Mihai Arghir
Manh-Hung Nguyen
Institut Pprime,
CNRS UPR3346,
Université de Poitiers,
86962 Futuroscope Chasseneuil, France
David Tonon
SNECMA Space Engine Division,
27208 Vernon, France
Jérôme Dehouve
Centre National d’Etudes Spatiales,
91023 Courcouronnes Evry, France
Analytic Modeling of Floating
Ring Annular Seals
In order to avoid contact between the vibrating rotor and the stator, annular seals are
designed with a relatively large radial clearance (100 lm) and, therefore, have an important leakage. The floating ring annular seal is able to reduce the leakage flow rate by using
a much lower clearance. The seal is designed as a ring floating on the rotor in order to
accommodate its vibrations. The pressure difference between the upstream and the downstream chambers is pressing the nose of the floating ring (secondary seal) against the stator. The forces acting on the floating ring are the resultant of the hydrodynamic pressure
field inside the primary seal, the friction forces in the secondary seal, and the inertia forces
resulting from the non-negligible mass of the ring. For proper working conditions, the ring
of the annular seal must be able to follow the vibration of the rotor without any damage.
Under the effect of the unsteady hydrodynamic pressure field (engendered by the vibration
of the rotor), of the friction force, and of the inertia force, the ring will describe a periodic,
a quasi-periodic, or a chaotic motion. Damage can come from heating due to friction in the
secondary seal or from repeated impacts between the rotor and the ring. The present work
presents an analytic model able to take into account only the synchronous periodic whirl
motion of the floating ring. [DOI: 10.1115/1.4004728]
Introduction
The floating ring annular seal is derived from the classical
dynamic annular seal. These seals are used in high-pressure turbomachinery and in order to avoid unwanted contact between the
vibrating rotor and the stator. They are designed with a relatively
large clearance (100 lm) and have therefore a rather large leakage. The floating ring annular seal is able to reduce the leakage
flow rate by using a much lower clearance. As described by Müller and Nau [1] the seal is designed as a ring floating on the rotor
in order to accommodate its vibrations. A schematic design is presented in Fig. 1. The seal normally separates the upstream chamber, where the pressure Pupstream is high, from a downstream
chamber, where the pressure Pdownstream is low. A leakage is present in what is called the main seal, and the forces engendered in
this seal are of a hydrodynamic/hydrostatic type. A secondary
sealing path occurs between the nose of the ring and the stator.
The pressure difference between the upstream and the downstream chambers are pressing the nose of the floating ring against
its casing (stator). It is supposed that a mixed friction regime
occurs in the secondary seal.
The forces acting on the floating ring are then the result of the
hydrodynamic pressure field inside the seal, the friction force in
the secondary seal, and (for unsteady working conditions) the
inertia force resulting from the non-negligible mass of the ring. It
is supposed that the friction torque is low and cannot entrain the
ring. This is the case of annular seals lubricated with compressible
(gaseous) fluids. However, floating ring annular seals working
with high viscous fluids (generally incompressible) are provided
with one or more pins on the outer circumference for preventing
rotation. General analyses of floating ring seals are presented in
Refs. [2–4]. Ha et al. [2] present an analysis based on the assumption that the floating ring seal is blocked in an eccentric position
and cannot follow the vibrations of the rotor. Shapiro [3] and
Kirk [4] present nonlinear analysis of the floating ring seal. Forces
in the primary seal are described by dynamic coefficients, and
forces in the secondary seal are estimated from a Coulomb model.
Contributed by the International Gas Turbine Institute (IGTI) of ASME for publication in the JOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER. Manuscript
received May 29, 2011; final manuscript received June 3, 2011; published online
March 1, 2012. Editor: Dilip R. Ballal.
Both references present the trajectories of the floating ring following the vibrations of the rotor.
In order to work properly, the floating ring must be under static
as well as under dynamic equilibrium. The static equilibrium is
described by the balance between the hydrostatic moment generated by an eccentric rotor and the contact moment stemming from
the pressure difference between the upstream and the downstream
chamber. The first moment is tilting the ring and tries to open it
while the second one has a contrary effect. Under static equilibrium and in order to ensure sealing, tilt must be avoided, and the
ring must be pressed against the stator. This is generally the case
for carbon rings when the annular seal is not too long. However,
for proper working conditions the floating ring must be able to
follow the vibration of the rotor without any damage. Under the
effect of the unsteady hydrodynamic pressure field (engendered
by the vibration of the rotor), of the friction force, and of the
inertia force, the ring will describe a periodic, a quasi-periodic, or
a chaotic motion. Damage can come from heating due to friction
in the secondary seal or from repeated impacts between the rotor
and the ring. The present work presents an analytic model able to
take into account only the synchronous periodic whirl motion of
the floating ring.
Theoretical Analysis of the Whirling Ring
It is considered that the floating ring has a planar movement in
plane OXY (Fig. 2). Its nose is permanently in contact with the sta! !
tor casing, and the tilting of the floating ring around OX or OY is
excluded. The equations of motion of the floating ring are
(
X€B
M
Y€B
)
(
~R=B þ
¼F
Ff X
)
Ff Y
(1)
The gravity forces are neglected. This is a reasonable assumptions
if, for example, one considers a ring of M ¼ 0.07 kg and an acceleration of 9 g. The resulting inertia force is then only 6.2 N and
can be neglected compared to the other forces. The hydrodynamic
and the friction forces are detailed in the following sections.
Hydrodynamic Forces. It is supposed that the axes of the ring
and of the rotor are perfectly aligned and forces are stemming
Journal of Engineering for Gas Turbines and Power
C 2012 by ASME
Copyright V
MAY 2012, Vol. 134 / 052507-1
Downloaded From: http://gasturbinespower.asmedigitalcollection.asme.org/ on 01/27/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
consider the steps of the approach used for calculating the
dynamic coefficients [5]:
— The rotor is in a steady state given by arbitrary X0, Y0, and
X_ 0 ¼ Y_0 ¼ 0. The thin film flow equations (either Reynolds
or the “bulk flow” system of equations) are then solved, and
this represents the steady (zero order) solution,
FX;Y X0 ; Y0 ; X_ 0 ¼ 0; Y_0 ¼ 0Þ.
— It is next supposed that the rotor has small amplitude
vibrations around this position, i.e., DX ¼ X X0 and
DY ¼ Y Y0 , where jDXj jX0 j and jDY j jY0 j. No
assumption is made on the magnitude of the perturbation
_
velocities, DX_ ¼ X_ and DY_ ¼ Y.
Fig. 1
Schematic design of a floating ring seal
only from eccentricity and squeeze effects (no misalignment).
This assumption is not too restrictive because floating ring annular
seals generally have a short length of an order of magnitude
smaller than the radius. Under these assumptions the variation of
hydrodynamic forces in the (main) annular seal is
8
@FX
@FX
@FX _ @FX _
>
>
>
< dFX ¼ @X dX þ @Y dY þ @ X_ dX þ @ Y_ d Y
>
>
@F
@F
@F
@F
>
: dFY ¼ Y dX þ Y dY þ Y dX_ þ Y dY_
@X
@Y
@ X_
@ Y_
It results from this last point that Eq. (3), resulting from the
integration of Eq. (2), must be replaced by
FX0
FX
¼
FY0 ðX0 ;Y0 ;0;0Þ
FY ðX;Y;X;_ YÞ_
KXX KXY
X X0
KYX KYY ðX0 ;Y0 ;0;0Þ Y Y0
" #
CXX CXY
X_
(4)
CYX CYY ðX ;Y ;0;0Þ Y_
0
(2)
By integrating these relation over a small time step one obtains
FX0
KXX KXY X X0
FX
¼
FY
FY0
KYX KYY
Y Y0
#
"
CXX CXY
X_ X_ 0
(3)
CYX CYY
Y_ Y_0
where partial derivatives were replaced by dynamic coefficients
Kij ¼ @Fi =@Xj and Cij ¼ @Fi =@ X_ j (i,j ¼ X,Y). This relation
enables the estimation of fluid force variations when dynamic
coefficients, displacements, and velocities are known. The use of
this relation for integrating the equations of motion Eq. (1) can be
subject to several interpretations. It is, therefore, necessary to
0
This equation may be written under the following form for underlining the contribution of the perturbation velocities:
FX
FX
¼
FY ðX;Y;X;_ YÞ_
FY ðX;Y;0;0Þ
" #
CXX CXY
X_
(5)
CYX CYY ðX ;Y ;0;0Þ Y_
0
0
The validity of this interpretation is verified by using the Jeffcott
rotor and the short bearing model (L/D < 0.25). Analytic relations
are thus available for the nonlinear unsteady hydrodynamic forces
and for the dynamic coefficients stemming from these forces.
The analyzed system is then a short rigid rotor of mass 2M
located at its midlength and supported by two identical short bearings. Each bearing supports a load Wg ¼ Mg. It is supposed that
the rotor is free of any imbalance. The equations of motion of the
rotor are then
M
) (
)
( ) (
FX ðtÞ
Wg
X€
þ
¼
0
FY ðtÞ
Y€
(6)
_ Y_ are the nonlinear unsteady
where FX;Y ðtÞ ¼ FX;Y X; Y; X;
hydrodynamic forces stemming from the integration of the Reynolds equation. These forces and the corresponding dynamic coefficients are given in most lubrication textbooks and are reproduced
in the Appendix. The mass of the rotor is M ¼ 1 kg, and the rotation speed is X ¼ 1000 rad/s. The geometric characteristics of the
bearings are R ¼ 37 mm, L ¼ 10 mm, and Cjeu ¼ 50 lm. They are
lubricated with water l ¼ 103 kg/m/s under isothermal conditions. The rotor is initially centered (X0 ¼ Y0 ¼ X_ 0 ¼ Y_0 ¼ 0),
and under the effect of Wg will find a static equilibrium position.
Equation (6) is integrated by using a second order Euler method
where the estimation of hydrodynamic forces between two time
steps is subject to different interpretations:
Fig. 2
Fixed coordinate system and forces on the floating ring
052507-2 / Vol. 134, MAY 2012
(1) By using relation Eqs. (A1) and (A2) for the unsteady nonlinear forces. The results are depicted by a black trajectory
on Fig. 3. This trajectory is the correct result and is used for
verifying the other approaches.
(2) At each time step the hydrodynamic forces are calculated
by using Eq. (5). This approach supposes two steps. In the
first step FX;Y ðX; Y; 0; 0Þ are estimated by considering only
Transactions of the ASME
Downloaded From: http://gasturbinespower.asmedigitalcollection.asme.org/ on 01/27/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Fig. 3. It superposes the black and blue trajectories thus validating the approach.
(4) The « steady » part of the hydrodynamic forces is estimated
as in Eq. (9) of the previous approach, i.e., by successively
adding the effect of the relative displacements of the rotor
center and of the dynamic coefficients Ki,j. The unsteady
effects are next added by taking into account the variation
of the velocity of the rotor center as it might be suggested
by Eq. (3).
FX
FY
¼
_ YÞ
_
ðX; Y; X;
FX
FY ðX; Y; 0; 0Þ
"
#
CXX CXY
X_ X_ 0
CYX CYY ðX0 ; Y0 ; 0; 0Þ Y_ Y_0
(10)
The results obtained by using this approach are depicted by
the green trajectory in Fig. 3 and show that this interpretation is incorrect.
(5) In this approach the hydrodynamic forces are calculated in
a single step as might suggest Eq. (3).The results depicted
by a violet trajectory in Fig. 3 show that this approach is
incorrect.
Fig. 3 Trajectory of the rotor center (Jeffcot rotor supported by
short bearings)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
the rotor eccentricity e ¼ e2X þ e2Y . They correspond to
the steady part of the nonlinear forces given by Eq. (A1)
and (A2) and estimated by imposing e_ ¼ /_ ¼ 0.
FX
FY
cos / sin / Fr
sin / cos /
Ft ðr; t; 0; 0Þ
|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
¼
ðX;Y;0;0Þ
(7)
½/
The unsteady effects are then added by using the damping
_
coefficients and the velocities of the rotor center X_ and Y.
"
FX
FY
#
"
¼
_ YÞ
_
ðX;Y;X;
FX
FY
#
"
ðX;Y;0;0Þ
CXX
CXY
CYX
CYY
#
ðX0 ;Y0 ;0;0Þ
" #
X_
Y_
(8)
The result obtained by using this estimation of the hydrodynamic forces is the blue trajectory in Fig. 3. The coincidence with the black trajectory proves the validity of the
approach.
(3) Hydrodynamic forces are calculated from relation Eq. (4)
in two steps. In the first step FX;Y ðX; Y; 0; 0Þ are calculated
by using the definition of stiffness coefficients.
FX
FX
¼
FY ðX;Y;0;0Þ
FY ðX0 ;Y0 ;0;0Þ
X X0
KXX KXY
(9)
KYX KYY ðX0 ;Y0 ;0;0Þ Y Y0
The second step is identical to Eq. (8) of the previous
approach. It is to be underlined that only the “steady” part
of the hydrodynamic forces is calculated by adding the displacement effects of the rotor center between two time steps.
The unsteady effects due to the absolute velocities of the
rotor center are added at each time step. The result stemming from this approach is depicted by the red trajectory in
Journal of Engineering for Gas Turbines and Power
In conclusion, the unsteady hydrodynamic forces can be estimated by using dynamic coefficients if one takes into account the
conditions and the assumptions used for defining and calculating
them. The estimation is made in two steps. The first step is the
evaluation of the steady part of the hydrodynamic forces (either by
integrating the relative displacements of the rotor center X X0
and Y Y0 and the stiffness coefficients or by direct estimation of
steady forces for an eccentricity given by X, Y). The second step is
to take into account the unsteady effects by using the damping
_
coefficients and the absolute velocities of the rotor X_ Y.
Friction Forces. Tribological Aspects. The tribological nature
of the contact inside the secondary seal must be taken into account
for correctly estimating the real value of the friction force.
This contact represents the normally closed secondary sealing
path shown in Fig. 1. Nevertheless, due to roughness effects, a
secondary leakage flow will lead to mixed lubrication conditions
(elastohydrodynamic). This regime is characterized by an average
distance h between the two surfaces of the order 1 < h/r < 3 [6]
where r is the combined standard deviation of the two surfaces,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
(11)
r ¼ r21 þ r22
Following Fig. 2, the normal load on the nose of the floating ring is
Fz ¼ Pupstream p R23 R21 Pdownstream p R22 R21
(12)
Part of the normal load is supported by the fluid pressure and the
rest by the contact pressure resulting from the elastic deformation
of asperities.
Fz ¼ Fz; fluid þ Fz; asp
(13)
The normal force supported by the fluid is estimated from a 1D thin
film model of the pressure driven flow in the secondary seal [7].
The flow path is supposed to be smooth and of average height h.
The normal force Fz,asp is estimated by considering the total
number of elastic contacts between asperities [8]. The contact
probability is estimated from a normal roughness distribution,
Probð1 > hÞ ¼
ð1
h
uð1Þd1 ¼
ðc
35 2
ðc 12 Þ3 d1
7
32c
h
(14)
MAY 2012, Vol. 134 / 052507-3
Downloaded From: http://gasturbinespower.asmedigitalcollection.asme.org/ on 01/27/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
8
>
< 35 ðc2 12 Þ3 if c < 1 < c
uð1Þ ¼ 32c7
>
:
0
otherwise
(15)
c ¼ 3r
(16)
It is next supposed that the roughness asperities have spherical
shapes and are subject only to elastic deformations. The contact
force corresponding to one asperity Fn,e is estimated from Hertz’s
theory,
4
Fn;e ¼ E0 b1=2 d3=2
3
(17)
1
1 12 1 22
¼
þ
0
E
E1
E2
(18)
The assumed situation takes into account only a periodic
motion of the floating ring and discards any quasi-periodic or limit
cycle response. Of course, the floating ring can be blocked by a
friction force larger than the fluid force, but this situation can
appear only if the amplitude of the whirl motion of the rotor is
less than the radial clearance. Otherwise, either the ring starts to
follow the rotor or an impact occurs.
The coordinate system Oxy depicted in Fig. 4 is rotating with
!
constant X and the axis Ox is always oriented towards the center
of the floating ring B. Both the rotor and the floating ring are
eB , respectively.
whirling around O but with different radii, ~
eR and ~
!
The vector ~
eBR ¼ BR rotates with the coordinate system Oxy, and
!
the angle c relative to Ox is constant. In the rotating coordinate
system, the position of the rotor is described by ~
eR or by two constant values eR and uR .
The equations of motion of the floating ring are
M
The total contact force [7] is
4
Fz;asp ¼ NE0 b1=2
3
ðc
35 2
ðc 12 Þ3 ð1 hÞ3=2 d1
7
h 32c
where N ¼ ðg1 þ g2 ÞSc is the total number of asperities and
Sc ¼ p R23 R22
(19)
(20)
The same assumption as is made for Eq. (13) is made for the friction force:
Ff ¼ Ff; fluid þ Ff; asp
Ff; fluid
eB X
Sc
¼l
h
(22)
¼
FR=B;x
FR=B;y
þ
Ff ;x
Ff ;y
xR; B ¼ eR; B cos uR; B
(25a)
x_ R; B ¼ e_ R; B cos uR; B eR; B u_ R; B sin uR; B
(25b)
x€R; B ¼ e€R; B cos uR; B 2e_R; B u_ R; B sin uR; B
€ R; B sin uR; B eR; B u_ 2R; B cos uR; B
eR; B u
(25c)
yR; B ¼ eR; B sin uR; B
(25d)
y_ R; B ¼ e_ R; B sin uR; B þ eR; B u_ R; B cos uR; B
(25e)
y€R; B ¼ e€R; B sin uR; B þ 2e_ R; B u_ R; B cos uR; B
€R; B cos uR; B eR; B u_ 2R; B sin uR; B
þ eR; B u
Ff; asp ¼ fFz; asp
(24)
The displacements, the velocities, and the accelerations of the
rotor and of the floating ring are
(21)
with
x€B
y€B
(25f)
(23)
where f is the dry friction coefficient between the nose of the carbon floating ring and its steel casing (stator). The main part of the
friction force will still be due to the contact forces between roughness asperities, but this is a reduced percentage because an important part of the normal load is supported by the fluid pressures
with insignificant contribution to the friction force. As known,
mixed lubrication conditions lead to much lower friction coefficients than the values for dry friction. The exact value of the friction force depends on the roughness distribution, on the physical
properties of the two contacting solids, and on the sealed fluid.
Analytical Solution of the Floating Ring Whirl. Under simplifying assumptions the dynamic response of the floating ring
can be analytically estimated. It is supposed that the rotor
describes a circular synchronous orbit around O with an amplitude
that can be larger than the radial clearance. The main simplifying
assumption leading to an analytic model is the hypothesis that the
floating ring follows synchronously the whirl motion of the rotor.
This assumption leads to the following consequences:
— The fluid force between the floating ring and the rotor is
constant.
— The friction force between the nose of the floating ring and
the stator is constant, and stick-slip phenomena are excluded.
— The floating ring can have only a synchronous whirl motion.
The fluid torque is neglected, and so is any rotation of the
floating ring around its own center.
— Impacts between the rotor and the floating ring are out of
the field of the present analysis.
052507-4 / Vol. 134, MAY 2012
Fig. 4
Whirling coordinate system
Transactions of the ASME
Downloaded From: http://gasturbinespower.asmedigitalcollection.asme.org/ on 01/27/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Following the formulated simplifying assumptions (both the rotor
and the floating ring describe centered uniform whirls):
uR 6¼ 0;
u_ R ¼ X;
€ R ¼ 0;
u
e_R ¼ e€R ¼ 0
(26)
uB ¼ 0;
u_ B ¼ X;
€ B ¼ 0;
u
e_B ¼ e€B ¼ 0
(27)
» nonlinear forces are calculated in two steps by using the
approach described in the previous paragraph.
yR ¼ eR sin uR ;
x B ¼ eB ;
x_R ¼ eR X sin uR
(28a)
y_R ¼ eR X cos uR
(28b)
x_ B ¼ 0;
yB ¼ 0;
x€B ¼ eB X
y_B ¼ eB X;
2
(28d)
The friction force between the nose of the floating ring and its casing (stator) is orthogonal to ~
eB , so
Ff ; x ¼ 0;
Ff ; y ¼ Ff
(29)
~R=B
The hydrodynamic force of the rotor on the floating ring F
depends on the eccentricity xR xB, yR yB and on the relative velocity x_ R x_B , y_R y_B .
The equations of motion Eqs. (24) are then
(
M
eB X2
)
0
(
¼
FR=B;x
FR=B;y
)
(
0
1þ
xR xB
@ R yBA
x_R x_B
y_R y_B
0
)
(30)
Ff
eBR
Cjeu
cðe; uÞ ¼ arctan
(31)
eR sin u
eR cos u e
R
x_ R x_B
y_R y_B
FR=B;x 0
1 þ Cxy
xR xB
FR=B;y
y
y
@ R BA
R=B
(35)
Crt
R
Ctt eðe;uÞ cðe;uÞ
Crr
Ctr
(36)
0
Crr
Ctr
Crt
Rcðe;uÞ
Ctt eðe;uÞ
0
eR X sin u
eR X cos u eX
(37)
The nonlinear system Eq. (30) of two algebraic equations with
two unknowns e, u is:
)
(
)
FR=B;r
eðe;uÞ
¼ Rcðe;uÞ
FR=B;t 0
Crt
T Crr
þ Rcðe;uÞ
Ctr Ctt eðe;uÞ
(38)
0
eR X sin u
Rcðe;uÞ
eR X cos u eX
þ
0
Ff
This system can be solved by any gradient type numerical
method.1
Remark. It is supposed that the rotor describes a centered, uniform, circular whirl. This assumption supposes that the rotor whirl
is not influenced by the response of the floating ring. The solution
of the nonlinear system Eq. (38) enables an a posteriori estimation
of the forces transmitted by the floating ring to the rotor.
(
FB=R;x
FB=R;y
)
(
¼
FR=B;x
FR=B;y
)
(
¼
MeB X2
)
(39)
Ff
This can be expressed in a coordinate system linked to the rotor
and defined by the vector ~
eR .
FB=R;r
FB=R;t
0
0
T
þ Rcðe;uÞ
(32)
x_ R x_ B
y_R y_B
B
y_R y_B
Taking into account the previous paragraph, the hydrodynamic
~R=B are
forces acting on the floating ring F
FR=B;x 0
1¼
xR xB
FR=B;y
y
y
@ R BA
T
sin cðe; uÞ
cos cðe; uÞ
By replacing in Eq. (33):
(
)
(
)
FR=B;r
FR=B;x
0
1 ¼ Rcðe;uÞ
xR xB
FR=B;y
FR=B;t eðe;uÞ
B yR yB C
0
@ x_ x_ A
eX2
M
0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼ e2 þ e2R 2e eR cos u=Cjeu
cos cðe; uÞ
sin cðe; uÞ
¼ Rcðe;uÞ
R=B
Cxy
(
The whirl motion of the rotor is an imposed excitation so its amplitude eR and velocity X are known. The unknowns of the system of
equations in Eq. (30) are the parameters describing the whirl of the
floating ring, namely its amplitude eB and the phase of the rotor
uR . For the ease of notation, the subscript R of the phase lag uR
and the subscript B of the floating ring amplitude eB are discarded
in the following, i.e., uR u, eB e. Equation (30) represents a
nonlinear system of two equations with two unknowns, e and u.
The relative eccentricity between the floating ring and the rotor
and the rotation angle c of the vector ~
eBR are
eðe; uÞ ¼
Rcðe;/Þ ¼
(28c)
y€B ¼ 0
(34)
0
0
This yields
xR ¼ eR cos uR ;
FR=B;r FR=B;t e0ðe;uÞ
FR=B;x 0
1 ¼ Rcðe;uÞ
xR xB
FR=B;y
y
y
R
B
@
A
¼
cos u sin u
sin u cos u
MeB X2
Ff
¼
KReff eR
CReff eR X
(33)
(40)
The « steady » hydrodynamic forces and the damping coefficients
are
!first calculated in the OrRBtRB coordinate system with axis
eBR . In this coordinate system
OrRB always aligned with vector ~
the steady forces and the dynamic coefficients depend only on the
eccentricity between the rotor and the floating ring, eðe; uÞ.
A rotation of angle cðe; uÞ is necessary for expressing forces and
dynamic coefficients in the coordinate system Oxy. The « unsteady
The underlined effective stiffness and damping coefficients stemming from the action of the ring on the rotor are
Journal of Engineering for Gas Turbines and Power
1
Due to the use of the gradient type algorithm for solving the nonlinear algebraic
system it is more correct to designate the present approach as quasi-analytic.
MAY 2012, Vol. 134 / 052507-5
Downloaded From: http://gasturbinespower.asmedigitalcollection.asme.org/ on 01/27/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
KReff ¼
Ff sin u MeB X2 cos u
eR
(41)
CReff ¼
Ff cos u þ MeB X2 sin u
eR X
(42)
Results
Calculations were performed for the floating ring depicted in
Fig. 5. The floating ring is made of carbon mounted in a steel
outer ring. Its dimensions are R1 ¼ 45.29 mm, R2 ¼ 45.5 mm,
R3 ¼ 47 mm, R4 ¼ 49.75 mm, R5 ¼ 53 mm, L ¼ 4 mm, and
Cjeu ¼ 30 lm. The mass of the floating ring is M ¼ 0.056 kg. The
working fluid is air with Pupstream ¼ 9 bars, Tupstream ¼ 300 C,
and Pdownstream ¼ 3 bars. The rotation speed is X ¼ 43 krpm.
The dynamic coefficients of the annular seal are depicted in
Fig. 6 and were estimated by solving the zero and the first order
“bulk flow” equations [9].
The friction force must also be estimated prior to any dynamic
analysis. It is supposed that the contact between the floating ring
nose and its casing is characterized by the following parameters:
g1 ¼ g2 ¼ 0.5 1010 m2, b ¼ 50 lm, r1 ¼ r2 ¼ 0.5 lm, E1 ¼
1.42 1010 Pa (carbon), E2 ¼ 2 1011 Pa (steel), 1 ¼ 0.22,
2 ¼ 0.29, and f ¼ 0.2 (carbon/steel). The results of the mixed
lubrication regime analyzes are
Fz ¼ 428 N;
Ff ¼ 28:9735 N;
Fz;fl ¼ 284 N;
Ff ;as ¼ 28:972 N;
Fz;as ¼ 145 N
Ff ;fl ¼ 1:51 103 N
This corresponds to an equivalent friction coefficient féq
¼ Ff/Fz ¼ 0.068. The value of the resulting distance between the
two contact surfaces is h ¼ 1.6 lm, which validates the assumption of the mixed friction regime because r < h < 3r, and the
combined standard deviation of the contact surfaces is r ¼ 0.7 lm.
Results obtained for these working conditions are depicted in
Figs. 7–10. Figure 7 shows that under the effect of the friction
force the floating ring remains blocked for excitation amplitudes
eR/Cjeu < 0.7. With increasing the excitation amplitude, the
floating follows the synchronous vibrations until contact occurs
for eR/Cjeu ¼ 3.
eBR is depicted in Fig. 8.
The angle c between the vectors ~
eB and ~
If one considers that the rotor is the exciting system and the floating ring the excited one, values 0 < c < 90 indicate an undercritical regime, c ¼ 90 corresponds to resonance and 90 < c < 180
characterizes a supercritical regime. Values of c > 110 depicted
on Fig. 8 show that the floating ring starts to slide directly in the
supercritical regime. The contact occurs due to the decrease of the
minimum film thickness following the progression of the floating
ring in the supercritical regime (Fig. 9). The angle u between the
eR is depicted in Fig. 10. Its progressive decrease
vectors ~
eB and ~
shows that the center of the rotor R tends toward a position located
between the center of the coordinate system and the center B of the
floating ring.
Fig. 5
Geometry of the floating ring seal
052507-6 / Vol. 134, MAY 2012
Fig. 6 Dynamic coefficients of the annular seal (Pupstream 5 9
bars, X 5 43 krpm)
Fig. 7
Amplitude of the floating ring whirl orbit
Transactions of the ASME
Downloaded From: http://gasturbinespower.asmedigitalcollection.asme.org/ on 01/27/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Fig. 8
Angle between the rotor and the floating ring
Fig. 11 Transmitted effective stiffness and damping
Fig. 9
Minimum film thickness
Figure 11 depicts the forces transmitted by the floating ring to
the rotor. These forces are expressed as effective stiffness and
damping coefficients. The effective stiffness coefficient is of the
order of the direct stiffness of the annular (main) seal but with
positive and negative values. The effective damping coefficient
has values an order of magnitude larger than the direct damping
of the annular seal, mainly due to the effect of the contact friction
forces acting on the nose of the floating ring.
It is natural to investigate now how the floating ring will work
under different working conditions. Therefore, the rotation speed
was varied between 1500 rad/s and 7500 rad/s and the upstream
pressure between 5 bars and 13 bars. The downstream pressure
was kept constant. The dynamic coefficients were recalculated for
each new working condition. Results are depicted in Fig. 12 in
terms of excitation amplitude versus rotation speed for different
values of the upstream pressure. Each Pusptream value consists of
two curves. The lower one corresponds to the case when the friction force blocks the floating ring. This is an acceptable working
condition for the floating ring if the amplitude of the rotor is lower
than the radial clearance of the annular seal. The upper curve
corresponds to the case when the rotor is in contact with the float-
Fig. 10 Phase angle of the rotor center
Journal of Engineering for Gas Turbines and Power
ing ring. Figure 12 shows that for Pupstream < 12 bars, the floating
ring can follow the synchronous vibrations of the rotor up to
6500 rad/s and has an optimum around Pupstream ¼ 9 bars where it
can work without contact for very large values of the excitation
amplitude. For Pupstream ¼ 13 bars, the floating ring can still whirl
without contact but only for synchronous vibrations higher than
2500 rad/s and for limited excitation amplitudes.
Conclusions and Perspectives
The present work presents a simplified analysis performed for
floating ring annular seals under the assumption that both the rotor
and the ring describe circular synchronous whirl motions.
The floating ring annular seal works well provided that the friction force is correctly accounted for. The results show for which
working conditions (upstream pressure and rotation speed) the
floating ring is able to follow the synchronous rotor whirl. The
main conclusion is that the floating ring annular seal must be carefully integrated into the rotating machine by taking into account
Fig. 12 Working conditions of the floating ring
MAY 2012, Vol. 134 / 052507-7
Downloaded From: http://gasturbinespower.asmedigitalcollection.asme.org/ on 01/27/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
all working conditions that can be encountered (start, stop, idle,
regime, etc.). The present work must be completed by a stability
analysis. A numerical counterpart of the present work must be
also performed for taking into account nonlinear dynamic
responses of the floating ring.
Ctt ¼ Crt ¼ Acknowledgment
The authors are grateful to Snecma Space Engine Division and
Centre National d’Etudes Spatiales for supporting this work.
In the rotating reference system depicted in Fig. 13, the
unsteady nonlinear hydrodynamic forces in a short bearing (L/
D 0.25) are [10]:
lRL3
2C2jeu ð1
e2 Þ2
3
Ft ðtÞ ¼
lRL e
2C2jeu ð1
pe_ð1 þ 2e2 Þ
X
pffiffiffiffiffiffiffiffiffiffiffiffiffi þ 4e2
/_
2
1 e2
e2 Þ2
lRL3
p
C3jeu 2ð1 e2 Þ3=2
cos /ðtÞ sin /ðtÞ Fr ðtÞ
FX ðtÞ
¼
sin /ðtÞ cos /ðtÞ
FY ðtÞ
Ft ðtÞ
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
(A8)
(A9)
(A10)
½uðtÞ
½KXY ¼ ½/ðtÞT ½Krt ½/ðtÞ
(A1)
(A11)
Nomenclature
(A2)
The corresponding dynamic coefficients are
Krr ¼ @Fr lRL3 X 2eð1 þ e2 Þ
¼
@e
C3jeu ð1 e2 Þ3
(A3)
Ktt ¼ @Ft
lRL3 X
e
¼
e0 @/
C3jeu ð1 e2 Þ2
(A4)
Krt ¼ @Fr
lRL3 X
p
¼
e0 @/
C3jeu 4ð1 e2 Þ3=2
(A5)
Ktr ¼ @Ft
lRL3 X pð1 þ 2e2 Þ
¼ 3
@e
Cjeu 4ð1 e2 Þ5=2
(A6)
@Fr lRL3 pð1 þ 2e2 Þ
¼ 3
@e
Cjeu 2ð1 e2 Þ5=2
(A7)
Crr ¼ ¼
@Fr
lRL3
2e
@Ft
¼ 3
¼ Ctr ¼ _
@ e_
Cjeu ð1 e2 Þ2
e0 @ /
pffiffiffiffiffiffiffiffiffiffiffiffiffi
X
/_
1 e2
4e_ þ p
2
e0 @ /_
Forces and dynamic coefficients in the fixed reference frame are
obtained by applying a multiplication matrix,
Appendix
Fr ðtÞ ¼ @Ft
C¼
Cjeu ¼
d¼
e¼
f¼
E¼
F¼
FR/B ¼
Ff ¼
g¼
K¼
L¼
M¼
P¼
R¼
Sc ¼
Wg ¼
x, y ¼
_ y_ ¼
x;
x€; y€ ¼
t¼
b¼
u, c ¼
X¼
r¼
g¼
1¼
¼
l¼
e¼
/¼
damping [Ns/m]
radial clearance [m]
elastic deformation (indentation) [m]
eccentricity
friction coefficients
elasticity modulus [Pa]
force [N]
hydrodynamic force in the primary seal [N]
friction force [N]
gravitational acceleration [m/s2]
stiffness [N/m]
length [m]
mass [kg]
pressure [Pa]
radius [m]
contact surface [m2]
gravitational load [N]
displacements [m]
velocities [m/s]
accelerations [m/s2]
time [s]
roughness asperity radius [m]
angles indicated in Fig. 2
whirl rotation speed [s1]
standard deviation [m]
roughness density [m2]
integration variable [m]
Poisson coefficient
dynamic viscosity [Pa s]
relative eccentricity, e=Cjeu
attitude angle
Subscripts
r, t ¼
R¼
B¼
x, y ¼
X, Y ¼
0¼
1, 2 ¼
cylindrical coordinate system
rotor
floating ring
whirling coordinate system
fixed coordinate system
initial conditions
carbon floating ring, casing (stator)
References
Fig. 13 Coordinate system and notations
052507-8 / Vol. 134, MAY 2012
[1] Müller, H. K., and Nau B. S., 1998, Fluid Sealing Technology-Principles and
Applications, Marcel Dekker, New York.
[2] Ha, T.-W., Lee, Y.-B., and Kim, C.-H., 2002, “Leakage and Rotordynamic
Analysis of a High Pressure Floating Ring Seal in the Turbo Pump Unit of a
Liquid Rocket Engine,” Tribol. Int., 35(1), pp. 153-161.
Transactions of the ASME
Downloaded From: http://gasturbinespower.asmedigitalcollection.asme.org/ on 01/27/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
[3] Shapiro, W., 2005, “Users’ Manual for Computer Code DYSEAL- Dynamic
Response of Seals,” Report No. NASA/CR-2003-212368.
[4] Kirk, R. G., 1988, “Transient Response of Floating Ring Liquid Seals,” J. Tribol., 110, pp. 572–578.
[5] Frêne, J., Nicolas, D., Deguerce, B., Berthe, D., and Godet, M., 1990,
Lubrification Hydrodynamique: Paliers et Butées, Editions Eyrolles,
Paris.
[6] Hamrock, B. J., Schmid, S. S., and Jacobson, B. O., 2004, Fundamentals of
Fluid Film Lubrication, 2nd Ed., Marcel Dekker, New York.
Journal of Engineering for Gas Turbines and Power
[7] Brunetière, N., 2010, Les Garnitures Mécaniques. Etude Théorique et Expérimentale, Habilitation à Diriger des Recherches, Université de Poitiers.
[8] Greenwood, J. A., and Williamson, J. B. P., 1966, “Contact of Nominally Flat
Surfaces,” Proc. R. Soc. London, Ser. A, 295, pp. 300–319.
[9] Arghir, M., and Frêne, J., 2001, “Numerical Solution of Lubrication’s Compressible Bulk Flow Equations. Applications to Annular Gas Seals Analysis,”
Paper No. 2001-GT-117.
[10] Szeri, A. Z., 1998, Fluid Film Lubrication. Theory and Design, Cambridge University Press, Cambridge, England.
MAY 2012, Vol. 134 / 052507-9
Downloaded From: http://gasturbinespower.asmedigitalcollection.asme.org/ on 01/27/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use