration is similar to conventional nonhydrostatic cylindrical SFDs

without the use of a mechanical centering spring. This journal

bearing type is probably the least expensive to manufacture but

requires a mechanical antirotation device to prevent rotation of

the supported bearing (Adams and Zahloul (10)).

By comparison, the multipad hydrostatic bearings rectangu-

lar pad conﬁguration (i.e., four-pad conﬁguration, three-pad con-

ﬁguration; see Fig. 1) is inherently without rotation and, conse-

quently, one may make the assumption that the bearing is not

allowed to tilt or become misaligned. These types of bearings do

not require the motion of the surfaces to generate the lubricant

ﬁlm. Hence, they can operate from very low to very high speeds.

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Analysis of a Rigid Rotors 719

Fig. 1—Three-pad hydrostatic squeeze ﬁlm damper (color ﬁgure available online).

Description of Hydrostatic Squeeze Film Damper

A cross section of a new three-pad HSFD in the eccentric case

is shown in Fig. 1. This ﬁgure shows a vertical rigid rotor sup-

ported by an HSFD composed of three-pads. All pad geometries

are identical and equally spaced around the journal. The indices

1, 2, and 3 refer to the characteristics of the lower, left, and right

hydrostatic bearing ﬂat pads, respectively. Each pad is fed by a

capillary restrictor through a recess, which is supplied with an ex-

ternal pressure Ps.

Hydrostatic Squeeze Film Damper Characteristics

Calculation of the characteristics of the HSFD can be ob-

tained through the juxtaposition of three hydrostatic bearing ﬂat

pads (Fig. 2). It is assumed that the ﬂuid is incompressible and

inertialess. The ﬂow is laminar and the regime is steady state and

isothermal.

Reynolds Equation

The Reynolds equation allows for the computation of the

pressure distribution Pi(xi,zi,t). This equation can be solved nu-

merically by applying the centered ﬁnite differences method or

analytically for speciﬁc cases such as inﬁnitely long or short bear-

ings. If we consider that there is no slip between the ﬂuid and pad

bearing, the boundary conditions associated with the speed will

be as follows (Fig. 3):

rOn a ﬂat pad:

U1i=0; V1i=0andW1i=0[1]

Fig. 2—Hydrostatic bearing ﬂat pads.

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720 A. BOUZIDANE AND M. THOMAS

yi

h

i

o

Pad bearing

W1i

i1

U

i

1

V

i

2

W

V2i

i

2

U

x

i

z

i

Runner

Fig. 3—Boundary conditions of hydrostatic squeeze ﬁlm dampers.

rOn a runner:

U2i=0; V2i=hiand W2i=0[2]

where U1i;V1iand W1iare the speeds of the surface of the ith

hydrostatic bearing pad, and U2i;V2iand W2iare the speeds of the

surface of the runner; hiis the squeeze velocity of the ith hydro-

static bearing pad (i=1, 2, and 3).

With these boundary conditions, and for an incompressible,

laminar, isoviscous, and inertialess ﬂuid, the Reynolds equation

may be written as (Bouzidane, et al. (12)):

∂

∂xi∂Pi(xi,zi,t)

∂xi+∂

∂zi∂Pi(xi,zi,t)

∂zi

=12 μ

h3

i

hi(i=1,2,and 3).[3]

Note that the cavitations are not neglected when the thickness

ﬁlm increasing.

r0≤xi≤Aand 0 ≤zi≤B;

rPi(xi,zi,t) is the hydrostatic pressure ﬁeld of the ith hydro-

static bearing pad;

rhiis the ﬁlm thickness of the ith hydrostatic bearing pad (hi=

f(xi,zi));

r(xi,zi,yi) is the coordinate system used in the Reynolds equa-

tion.

The ﬁlm thickness hi(hi=f(x,y)= f(xi,zi)) is obtained as fol-

lows:

⎧

⎪

⎪

⎨

⎪

⎪

⎩

h1=h0−x∗

1

h2=h0−x∗

2

h3=h0−x∗

3

,[4]

where x∗

1,x∗

2,andx∗

3are obtained as follows (Fig. 4):

⎧

⎪

⎪

⎨

⎪

⎪

⎩

x∗

1=x

x∗

2=−xcos(π/6) +ysin(π/6)

x∗

3=−xcos(π/6) −ysin(π/6)

,[5]

where (x,y) is the coordinate system used to describe the rotor

motion.

The squeeze velocity hi(dhi

dt )oftheith hydrostatic bearing pad

is obtained as follows:

⎧

⎪

⎨

⎪

⎩

h1=−x

h2=xcos(π/6) −ysin(π/6)

h3=xcos(π/6) +ysin(π/6)

.[6]

O

∗

1

x

∗

2

x

∗

3

x

x

6/

π

6/

π

y

Fig. 4—Journal coordinates system (color ﬁgure available online).

Note that the starting operating condition (hi=h0,Pri =Pr0,

βi=β0) is deﬁned as the HSFD center position (neutral load),

where

rh0and β0are the ﬁlm thickness and pressure ratio, respec-

tively, at the centered position of the HSFDs, and

rPri and βiare the recess pressure and pressure ratio of the ith

hydrostatic bearing pad (i=1, 2, and 3).

It is assumed that the recess depth is considered very deep

and the pressure in the recess of the ith hydrostatic bearing pad

is constant and equals Pri and the ambient pressure is null. Thus,

the boundary conditions for Eq. [3] will be as follows (Fig. 2):

Pi(0 ≤xi≤A;zi=0,B;t)=0; Pi(xi=0,A;0 ≤zi≤B;t)=0

Pi(x1≤xi≤x2;z1≤zi≤z2;t)=Pri .[7]

Pi(xi,zi,t)<0,setPi(xi,zi,t)=0

The resolution of Eq. [3] allows one to obtain the pressure

ﬁelds Pi(xi,zi,t). The Reynolds equation can be solved using a

variety of numerical methods. The ﬁnite difference method was

considered in this study. Note that the ﬂow of lubricant through

the restrictor was equal to the journal bearing input ﬂow and neg-

ative pressure was set to zero during the interactive process to

deal with oil ﬁlm cavitations (when the ﬁlm thickness increased).

Recess Pressure

The recess pressure for each hydrostatic bearing pad is deter-

mined by resolving the following ﬂow continuity equation:

Qri =Qoi [8]

where

Qri =πd4

c

128μlc

(PS−Pri)[9]

Qoi =Qvi +Qoxi +Qozi [10]

Qvi =Srhi[11]

Qoxi =2B

0

dzihi

0

uxidy i;Qozi =2A

0

dxihi

0

uzi dyi[12]

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Analysis of a Rigid Rotors 721

Fig. 5—Rigid rotor and orbital motion of journal relative to three-pad HSFD (color ﬁgure available online).

uxi =1

2μ

∂Pi

∂xi

(yi−hi)yi;uzi =1

2μ

∂Pi

∂zi

(yi−hi)yi,[13]

where dcis the capillary diameter and lcis its length; Qvi repre-

sents the squeeze ﬂow of the ith hydrostatic bearing pad; Qoxi and

Qozi are the oil ﬂow of the ith hydrostatic bearing pad in the xiand

zidirections, respectively; Qri represents the ﬂow through a cap-

illary restrictor-type hydraulic resistance; and uxi ,uyi,anduzi are

the ﬂow velocities in the xi,yi,andzidirections, respectively.

Flow Rate Requirement

The total volumetric ﬂow rate that must be supplied to the

HSFDs is

QT=

3

i=1

Qoi =Qo1+Qo2+Qo3[14]

ROTOR DYNAMICS BEHAVIOR

In order to reduce the excessive high amplitudes of forced vi-

brations and the forces transmitted to the base, caused by rotor

imbalance and passage through critical speeds, a study on the dy-

namic behavior of a rotor supported by HSFDs comparing both

linear and nonlinear methods was conducted.

A symmetrical rigid rotor with two identical three-pad HJBs is

shown in Fig. 5. The rotor-bearing system was oriented in the ver-

tical direction and all related components (housings, shaft) were

considered to be rigid; in this case, the shaft bending stiffness was

considered to be much greater than the HJB stiffness. Note that

the shaft turned with constant angular velocity (ω) and underwent

translational motion only; that is, no tilt motion occurred. Thus,

the movement of the rotor center, Gr, was identical to that of

the HJB centers, Oj.Let(Oj,x,y) be an inertial coordinate frame

originating at the housing center. The yand xmotions of the rotor

were assumed to be uncoupled. The rotor was unbalanced with a

center of inertia, G, at a distance, e, from the geometrical center

Oj(Fig. 5).

The equations of the rotor motion can be expressed in Carte-

sian coordinates as follows:

mx =Fx+mexω2cos ωt

my =Fy+meyω2sin ωt

,[15]

where mis the mass of the rotor, eis the eccentricity, ωis the

excitation frequency, and Fxand Fyare the hydrostatic forces in

the xand ydirections, respectively.

Forces Hydrostatics Bearings

Nonlinear Model

The nonlinear ﬂuid ﬁlm forces on the three-pad HSFD in

Cartesian coordinates (Oj,x,y) were determined by the nonlin-

ear model as follows (Fig. 5):

Fx=−(FP1−(FP2+FP3)sin(π/6))

Fy=−(FP2−FP3)cos(π/6)

,[16]

where FPi represents the hydrostatic force of the ith hydrostatic

bearing pad (i=1, 2, and 3), which is obtained by integrating the

pressure over the bearing area:

Fpi =Si

Pi(xi,yi,t)dsi= Pi(xi,yi,t)dxidzi,[17]

Fig. 6—Stiffness and damping relative to the

i

th hydrostatic bearing pad

(color ﬁgure available online).

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722 A. BOUZIDANE AND M. THOMAS

4 8 12 16 20

Rotational Speed (krpm)

0.00

0.05

0.10

0.15

0.20

0.25

Dimensionless Vibration Amplitude

Nonlinear Method

Linear Method

48121620

Rotational Speed (krpm)

0.00

0.05

0.10

0.15

0.20

0.25

Transmitted Force (kN)

Nonlinear Method

Linear Method

Fig. 7—Comparison of linear and nonlinear models: vibratory response and transmitted force for ε=0.05 (

dc

=1.2 mm,

A

/

B

=6) (color ﬁgure available

online).

Fig. 8—Comparison of three-pad and four-pad HSFD: vibratory response, transmitted force, and ﬂow rate for ε=0.25 (

dc

=1.2 mm,

A

/

B

=6) (color ﬁgure

available online).

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Analysis of a Rigid Rotors 723

TABLE 1—SIMULATION PARAMETERS

Bearing Characteristics SI

Sb=A×B135.10−5m2

a/A =b/B 0.5

β00.67

Ps10 bar

lc58 mm

μ0.0117 Pa.s.

ρ860 kg/m3

m10 kg

where Siand dSiare the contact surface and element on the sur-

face of the ith bearing pad, respectively.

Linear Model

The linear model is based on a small displacement and small

speed hypothesis (Adams and Zahloul (10); Bouzidane, et al.

(12)) and is presented by linearizing the behavior around an equi-

librium state. The linear ﬂuid ﬁlm forces on the three-pad HSFD

in Cartesian coordinates (Oj,x,y) were obtained as follows:

Fx

Fy=−

Cxx Cxy

Cyx Cyy

[CP]

x

y−Kxx Kxy

Kyx Kyy

[KP]

x

y,[18]

where CPand KPrepresent the total hydrostatic bearing damp-

ing matrix and stiffness matrix, respectively, which are given as

follows (Adams and Zahloul (10)):

[CP]=

i=3

i=1

CPi cos2(γi)−cos(γi)sin(γi)

−cos(γi)sin(γi)sin

2(γi)[19]

[KP]=

i=3

i=1

KPi cos2(γi)−cos(γi)sin(γi)

−cos(γi)sin(γi)sin

2(γi)[20]

with

KPi =−

∂FPi

∂hi0

[21]

CPi =−

∂FPi

∂hi0

,[22]

where Kpi and Cpi represent the stiffness and damping of the ith

hydrostatic bearing pad, and Fpi is the hydrostatic force of the ith

hydrostatic bearing pad (Fig. 6). The partial derivatives were cal-

culated numerically using the numerical differentiation method.

Linear and Nonlinear Simulations

The HSFD effects on rotor dynamics are characterized by the

hydrostatic forces generated by a pressure ﬁeld. These forces can

be determined by using either nonlinear or linear methods. It

is important to mention that these forces vary according to the

position and velocity of the shaft center in the journal bearing.

Calculation of the ﬂow rate, vibratory responses, and amplitude

of transmitted forces due to a rotating unbalance vary depend-

ing on the rotational speed and are determined by resolving the

equations of rotor motion (Eq. [15]) by using either nonlinear or

linear methods. The computed amplitudes are determined from

direct numerical integration of the equations of motion using

a step-by-step method as follows: for each frequency of excita-

tion ω(rad/s), the ﬁrst ﬁve periods of the temporal response are

determined through the Newmark method. The results showed

that the calculations of amplitudes for the ﬁrst ﬁve periods were

largely sufﬁcient. It was seen that using a constant time step was

not helpful, and to ensure convergence it would be more judi-

cious to choose a time step based on the frequency. A time step

equal to t=15.10−4(2/ω) was found to be optimal, and us-

ing a ﬁner time step did not result in any improvement. The am-

plitudes of the ﬂow rate, unbalance response, and transmitted

forces were, however, only reproduced for the last period in or-

der to avoid the transitory response. The hydrostatic forces and

ﬂow rate were determined at each step. The hydrostatic forces,

calculated with the nonlinear method, were determined by the

application of the boundary conditions (Eq. [7]) and integration

of the pressure ﬁeld determined by resolution of the Reynolds

equation (Eq. [3]), by the ﬁnite difference method, and solved by

successive overrelaxation method. Note that to deal with oil ﬁlm

cavitations, negative pressure was set to zero during the interac-

tive process. However, for the linear model, the hydrostatic forces

are determined from Eq. [8], which are based on the dynamic co-

efﬁcients. The ﬁlm thickness h0was determined by resolving the

ﬂow continuity equation from a given pressure ratio β0and us-

ing an iterative secant method, and the pressure was determined

by resolving the ﬂow continuity (Eq. [8]) by applying an iterative

secant method.

Computation of the ﬁlm thickness and recess pressure was

performed using an iterative secant method after bounding the

roots. The convergence tolerances of these computations were

deﬁned as follows:

rPressure: 0.01(( Pr

i,j−Pr−1

i,j

100Pri )max =0.01)

rFilm thickness: 10−6

rRecess pressure: 10−6,

Fig. 9—Variation in ﬁlm thickness with bearing pad dimension ratios

for various capillary diameters (β0=0.67) (color ﬁgure available

online).

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724 A. BOUZIDANE AND M. THOMAS

4 8 12 16 20

Rotational Speed (krpm)

0.00

0.25

0.50

0.75

1.00

Dimensionless Vibration Amplitude

A/B=1

A/B=3

A/B=6

48121620

Rotational Speed (krpm)

4.25

4.38

4.50

4.63

4.75

Flow Rate (l/min)

A/B=1

A/B=3

A/B=6

4 8 12 16 20

Rotational Speed (krpm)

0.00

0.25

0.50

0.75

1.00

Transmitted Force (kN)

A/B = 1

A/B = 3

A/B = 6

Fig. 10—Inﬂuence of bearing pad dimension ratios on vibratory response, transmitted force, and ﬂow rate (color ﬁgure available online).

where Pr

i,jrepresents the computed pressure at each mesh point

(i,j)andris the iteration number of the computation.

NUMERICAL RESULTS

The dynamic behavior of the rigid rotor supported by a three-

pad HSFD was investigated. Both linear and nonlinear solving

procedures were used as described below:

rIn the ﬁrst step, a comparison of linear and nonlinear results

for computing the vibratory responses and transmitted forces

for small vibration around the equilibrium position (ε=0.05)

is presented. The values of the dynamic coefﬁcients (Kpi =

7,336,697 N/m; Cpi =4,419.835 N.s/m) were determined nu-

merically (numerical differentiation method) using small per-

turbations of the shaft while located at its equilibrium position

(Bouzidane, et al. (11)).

rIn the second step, the effects of the hydrostatic bearing di-

mension ratios, capillary diameters, and rotational speed on

the ﬂow rate, unbalance response, and transmitted forces were

investigated using a nonlinear method.

Table 1 shows the numerical parameters applied for the com-

putation.

Comparison between Linear and Nonlinear Models

To check the validity of numerical analysis and stability of

the rigid rotor–HJB system, the results of nonlinear models

were computed and compared with those obtained by the linear

model for small vibrations around the static equilibrium position

with ε=0.05 (ε=e/h0) (the unbalance eccentricity). Figure 7

shows a comparison between the linear and nonlinear results by

computing the dimensionless vibration amplitudes and transmit-

ted forces. The dimensionless vibratory amplitude is determined

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Analysis of a Rigid Rotors 725

4 8 12 16 20

Rotational Speed (krpm)

0.00

0.25

0.50

0.75

1.00

Dimensionless Vibration Amplitude

dc = 1.2 mm

dc = 1.0 mm

dc = 0.8 mm

4 8 12 16 20

Rotational Speed

(

krpm

)

0.00

0.25

0.50

0.75

1.00

Transmitted Force (kN)

dc = 1.2 mm

dc = 1.0 mm

dc = 0.8 mm

48121620

Rotational Speed

(

krpm

)

0.0

1.0

2.0

3.0

4.0

5.0

Flow Rate (l/min)

dc= 1.2 mm

dc= 1.0 mm

dc= 0.8 mm

Fig. 11—Inﬂuence of capillary diameter on vibratory response, transmitted force, and ﬂow rate (color ﬁgure available online).

as follows:

˜

A=(|x|/h0)2+(|y|/h0)2.[23]

where h0is the ﬁlm thickness and

|Ft|=|Fx|2+|Fy|2.[24]

It can be seen that the results obtained using the nonlinear

method were in very good agreement with those obtained by

the linear methods when small vibrations were considered be-

cause they were almost identical to those predicted by the linear

method. It should be noted that the results obtained showed that

the dynamic behavior of the rotor was always stable (there are no

oscillations in the value of transmitted forces).

Comparisons between Three-Pad HSFD

and Four-Pad HSFD

Figure 8 shows a comparison of the results for a three-pad

HSFD and a four-pad HSFD (Adams and Zahloul (10); Bouzi-

dane, et al. (12)) for computing the dimensionless vibration am-

plitudes, transmitted forces, and ﬂow rates for ε=0.25 (dc=

1.2 mm, A/B=6). These graphs show that the disadvantages

of a four-pad HSFD relative to a three-pad HSFD are as fol-

lows: the large transmitted force amplitudes, especially for high

speeds (N (krpm) <11) due to the high damping factor and

stiffness and the high ﬂow rates, because four thrust bearings

are required instead of three thrust bearings for a three-pad

HSFD. However, for low speeds (N (krpm) <11), the four-pad

HSFD had better dynamic characteristics and stability than the

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726 A. BOUZIDANE AND M. THOMAS

three-pad HSFD due to its higher stiffness, damping, and zero

cross-coupling terms.

Effect of Bearing Pad Dimension Ratio and Capillary

Diameter on Film Thickness

Once the accuracy of the nonlinear approach has been estab-

lished, it may be used for all the numerical simulations. Figure 9

shows the variation in ﬁlm thickness with bearing pad dimension

ratio for various capillary diameters (dc). These results show that

the ﬁlm thickness decreased when the bearing pad dimension ra-

tio increased and the capillary diameter decreased. The reason

for this decrease can be explained as follows:

rIncreasing the bearing pad dimension ratio can decrease the

pressure ratio to values lower than 0.67; in order to increase

the pressure ratio to 0.67, the ﬁlm thickness must decrease.

rWhen increasing the diameter of capillary, the pressure ratio

takes values higher than 0.67; in order to reduce the pressure

ratio to 0.67, the ﬁlm thickness must increase.

The ﬁlm thickness (h0) is determined by resolving the ﬂow

continuity equation from a given pressure ratio (β0=0.67) at the

center position of HSFDs. It should be noted that the pressure

ratio was chosen as 0.67 to obtain a maximum stiffness value of

the HSFD (Bouzidane and Thomas (11)).

Effect of the Bearing Pad Dimension Ratios

and Capillary Diameter on Flow Rate, Rotor Response,

and Transmitted Force

Figures 10 to 11 show the effect of bearing pad dimension ratio

(A/B), capillary diameter (dc), and rotational speed on the ﬂow

rate, unbalance response, and transmitted force of an HSFD as

computed using the nonlinear method.

The inﬂuence of the bearing pad dimension ratio and rota-

tional speed on the dimensionless vibratory responses, transmit-

ted forces, and ﬂow rates for a capillary diameter of 1.2 mm and

unbalance eccentricity of 0.25 are presented in Fig. 10. This ﬁgure

shows that the increase in the bearing pad dimension ratio from

1 to 6 decreased the ﬂow rate, dimensionless vibration responses,

and amplitude of transmitted forces, especially close to the criti-

cal speed. Two phenomena were observed: an increase in critical

speed and an increase in damping. The increase in damping of the

HSFD was due to an increased pressure in the recesses because

the ﬁlm thickness decreased when the bearing pad dimension ra-

tio increased (Fig. 9). The critical speed shifted to higher values

when the bearing pad dimension ratio increased as the ﬁlm thick-

ness decreased due to an increase in stiffness. As the rotational

speed became further away from the critical speed, variations in

bearing pad dimension ratio had no signiﬁcant effect on the am-

plitude of the ﬂow rate and dimensionless vibratory responses.

Figure 11 shows the inﬂuence of the capillary diameter and

rotational speed on the dimensionless vibratory responses, trans-

mitted forces, and ﬂow rate amplitudes for a bearing pad dimen-

sion ratio A/Bof 6, with an eccentricity unbalance εset to 0.25 and

the pressure ratio (β0) set to 0.67. This ﬁgure shows that the in-

crease in capillary diameter from 0.8 to 1.2 mm led to a signiﬁcant

increase in the ﬂow rate regardless of the speed and an increase

in vibration responses and transmitted forces in the vicinity of the

critical speed due to decreased damping. Consequently, the trans-

mitted force decreased at speeds higher than the critical speeds.

The decrease in the critical speed was due to a decrease in stiff-

ness. This decreased damping and stiffness can be explained by

the increase in the ﬁlm thickness (ho), because the pressure ratio

was set to 0.67 (Fig. 9). When increasing the squeeze ﬁlm thick-

ness in hydrostatic bearing due to the increase in the diameter of

the capillary, the pressure inside the recesses decreases and the

damping hydrostatic bearing decreases (Bouzidane and Thomas

(11)). This causes large vibrations and transmitted force ampli-

tudes around the critical speed.

CONCLUSION

In this article, a new type of three-pad HSFD was designed

and proposed to control the rotor vibrations and bearing trans-

mitted forces caused by rotor imbalance. Nonlinear modeling was

performed in order to investigate the effects of hydrostatic bear-

ing dimension ratios, capillary diameters, and rotational speed

on the ﬂow rates, unbalance responses, and bearing transmitted

forces of a rigid shaft supported by three-pad HSFDs.

The conclusions may be summarized as follows:

rIn comparison to the four-pad HSFD, the advantages of us-

ing a three-pad HSFD include reducing the cost due to the

need of a feeding system and the volumetric ﬂow rate used

in hydrostatic lubrication because it requires only three thrust

bearings.

rWhen the bearing pad dimension ratio increased, the dynamic

characteristics (damping and stiffness) increased because the

ﬁlm thickness decreased. Due to an increase in damping, the

amplitude of vibratory responses and transmitted forces de-

creased. The critical speed increased due to an increase in

stiffness. The opposite behavior was observed at speeds higher

than the critical speeds.

rAn increase in bearing pad dimension ratio led to a decrease

in the ﬂow rate in the vicinity of its critical speed due to an

increase in the damping factor.

rAn increase in capillary diameter led to an increase in the ﬁlm

thickness. Therefore, the ﬂow rate increased regardless of the

speed. When increasing the ﬁlm thickness in hydrostatic bear-

ings, the pressure inside the recesses decreases and the damp-

ing hydrostatic bearing decreases. This causes large vibrations

and transmitted force amplitudes around the critical speed.

Furthermore, the bearing transmitted forces decreased at very

high speed due to a decrease in damping.

rThis new hydrostatic journal is suitable for controlling rotor

vibration and bearing transmitted forces caused by rotor im-

balance, especially at high speeds. In light of the important

conclusions drawn from the present study, an experimental

veriﬁcation of three-pad HSFDs is recommended.

ACKNOWLEDGEMENTS

The authors thank Dr. Azzedine Dadouche, National Re-

search Council/Institute for Aerospace Research, for his helpful

advice on various technical issues examined in this article.

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Analysis of a Rigid Rotors 727

REFERENCES

(1) Allaire, P. E. (1979), “Design of Journal Bearings for High-Speed Rotating

Machinery,” In Fundamentals of the Design of Fluid Film Bearings, Rhode,

S. M., Maday, C. J., and Allaire, P. E. (Eds.), pp 45–83, American Society

of Mechanical Engineers: New York.

(2) San Andres, L. A. and De Santiago, O. (2004), “Forced Response

of a Squeeze Film Damper and Identiﬁcation of Force Coefﬁcients

from Large Orbital Motions,” Transactions of the ASME,126, pp 292–

300.

(3) Burrows, C. R., Sahinkaya, M. N., and Turkaya, O. S. (1983), “An Adap-

tive Squeeze-Film Bearing,” ASME Paper No. 83-Lub-23.

(4) Mu, C., Darling, J., and Burrows, C. R. (1991), “An Appraisal of a Pro-

posed Active Squeeze Film Damper,” Journal of Tribology,113(4), pp

750–754.

(5) San Andres, L. A. (1992), “Analysis of Hydrostatic Journal Bearings with

End Seals,” Journal of Tribology,114(7), pp 802–811.

(6) Braun, M. J., Zhou, Y. M., and Choy, F. K. (1994), “Transient Flow Pat-

terns and Pressures Characteristics in a Hydrostatic Pocket,” Journal of

Tribology,116, pp 139–146.

(7) Braun, M. J., Choy, F. K., and Zhu, N. (1995), “Flow Patterns and Dynamic

Characteristics of a Lightly Loaded Hydrostatic Pocket of Variable Aspect

Ratio and Supply Jet Strength,” Tribology Transactions,38(1), pp 128–

136.

(8) Hathout, J. P., El-Shafei, A., and Youssef, R. (1997), “Active Control of

Multi-Mode-Rotor–Bearing Systems Using HSFDs,” Journal of Tribol-

ogy,119(7), pp 49–56.

(9) Sawicki, J. T., Capaldi, R. J., and Adams, M. L. (1997), “Experimental and

Theoretical Rotordynamic Characteristics of a Hybrid Journal Bearing,”

Journal of Tribology,119, pp 132–141.

(10) Adams, M. L. and Zahloul, H. (1987), “Attenuation of Rotor Vibration

Using Controlled-Pressure Hydrostatic Squeeze Film Dampers,” Paper

presented at the Eleventh Biennial ASME Vibrations Conference, Boston,

MA, September.

(11) Bouzidane, A. and Thomas, M. (2007), “Equivalent Stiffness and Damping

Investigation of a Hydrostatic Journal Bearing,” Tribology Transactions,

50(2), pp 257–267.

(12) Bouzidane, A., Thomas, M., and Lakis, A. (2008), “Non Linear Dy-

namic Behaviour of a Rigid Rotor Supported by Hydrostatic Squeeze Film

Dampers,” Tribology Transactions,130(4), pp 041102-1-041102-9.

(13) Shen, G., Xiao, Z., Zhang, W., and Zheng, T. (2006) “Nonlinear Behavior

Analysis of a Rotor Supported on Fluid-Film Bearings,” Journal of Vibra-

tion and Acoustics,128, pp 35–40.

(14) Qin, W., Zhang, J., and Ren, X. (2009), “Response and Bifurcation of Ro-

tor with Squeeze Film Damper on Elastic Support,” Chaos, Solitons &

Fractals,39, pp 188–195.

(15) Chang-Jian, C. W., Terng Yau, H., and Lin Chen, J. (2010), “Nonlinear Dy-

namic Analysis of a Hybrid Squeeze-Film Damper-Mounted Rigid Rotor

Lubricated with Couple Stress Fluid and Active Control,” Applied Mathe-

matical Modelling,34(9), pp 2493–2507.

(16) Rattan, S. S., Mehta, N. P., and Bhushan, G. (2011), “Effect of Load Orien-

tation on the Stability of a Three-Lobe Pressure Dam Bearing with Rigid

and Flexible Rotors,” Journal of Engineering Technology,1, pp 10–15.

Downloaded by [ÉTS - École de technologie supérieure] at 12:42 14 June 2013