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Nonlinear Dynamic Analysis of a Rigid Rotor Supported by a Three-Pad Hydrostatic Squeeze Film Dampers

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The aim of this paper is to study the non linear dynamic behaviour of rigid rotors, supported by a new hydrostatic squeeze film damper (HSFD). A non linear model of a hydrostatic squeeze film damper has been presented. The results are compared with those obtained from a linear approach which is only valid for small vibrations around the equilibrium position. Comparing with the four-pad HSFD, the advantage of using a three-pad HSFD consists of reducing the cost coming from the need of a feeding system, and the quantity of volumetric flow rate used in hydrostatic lubrication. In this study, the effects of the pad dimensions ratios, capillary diameters, and rotational speed on the flow rate, unbalance responses, and the transmitted forces are investigated using a nonlinear method and the results are analysed and discussed. In fact, the results obtained show that this type of hydrostatic squeeze film damper provides to hydrostatic designers a new bearing configuration suitable to control rotor vibrations and bearing transmitted forces for high speed.
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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
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)):
xiPi(xi,zi,t)
xi+
ziPi(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.
r0xiAand 0 ziB;
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=h0x
1
h2=h0x
2
h3=h0x
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 xiA;zi=0,B;t)=0; Pi(xi=0,A;0 ziB;t)=0
Pi(x1xix2;z1ziz2;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
(PSPri)[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
(yihi)yi;uzi =1
2μ
Pi
zi
(yihi)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=−(FP2FP3)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×B135.105m2
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
yKxx 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=15.104(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,jPr1
i,j
100Pri )max =0.01)
rFilm thickness: 106
rRecess pressure: 106,
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
Downloaded by [ÉTS - École de technologie supérieure] at 12:42 14 June 2013
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
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A fast and accurate model to calculate the fluid-film forces of a fluid-film bearing with the Reynolds boundary condition is presented in the paper by using the free boundary theory and the variational method. The model is applied to the nonlinear dynamical behavior analysis of a rigid rotor in the elliptical bearing support. Both balanced and unbalanced rotors are taken into consideration. Numerical simulations show that the balanced rotor undergoes a supercritical Hopf bifurcation as the rotor spin speed increases. The investigation of the unbalanced rotor indicates that the motion can be a synchronous motion, subharmonic motion, quasi-period motion, or chaotic motion at different rotor spin speeds. These nonlinear phenomena are investigated in detail. Poincaré maps, bifurcation diagram and frequency spectra are utilized as diagnostic tools.
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This paper investigates the nonlinear response and bifurcation of rotor with Squeezed Film Damper (SFD) supported on elastic foundation. The motion equations are derived. To analyze the bifurcation of nonlinear response of SFD rotor, the Floquet Multipliers is obtained by solving the perturbation equations with numerical method. For computing Floquet Multipliers, a novel method is presented in this paper, which can begin integration at the stable solution. Simulation results are given in two figures. One figure, which consists of eight subfigures, gives the effect of rotating speed on the response of SFD damper supported on elastic foundation: with increasing rotating speed, the nonlinear response evolves from quasi-period to period, then jumps between different periods, and finally returns to quasi-period; the corresponding bifurcations are saddle-node bifurcation and secondary Hopf bifurcation. The second figure, which consists of six subfigures, shows that: the support stiffness has large influence on the response of bearings and film force in SFD; large support stiffness can lead to oil whirl in SFD.
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This paper describes an experimental and theoretical investigation of a four-pecker, oil-fed, orifice-compensated hydrostatic bearing including the hybrid effects of journal rotation. The test apparatus incorporates a double-spool-shaft spindle which permits independent control over the journal spin speed and the frequency of an adjustable magnitude circular orbit, for both forward and backward whirling. This configuration yields data that enables determination of the full linear anisotropic rotordynamic model. The dynamic force measurements were made simultaneously with two independent systems, one with piezoelectric load cells and the other with strain gage load cells. Theoretical predictions are made for the same configuration and operating conditions as the test matrix using a finite-difference solver of Reynolds lubrication equation. The computational results agree well with test results, theoretical predictions of stiffness and damping coefficients are typically within thirty percent of the experimental results.