Chinese Journal of Aeronautics, (2018), 31(5): 949–964
Chinese Society of Aeronautics and Astronautics
& Beihang University
Chinese Journal of Aeronautics
cja@buaa.edu.cn
www.sciencedirect.com
Variable load failure mechanism for high-speed
load sensing electro-hydrostatic actuator
pump of aircraft
Cun SHI, Shaoping WANG, Xingjian WANG *, Yixin ZHANG
School of Automation Science and Electrical Engineering, Beihang University, Beijing 100083, China
Received 14 March 2017; revised 7 June 2017; accepted 30 September 2017
Available online 2 February 2018
KEYWORDS
Coupling lubrication model;
Electro-Hydrostatic Actuator (EHA);
High-speed pump;
Partial abrasion;
Slipper pair;
Variable load
Abstract This paper presents a novel transient lubrication model for the analysis of the variable
load failure mechanism of high-speed pump used in Load Sensing Electro-Hydrostatic Actuator
(LS-EHA). Focusing on the slipper/swashplate pair partial abrasion, which is considered as the
dominant failure mode in the high-speed condition, slipper dynamic models are established. A forth
sliding motion of the slipper on the swashplate surface is presented under the fact that the slipper
center of mass will rotate around the center of piston ball when the swashplate angle is dynamically
adjusted. Besides, extra inertial tilting moments will be produced for the slipper based on the theorem on translation of force, which will increase rapidly when LS-EHA pump operates under highspeed condition. Then, a dynamic lubricating model coupling with fluid film thickness field, temperature field and pressure field is proposed. The deformation effects caused by thermal deflection and
hydrostatic pressure are considered. A numerical simulation model is established to validate the
effectiveness and accuracy of the proposed model. Finally, based on the load spectrum of aircraft
flight profile, the variable load conditions and the oil film characteristics are analyzed, and series of
variable load rules of oil film thickness with variable speed/variable pressure/variable displacement
are concluded.
Ó 2018 Chinese Society of Aeronautics and Astronautics. Production and hosting by Elsevier Ltd. This is
an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction
* Corresponding author.
E-mail address: wangxj@buaa.edu.cn (X. WANG).
Peer review under responsibility of Editorial Committee of CJA.
Production and hosting by Elsevier
More Electric Aircraft (MEA) is the future development trend
for general aircraft, which partially replace the conventional
central hydraulic system by local electrically Power-By-Wire
(PBW) system. The successful application of PBW technology
brings less energy loss and higher efficiency. Among them,
Electro-Hydrostatic Actuator (EHA) is the key component
of the PBW system, which has already been applied in the large
civil aircraft, such as A380 and A350.1–3 However, because of
https://doi.org/10.1016/j.cja.2018.01.005
1000-9361 Ó 2018 Chinese Society of Aeronautics and Astronautics. Production and hosting by Elsevier Ltd.
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
950
C. SHI et al.
Nomenclature
vp ; ap
linear reciprocating motion velocity and axial
acceleration
r, h
polar radius and polar angle
Fg
slipper gravity
Fsa
slipper inertial force of linear reciprocating motion
FaDd
slipper inertial force of sliding motion
Mgx ; Mgy extra tilt moments produced by slipper Gravity
Msax ; Msay extra tilt moments produced by slipper Inertial
force of linear reciprocating motion
MaDd x ; MaDd x extra tilt moments produced by slipper Inertial force of sliding motion
FTs ðp; hÞ friction force between the slipper and swash plate
Ts ; Ttx , Tty friction moments between the piston and the
slipper socket along the x-, y- and zfs
friction coefficient of piston ball
bearing force between the slipper and the swash
plate along z-axis
M2 ; M3 ; M4 anti-overturning torques of the slipper on x-,
y- and zq
oil density
l0
oil viscosity at the initial condition
ap ; aT
pressure coefficient, temperature coefficient
T0
reference temperature
d1 , l1
length and diameter of the damping orifices of piston
d2 , l2
length and diameter of the damping orifices of slipper
oil film thickness shifting rates vector in kth iterah_ ðkÞ
tion
_
_
F0 ðhÞ
jacobi matrix of FðhÞ
Ek
diagonal matrix to revise Newton iterative method
the heating problem, they are just used as standby systems.4 To
solve the motor heating problem in EHA, a Load Sensing
Electro-Hydrostatic Actuator (LS-EHA) scheme has been proposed, which can solve the heating problem and the dynamic
problem simultaneously. The schematic diagram of the LSEHA is shown in Fig. 1. When the LS-EHA works in high load
and slow rate conditions, the pump displacement is reduced
through decreasing the angle of the swashplate, and the motor
speed is improved at the same time to maintain a stable output
of the actuator. Hence, the armature current will also be
decreased due to the reduction of the motor torque. Consequently, the energy loss can be effectively reduced and the system heating can be limited greatly.5,6 Based on the insightful
advantages, LS-EHA should be the future development trend
for the next generation of actuation system of aircraft.
The LS-EHA system consists of a brushless DC motor, a
LS-EHA pump, a Load Sensing Mechanism (LSM), etc.
Among them, the LS-EHA pump plays a core role for converting mechanical power into hydraulic power, which directly
determines the service life and reliability of the LS-EHA.
Fig. 2 shows the cross section of the high-speed pump used
in the LS-EHA system, which comprises nine pistons mounted
within the cylinder at an equal angular interval around the centerline of the cylinder, the cylinder is pushed towards the fixed
valve plate by a compressed cylinder spring, and the compressed spring force is transferred to the spherical cup through
several pins so that a hold-down force can be applied on the
retainer. The shaft and cylinder are connected by means of a
spline mechanism and they are driven around the axis of the
main shaft.7 When the shaft is driven by the brushless DC
motor, each piston periodically reciprocates within its corresponding cylinder bore due to the inclined swashplate and
the retainer. Compared to traditional aircraft hydraulic pump,
EHA pump is famous for its high-speed property and smaller
displacement to improve power mass ratio. To meet the
demands of the MEA era, it usually adopts high rotation speed
more than 10000 r/min for the pump. According to MessierBugatti and Parker, they have realized EHA pump with
15000 r/min rotation speed.8 High rotational speed means
higher power density, which is the significant feature of the
LS-EHA pump. Unfortunately, it will also bring out new severe problems for the LS-EHA pump, in which the most challenging problem is the abrasion of the critical friction
lubricating interfaces.
Fig. 1
F1
Schematic of LS-EHA system.
Variable load failure mechanism
Fig. 2
Cross section of high-speed pump used in LS-EHA.
There are three key friction pairs in the LS-EHA pump,
including the slipper/swashplate pair, the piston/cylinder pair
and the cylinder/valve plate pair.9 The oil films of these friction
pairs work as lubrication, sealing and bearing, which have a
great influence on the service life and reliability of the axial
pump. Among these above lubricating interfaces, the
cylinder/valve-plate pair is the key friction pair for oil film
hydrostatic bearing and produce high-pressure and lowpressure switch, and this friction pair prone to cavitation; piston/cylinder pair is the key factor of the maximum pressure for
the pump because of the oil film pressure squeezing point on
both ends of the bushing; the lubricating interface between
the slipper and the swashplate plays an important role in
proper machine operation. On the one hand, the slipper/
swashplate interface serves to seal the gap between the slipper
and the swashplate. On the other hand, the entire displacement
chamber pressure load exerted on the piston face must be
borne by the slipper and transmitted through a fluid film to
the swashplate face, which is considered as the weakest link
of the LS-EHA pump in high-speed condition. In other words,
during high rotation speed, the performance of the slipper/
swash-plate lubricating interface directly determines the performance degradation of LS-EHA pump.
In fact, the slipper usually tilts during rotation due to the
tilting moment produced by the centrifugal force and the slipper friction force, which will result in a wedged oil-film.10
Especially, the inertial force, including inertial force of the
axial reciprocating motion, the inertial force of sliding motion
on the swashplate surface as well as the centrifugal force of the
elliptical motion, will be rapidly increased when the pump
operates at high speed, which will further increase the slipper
tilting state. Consequently, the slipper oil film is usually
unevenly distributed, and partial abrasions will take place at
the points where the oil-film is thinner than the threshold value
determined by the surface roughness degree,11 which is the
main failure mode for the LS-EHA pump in high-speed
condition.
To reveal the fluid oil film characteristics of the slipper pair,
some theoretical and experimental researches have been carried out to study the lubrication oil film between slipper and
swashplate. Xu et al.10–12 developed numerical simulation
951
models for slipper/swash-plate and piston-cylinder pairs based
on the elastohydrodynamic lubrication theory. Koc and
Hooke13–15 have developed analytical and simple numerical
models, coupled with experimental analysis, to investigate slipper operation under different operating conditions, design
modifications, and nonflatness. However, the kinematics of
these test rigs is different from the practical friction pair, so
the slipper tilting state cannot be simulated correctly. Kazama
and Yamaguich16 presented a mixed friction model for water
hydraulics pump. Bergada et al.17,18 developed full Computational Fluid Dynamics (CFD) models to predict slipper leakage as part of a whole axial piston machine model. The
lubrication mechanism and the dynamic behavior of axial piston pumps and motors’ slipper bearings have been studied in
Ref.19. Due to the compact structure of the EHA pump, the
heating problems cannot be ignored. Especially in high speed,
the magnitude of heat generated by viscous friction can be
quite significant. To improve the fidelity of the lubricating
model, the lubrication characteristics of the slipper pair considering oil thermal effect have been investigated in Ref.20.
And a transient thermoelastohydrodynamic lubrication model
for the slipper/swashplate has been developed.21 The model
considers a nonisothermal fluid model, microdynamic motion
of the slipper, as well as pressure and thermal deformations
of the bounding solid bodies through a partitioned solution
scheme. Also, similar researches for both the piston-cylinder
interface22,23 and the cylinder block/valve plate interface24,25
are developed by considering higher fidelity pressure deformation models, stronger coupled fluid structure interaction, as
well as solid body temperature distributions/thermal deformations. However, the existing literatures have hardly focused on
the slipper/swashplate friction pair of the high-speed pump,
which may have different failure mechanisms compared to
conventional conditions.
The goal of this study is to investigate the variable load failure mechanism of high-speed pump used in LS-EHA based on
the lubrication analysis of the slipper pair. In the present work,
the detailed motion and force analysis for the slipper are performed. A transient lubricating model considering coupling
effects among the fluid film thickness field, temperature field
and pressure field is developed. The effects between the variable load and the slipper oil film characteristics are analyzed
based on the load spectrum, and a numerical simulation model
is established to verify the effectiveness and accuracy of the
analysis results. The main contributions of this paper are as
follows:
(1) This paper focuses on the high-speed pump used in LSEHA, paying much attention to the inertia force and
moments, which will be increased rapidly and play an
important role in the oil film performance of the slipper
pair.
(2) A multi-field coupling lubricating model has been proposed, and the elastic deformation as well as the
thermal expansion are considered in the proposed
model.
(3) Based on the load spectrum of the aircraft flight profile,
variable load lubrication characteristics are analyzed
with respect to variable speed/variable pressure/variable
displacement conditions.
952
C. SHI et al.
dsp
_
¼ xR sin u tan b þ bRð1
cos uÞ sec2 b
dt
2. Slipper kinematics analysis
vp ¼
In this section, a precise kinematic analysis for the slipper will
be carried out, which are the dynamic boundary conditions for
the solution of the proposed transient lubrication model.
Firstly, three coordinate systems should be introduced: XYZ
system, X1Y1Z1 system as well as xyz system. The XYZ system
and X1Y1Z1 system are inertial systems, and X1Y1Z1 system
can be obtained by a rotation of XYZ system around the Xaxis clockwise through an angle b: The xyz system is attached
to the center of slipper socket and will rotate as slipper rotates
along the surface of swashplate, its x-axis is tangential to the
trajectory of the center of slipper, and y-axis remains directly
radially outward.
The axial acceleration of the linear reciprocating motion
can be calculated as
2.1. Motion analysis
As shown in Fig. 3, the slippers are held tightly against the
swashplate by the retainer, and are directly connected with
the piston through spherical joints. During rotation, the
motions of the slipper are very complicated, which may contain four microdynamic motions at the same time: (A) linear
reciprocating motions with piston, which are paralleled to Z
axis; (B) Elliptical rotational motion in X1Y1 plane; (C) Spin
around z axis; (D) Sliding motion on the surface of swashplate
(when the angle of the inclined swashplate has changed Db; all
the slippers will overall move a distance as Dd along Y1 axis).
(1) Linear reciprocating motion
By taking the Outer Dead Center (ODC) where the volume
of the piston chamber is the largest as the starting motion
point, the linear reciprocating motion velocity of the slipper
can be described as
Fig. 3
ap ¼
ð1Þ
dvp
_ sin u sec2 b
¼ x2 R cos u tan b þ 2xbR
dt
_ 2 Rð1 cos uÞ sec2 b tan b
€
þ bRð1
cos uÞ sec2 b þ 2ðbÞ
ð2Þ
where sp is the linear reciprocating motion displacement of the
slipper, x represents the angular speed of the cylinder block, R
represents pitch radius of the piston bores, u is the rotation
_ b
€ stand for the swashplate angle,
angle of the piston, and b; b;
angular velocity of swashplate, and angular acceleration of
swashplate respectively.
Remark 1. According to the working principle of LS-EHA,
the swashplate angle of the pump is changed dynamically.
Thus, the angular velocity and the angular acceleration of the
swashplate cannot be ignored any more during motion
analysis, especially when the LS-EHA pump is working in
large dynamic phase (such as, aircraft take-off and landing
phase), which may be the new problem brought by the pump
displacement change.
(2) Elliptical rotational motion
Driven by the circular motion of the piston, the trajectory
of the slipper on the swashplate surface is an ellipse, which is
centered at the origin of the X1Y1 plane as shown in Fig. 3.
X21
Y2
þ 2 12 ¼1
2
R
R sec b
Forces/moments acting on piston/slipper of LS-EHA pump.
ð3Þ
Variable load failure mechanism
953
When the piston rotates an arbitrary angle u from A to B,
the rotation angle of the center of slipper is
8
arctanðtan u cos bÞ
0 < u < p2
>
>
>
>p
>
u ¼ p2
>
<2
us ¼ arctanðtan u cos bÞ þ p p2 < u < 3p
ð4Þ
2
>
>
3p
3p
>
u¼ 2
>
2
>
>
:
arctanðtan u cos bÞ þ 2p 3p
< u 6 2p
2
The elliptical motion angular velocity of the slipper center
of mass is
du
cos b
tan u sin b
x
b_
xs ¼ s ¼
dt
1 þ tan2 u cos2 b
cos2 u þ sin2 u cos2 b
ð5Þ
Considering the effect of the slipper spin motion, the kinematic parameters of an arbitrary point ðr; hÞ, which are derived
in Appendix A according to Fig. 4, under the slipper bottom
surface can be calculated as
vsr ¼ vsr ðr; hÞ ¼ xs qo sin h
ð6Þ
vsh ¼ vsh ðr; hÞ xz r ¼ ðxs xz Þr þ xs qo cos h
where vsr and vsh are the radial velocity and circumferential
velocity component of an arbitrary point, qo is the radial distance between slipper center and the origin of the X1Y1Z1 syspffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
tem, qo ¼ R ð1 þ cos2 u tan2 bÞ; and xz is the spin velocity of
the slipper.
(3) Sliding motion on surface of swashplate
When the angle of the inclined swashplate has changed, the
slipper center of mass will rotate around the piston ball center
under the effect of the slipper gravity, slipper axial inertial
force, etc. As shown in Fig. 5, relative to the center of the piston
ball o1 , the slipper center of mass o will slide a distance Dd as
Dd ¼ Ddmax lsg tan b
ð7Þ
where Ddmax is the maximum offset distance (when the angle of
the swashplate is the maximum value bmax ), lsg is the length
between the slipper center of mass and the piston ball center.
If the swashplate angle is dynamically changed, the sliding
velocity of the slipper vDd can be expressed as
vDd ¼
dDd
_ sg sec2 b
¼ bl
dt
ð8Þ
Then the sliding acceleration of the slipper aDs on the surface of the swashplate is
aDd ¼
dvDd
€ sg sec2 b 2b_ 2 lsg sec2 b tan b
¼ bl
dt
ð9Þ
Remark 2. The sliding motion on the surface of swashplate
will generate an inertial force FaDd along the axis of Y1 . Besides,
FaDd will derive a tilt moment. When the swashplate angle of
the LS-EHA pump is changed with high frequency, this
inertial tilting moment will increase rapidly and will have an
inevitable effect on slipper tilt.
2.2. Force analysis
The LS-EHA pump is usually characterized as high speed, so
the inertia forces of the slipper should not be ignored as many
previous references have done. Besides, the slipper and the piston are not regarded as an assembly because the slipper center
of mass will rotate around the piston ball center. The internal
force Fps between the piston and the slipper can be obtained by
the kinetic equation of the piston
Fps ¼
pR2p ps þ mp ap þ Ff
cos b
ð10Þ
where Rp is the piston radius, ps is the instantaneous pressure
in the piston chamber, mp is the piston mass, and Ff is the friction force. According to the simplification in Ref.10, the friction force Ff can be generated as
vp ðps pL Þhp
Ff ¼ 2
pRp lk
þ
hp
2lk
ð11Þ
where pL is the case drain pressure, hp is the oil film thickness
of the piston/cylinder friction pair, and lk is the instantaneous
contact length between piston and cylinder bore at a certain
swashplate.
Forces acting on the center of the slipper can be calculated
as
8
>
< Fg ¼ ms g
Fsa ¼ ms ap
>
:
FaDd ¼ ms aDd
Fig. 4 Diagram of motion speed at an arbitrary point under
slipper bottom surface.
ð12Þ
As shown in Fig. 6(a) and (b), three extra tilt moments,
which are ignored in previous research, can be generated based
on the theorem on translation of force in xyz system as
954
C. SHI et al.
Fig. 5
Fig. 6
Slipper sliding motion diagram on surface of swashplate.
Detailed force analysis for slipper in LS-EHA pump.
Mgx ¼ Fg lsg cos b cos us
Mgy ¼ Fg lsg cos b sin us
ð13Þ
Msax ¼ Fsa lsg sin b cos us
Msay ¼ Fsa lsg sin b sin us
ð14Þ
MaDd x ¼ FaDd lsg cos us
MaDd x ¼ FaDd lsg sin us
When slipper rotates on the swashplate under a high-speed
condition, there will be friction force between the slipper and
swashplate, and a tilting moment Md along y-axis can be calculated as
Md ¼ ls FTs ðp; hÞ
ð15Þ
The tilting moment of the slipper Mws along x-axis direction caused by the centrifugal force can be written by
2
cos b
tan u sin b
_
x
b
Mws ¼ ms Rlsg
1 þ tan2 u cos2 b
cos2 u þ sin2 u cos2 b
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1 þ cos2 u tan2 bÞ
ð16Þ
ð17Þ
where ls is the length between the slipper bottom surface and
the piston ball center. As shown in Fig. 6(c), the friction
moment between the piston ball and the slipper ball socket will
make the slipper spin and tilt, which can be obtained as
8
>
< Ts ¼ fs F1 Rb cos b
Ttx ¼ fs F1 Rb sin b sin us
>
:
Tty ¼ fs F1 Rb sin b cos us
ð18Þ
Variable load failure mechanism
955
The bearing force and anti-overturning torques between the
slipper and the swashplate can be obtained by the integration
of the pressure distribution of the lubrication oil film of slipper
pair pðr; hÞ as
8
R 2p R R0
2
>
>
> F1 ¼ pr0 pr þ 0 r0 pðr; hÞrdrdh
>
>
R R
>
< M2 ¼ 2p R0 pðr; hÞr2 cos hdrdh
r0
0
ð19Þ
R 2p R R0
>
>
M3 ¼ 0 r0 pðr; hÞr2 sin hdrdh
>
>
>
>
: M ¼ R 2p R R0 s r2 drdh
4
0
r0
sh
After all the related forces and moments acting on the slipper are obtained, dynamic models of the slipper can be
described by equilibrium equations about x-axis, y-axis and
z-axis
8
F1 Fps Fs Fg sin b Fsa cos b ¼ ms z€
>
>
>
<
M2 þ Mgx þ Msax þ MaDd x þ Ttx þ Mws ¼ Ix €dx
>
M3 þ Mgy þ Msay þ MaDd y þ Tty þ Md ¼ Iy €dy
>
>
:
M4 Ts ¼ Iz x_ z
ð20Þ
dy
where z€ is the acceleration of slipper along z-axis, and €
dx , €
and x_ z are the angular accelerations along x-axis, y-axis and
z-axis respectively. In high-speed pump, the inertia force and
moments will be increased rapidly. Inertial tilting moments,
including Msax ; Msay ; MaDd x ; MaDd y ; Mws ; etc., will play an
important role in slipper tilting. Therefore, Ix , Iy , and Iz are
used to describe the moments of inertia of the slipper about
x-, y-, z- in xyz system. The following Fig. 7 shows the slipper
axial resultant force Fz and moments on x-axis Mx , y-axis My
and z-axil Mz .
3. Multi-physics field coupling lubricating model
3.1. Slipper coupling lubricating model
The heat conduction in the slipper axial direction can be
neglected compared to the heat conduction in the radial direction. The slipper oil temperature distribution can be calculated
by solving the energy equation as
@T
cp q
ð21Þ
þ v gradT ¼ k divðgradTÞ þ lUD ðvÞ
@t
where cp is the fluid heat capacity, T is the slipper oil temperature, v is the fluid velocity vector, k is the fluid thermal con-
Fig. 7
ductivity, UD is the energy dissipation, and l is the oil
viscosity.
Viscosity is one of the most important factors affecting the
oil drive characteristics. Temperature has a significant influence on the oil viscosity,26,27 and the viscosity-temperature
relationship can be described as9
l ¼ l0 exp ap p aT ðT T0 Þ
ð22Þ
To calculate the precise height of the slipper lubrication oil
film, deformation caused by the thermal and the hydrostatic
pressure loads should be considered. The thermal expansion
DhT of a finite control volume can be described as
Z
ð23Þ
DhT ¼ ½H1 ½DT ½Ef½aT Tðx; yÞgdV
V
where H is the rigidity matrix, D is the nodal shape function
matrix, E is the elasticity modulus matrix, and aT is the thermal expansion coefficient matrix.
The elastic deformation Dhp caused by the hydrostatic pressure loads can be
Z Z
1 m2
pðn; fÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dndf
Dhp ¼
ð24Þ
pE
X
ðx nÞ2 þ ðy fÞ2
where E is the Young’s moduli, m is the Poisson’s ratios and X
is the computation domain.
Due to the tilting moments, the oil film between the slipper
and swashplate is usually wedge-shaped, as shown in Fig. 8.
The film thickness at an arbitrary point ðr; hÞ can be calculated
as
h2 h3
2h1 h2 h3
hðr; hÞ ¼ pffiffiffi
r sin h þ
r cos h
3R0
3R0
h1 þ h2 þ h3
þ DhT þ Dhp
þ
3
ð25Þ
where hðr; hÞ is the slipper oil film thickness, h1 , h2 and h3 are
the film thickness of three fixed points on the outer edge at an
interval of 2=3p, and r0 , R0 are the slipper inside radius and
outside radius.
Since thickness of the lubricating oil film between the slipper and the swashplate is usually in the range of several
micrometers, the flow can be considered as laminar flow, and
the fluid can be assumed to be Newtonian. Therefore, the pressure distribution of the slipper oil film can be described by the
well-known Reynolds equation. In the cylindrical coordinates,
the Reynolds equation can be formulated with respect to the
frame as28
Resultant force and moments of slipper in xyz system.
956
C. SHI et al.
1 @p
vr ¼ 2l
ðz2 hzÞ þ vsr hz is the radial flow velocity distribution
@r
along z-axis.
From Eqs. (21) and (26), one may note that the energy
equation and the Reynolds equation are partial differential,
and to discretize the partial differential equation, finite difference method has been adopted to obtain the numerical iterative formulations.
3.2. Simulation procedure
Fig. 8
Wedge-shaped oil film between slipper and swashplate.
3 1 @
h @p
1 @ h3 @p
r
þ 2
r @r
r @h l @h
l @r
vsh @h
@h
@h
þ6
þ 12
¼ 6
vsr
@r
@t
r @h
ð26Þ
where p is the oil film pressure, and the slipper tilting state can
be represented by the terms @h=@r and @h=@h. The pressure
boundary conditions of the Reynolds Eq. (26) can be expressed
as
(
pðr0 ; hÞ ¼ pr
pðR0 ; hÞ ¼ pL
ð27Þ
pðr; 0Þ ¼ pðr; 2pÞ @p
¼ @p
@h ðr;0Þ
@h ðr;2pÞ
where pr is the pocket pressure, and pL is the case drain pressure. Neglecting the complex flow field in the LS-EHA pump
due to high rotation speed, the pressure in the case can be considered as a constant. The slipper pocket pressure is determined by the flow continuity equation. The fluid flows into
the slipper pocket from the piston chamber via the piston
and slipper orifices should be equal to the flows out of the
pocket through a variable annular narrow clearance. Thus,
the pocket pressure pr can be determined by
pr ¼ ps Cpb þ Csb
Cpb Csb
pd4
Z 2p Z h
ð28Þ
vr R0 dzdh
0
0
pd4
where Cpb ¼ 128ll1 1 and Csb ¼ 128ll2 2 are leakage flow coefficients
of the damping orifices of piston and slipper, and
Table 1
The operating parameters of the LS-EHA pump are listed in
Table 1, and the pump rotation speed, delivery pressure as well
as the swashplate angle have no rated values, but variation
ranges. In order to analyze the variable load failure mechanism
for the high-speed LS-EHA pump, a numerical simulation
model was built in the MATLAB environment to verify the
proposed transient lubrication model. As shown in Fig. 9,
there are three calculation loops in the simulation procedure.
The initial oil film thickness can be random chosen in simulation, and then the pressure distribution and the temperature
distribution can be obtained in the first loop. In the second calculation loop, the nonlinear equations are solved by the Newton iterative method. Finally, when the slipper rotates along
the swashplate surface, the slipper oil film thickness h1 , h2 , h3
and the slipper spin velocity wz can be updated by the third
loop.
Remark 3. In the second calculation loop, the classical
Newton iterative method is usually used to obtain the oil film
thickness shifting rates. However, it is very easy to cause
matrix close to singular for Jacobi matrix inversion in
MATLAB environment. Therefore, to improve the simulation
accuracy, a revised Newton iterative method is proposed to
avoid matrix singularity as follows:
1
h_ ðkþ1Þ ¼ h_ ðkÞ F0 ðh_ ðkÞ Þ Fðh_ ðkÞ Þ
ð29Þ
In particular, the above functions are defined as
8
_ ¼ Ek FðhÞ
_
>
FðhÞ
>
<
_ ¼ ½Ek FðhÞ
_ 0 ¼ Ek F0 ðhÞ
_
F0 ðhÞ
>
>
1
: 0 _ 1
_
_ 1 E1
F ðhÞ ¼ ½Ek F0 ðhÞ
¼ F0 ðhÞ
k
ð30Þ
where Ek ¼ diagð1; 0:001; 0:001Þ is a diagonal matrix. Its diagonal elements can be adjusted for different conditions.
Parameters of simulation of LS-EHA pump.
Parameter
ms ðkgÞ
mp ðkgÞ
r0 ðmÞ
R0 ðmÞ
Rb ðmÞ
lsg ðmÞ
ls ðmÞ
ap ðm2 =NÞ
cp ðJ=ðkg CÞÞ
Value
3
4.68 10
11.82 103
3.25 103
6.75 103
6.75 103
3.5 103
3.5 103
1.8 108
1884
Parameter
Value
Parameter
Value
z
Rp ðmÞ
R ðmÞ
d1 ðmÞ
l1 ðmÞ
k ðN=mÞ
xz ðmÞ
aT ð CÞ
Tin ð CÞ
9
5.0 103
20.0 103
1.0 103
9.8 103
64 103
15.0 103
0.03
50
lðPa sÞ
pL ðMPaÞ
k ðw=m CÞ
q ðkg=m3 Þ
E ðPaÞ
m
x ðr=minÞ
ps ðMPaÞ
b ð Þ
0.046
0.06–0.08 bar
80
860
4.4 1012
0.32
1000–10000
5–28
1.23–14.5
Variable load failure mechanism
957
Fig. 9
Flowchart of proposed numerical simulation model.
4. Simulation results and analyses
4.1. Simulation results
Assumption 1. Based on previous research on the slipper spin,
it can be assumed that the slipper spin speed is equal to the
shaft speed in low rotation speed condition. In high-speed
condition (around 10000 r/min), for simplification, the spin
velocity of the slipper can be approximatively selected as
xz ¼ 0:25x.
Fig. 10 shows the evolution of maximum value, average
value and minimum value of the slipper oil film thickness
(hMaximum , hAverage and hMinimum ), axial resultant force Fz and
moments on x-axis and y-axis Mx and My of the slipper in
the first five calculation cycles. After the third cycle, the oil film
thickness exhibits a periodic change law, the combined force
and moments acting are close to zero. Therefore, the simulation model has been basically converged after three cycles of
operation. For the analysis accuracy of oil film characteristics,
the simulation results of the fifth cycle are taken for oil film
performance analysis.
Based on the above analysis, the simulation results of the
fifth simulation cycle are taken to be observed, and film thickness, pressure distribution and temperature distribution of the
slipper/swashplate pair are presented in Fig. 11. The oil film
thickness is unevenly distributed. The maximum value of the
oil film thickness always appears in the third quadrant area
of xyz system, and the lowest point occurs in the first quadrant. Particularly, with the consideration of the deformation
caused by thermal expansion and hydrostatic pressure loads,
the oil film thickness field is not a strict wedge-shaped plane.
The pressure and the temperature are higher close to the inner
edge of the sealing land, accompanied by bigger deformation
of the oil film thickness. Therefore, the oil film thickness close
to the inner edge of the seal land may be convex outward or
concave. Due to the uneven distribution of oil film thickness
and the slipper spin phenomenon, the outer edge and the inner
edge of the sealing land wear will take place once the minimum
oil film thickness is thinner than the surface roughness degree.
Especially as the slipper transitions from low to high pressure
(around ODC), the oil film thickness is minimal and the slipper
tilting is the biggest.
To illustrate the indispensable effects of inertia force in the
proposed coupling lubrication model, the axial inertia forces,
958
C. SHI et al.
Fig. 10
Evolution of oil film between slipper and swashplate (10000 r/min, 28 MPa, 14.5°).
centrifugal moments as well as total normal forces under different pump rotation speeds are presented in above Fig. 12.
The inertia force and centrifugal moment will increase rapidly
when the LS-EHA pump operates in high speed. The maximum value of inertia force can reach 100-400 N, which will
have a significant effect on fluid film performance. The large
centrifugal moment can further increase the slipper tilting state
in high-speed conditions.
It can also be seen from Fig. 12(c) that the total normal
force will decrease in delivery pressure section, which means
that the slipper oil film thickness will gradually increase in
the high-pressure area. Similarly, the oil film thickness is going
to decrease in the suction pressure area due to the increase of
the total normal force.
4.2. Comparison with published results
A validation case is chosen from the published papers to verify
the developed coupling lubrication model. In Ref.21, Ivantysynova et al. proposed a transient thermoelastohydrodynamic model to predict the lubricating performance between
the slipper and the swashplate. It can verify the model accuracy by comparing the simulation results with the results in
Ref.21, which plots the minimum, mean, and maximum fluid
film thickness between the slipper and swashplate over a shaft
revolution as shown in Fig. 13(a). It should be noticed from
Fig. 13 that the results in Ref.21 is under 1000 r/min condition.
However, the proposed model in this paper is focused on the
high-speed pump used in LS-EHA, which is a new structure
in aviation field, and the results in this paper are under
10000 r/min. However, it can still be seen that the dynamic tendency of the oil film thickness in this paper is consistent with
that in Ref.21.
Compared to many previous researches for low-speed
pumps, the EHA pump usually operates at high rotation speed
(e.g., more than 10000 r/min). In such a high rotation speed, it
is extremely difficult to directly measure the slipper fluid film
thickness and the leakage flow from slipper/swashplate interface. Alternatively, the pump case drain leakage flow can be
measured to indirectly reflect the leakage flow from the slipper
interface. Based on the simulated oil film thickness in this
paper and Eq. (28), we can get the leakage flow through the
slipper/swashplate pair. Comparing the leakage through slipper with the experimental results in Ref.8, we can see that
the growth trend of the leakage with an increased pump speed
is consistent with that in Ref.8 Furthermore, it can be seen
from Fig. 14 that the leakage flow through slipper/swashplate
interface at a speed higher than 5000 r/min increased sharply
than that at lower speed. According to the research in Ref.9,
it has been found that more than 70% of the pump leakage
is caused by the friction between valve plate and cylinder block
in low-speed pump. However, it can be concluded from Fig. 14
that the leakage through slipper pair will play a dominant role
in the total pump case drain leakage flow after 8000 r/min.
4.3. Analysis of variable load characteristics
To further analyze the dynamic performance of oil film and to
reveal its failure mechanism, as shown in Fig. 15, variable load
conditions of LS-EHA pump based on flight profile are going
to be analyzed. The aircraft longitudinal control is taken as an
example, and the flight profile consists of: Take-off, Initial
Climb, Level Accelerating, Climb, Cruise Flying, Descent,
Approach, Level Decelerate Flight, and Land.
As shown in Fig. 15, combined with the new principles of
LS-EHA: (A) in the dynamic process for aircraft attitude
adjustment (Fig. 15 (a) bold part), LS-EHA operates in the
large dynamic output status. The EHA pump has big swashplate angle for a large pump displacement to improve the
dynamic performance of the actuator. Therefore, the LS-
Variable load failure mechanism
Fig. 11
959
Thickness, pressure and temperature distribution between slipper and swashplate pair (10000 r/min, 28 MPa, 14.5°).
EHA pump is in the ‘high pressure/large pump displacement/
high speed’ operating condition, as illustrated in Fig. 15(d)–
(f). On the other hand, (B) when the aircraft is in a steady state
of constant attitude (such as, Level Accelerating, Climb and
Cruise Flying), the swashplate angle should be reduced and
the pump speed needs to be increased simultaneously to make
the output power remain unchanged. Thus, the motor of LSEHA works in the high efficiency area and the system heating
can be limited effectively. Consequently, the EHA of the eleva-
tor should be in the ‘low pressure/low displacement/high
speed’ operating state.
In fact, the EHA operating conditions may be adjusted in
real time during flight process. Therefore, the LS-EHA pump
is always in the variable load conditions of ‘variable pressure/variable displacement/variable speed’. To reflect the real failure mechanism of LS-EHA pump, rules of the variable load
characteristics for the oil film performance of the slipper/
960
C. SHI et al.
Fig. 12 Inertia forces, centrifugal moments and total normal
forces under different pump rotation speeds.
Fig. 13
swashplate pair should be implemented based on the proposed
coupling lubrication model and the variable load conditions.
Fig. 16 plots the maximum, mean and minimum oil film
thickness of the slipper/swashplate pair compared to different
rotation speeds. The oil film thickness varies with the angular
position. Based on the analysis of Fig. 12(c), the normal pressing force will decrease during delivery high pressure area. It
can be seen from Fig. 16 that the slipper oil film thickness
increases gradually and reaches the biggest oil film thickness
around the Inner Dead Center (IDC). After that, slipper oil
film thickness will decrease due to the increase of the total normal force. The fluid film thickness is the thinnest when the displacement chamber pressure transitions from low to high
(around ODC). Therefore, the fluid film thickness of slipper
at IDC and ODC are representative throughout the cycle.
Fig. 17 shows the slipper fluid film thickness variation with
respect to the angular position under different displacement
chamber delivery pressure. The average film thickness
decreases as the pressure increases, especially around the
IDC. Fig. 18 illustrates the oil film thickness changes under different swashplate angles. The average oil film thickness will
decrease obviously around ODC. However, it cannot see the
change rules near the IDC.
To further quantitatively analyze the nonlinear relationship
between slipper oil film thickness characteristics and variable
load conditions, based on controlling variables method, the
pump speed, delivery pressure as well as the swashplate angle
(corresponding to the changes of pump displacement) are
changed. Taking the IDC and ODC as specific locations,
where slipper is in the transition zone from high to low pressure and transitions from low to high area, we analyze the variable load performance for fluid film thickness.
As indicated in Fig. 19, when the pump speed increases
from 5000 r/min to 10000 r/min, the average film thickness
increases from 3 lm to 23 lm at IDC. In this position, the
oil film thickness approximately increases linearly with respect
to the pump speed. At ODC, when the displacement chamber
pressure transitions from low to high, the fluid film thickness
has relatively small increase with the increase of pump speed.
However, the thickness differences of the maximum value
and the minimum value increase as the speed increases at this
transition point.
It can be seen from Fig. 20 that the slipper oil film thickness
at IDC is gradually reduced as the delivery pressure increases,
Slipper film thickness comparison between current results and results in Ref.21.
Variable load failure mechanism
961
Fig. 14
Slipper interface leakage compared with pump case drain leakage flow in Ref.8.
Fig. 15
Analysis of variable load conditions of LS-EHA pump based on flight profile.
which are nonlinear decreasing relationships, and the thickness
differences are also decreased. The fluid film thickness at ODC
is basically unchanged with the pressure change. It means that
the effects on the oil film characteristics of pressure change
mainly focus on the transition zone from high to low pressure.
Fig. 21 shows the relationship between oil film thickness
and pump swashplate angle at transition zone. The slipper
oil film thickness increases first and reaches the maximum at
about 8°, and then decreases with the increase of the pump
swashplate angle. The fluid film thickness will always decrease
as the swashplate angle increases, and the thickness difference
will also decrease with the increase of swashplate angle at
ODC.
5. Conclusions
A coupling lubrication model for simulating the dynamic
lubrication performance between the slipper and swashplate
962
C. SHI et al.
Fig. 16
Slipper oil film thickness under different rotation speeds.
Fig. 17
Slipper oil film thickness under different delivery pressure.
Fig. 18
Slipper oil film thickness under different swashplate angles.
pair in the LS-EHA high-speed pump has been developed.
Four slipper dynamic models are presented based on the
motion analysis and detailed force analysis. The fourth sliding
motion on the surface of swashplate is considered due to the
dynamic changes of the swashplate angle (pump displacement)
in LS-EHA. The inertia forces and moments are specially paid
attention to because of the high operation speed conditions.
The oil film thickness fields, pressure field as well as temperature field of the slipper pair are established, and the deformation effects caused by thermal deflection and hydrostatic
pressure are considered by means of the coupling model.
Finally, a series of simulation experiments are performed
based on the aircraft flight profile and the following conclusions are drawn.
(1) The inertia force increases rapidly in high speed and has
a significant impact on the lubrication performance.
(2) The slipper fluid film thickness will increase rapidly as
the pump speed increases, especially around IDC. At
ODC, the film difference will increase as the rotation
speed increases.
(3) As the delivery pressure increases, oil film thickness
around IDC will be gradually reduced, and basically
remains unchanged at ODC.
(4) With the increase of pump displacement, the average oil
film thickness increases first and then decreases at IDC,
and the fluid film thickness is always reduced around
ODC.
Variable load failure mechanism
963
Acknowledgements
This work was supported by the National Natural Science
Foundation of China (Nos. 51620105010, 51675019 and
51575019), the National Basic Research Program of China
(No. 2014CB046402), the ‘‘111” Project, and the Excellence
Foundation of BUAA for PhD Students. This work was partially done when the author was visiting DHAAL lab in Duke
University with professor Trivedi.
Appendix A.
Fig. 19 Relationship between oil film thickness and pump speed
at transition zone.
As shown in Fig. 4, according to the cosine theorem, the equivalent radius qs ðr; hÞ of the arbitrary point is
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qs ðr; hÞ ¼ q2o þ r2 2qo r cosðp hÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼ q2o þ r2 þ 2qo r cos h
ðA1Þ
The sliding velocity vs ðr; hÞ of an arbitrary point on the bottom surface of the slipper can be calculated as
vs ðr; hÞ ¼ xs qs ðr; hÞ
ðA2Þ
(1) First quadrant
According to geometric relationships in Fig. 4(a), an arbitrary point in the first quadrant is selected, and then the angle
a between vs and the radius under the slipper radially outward
can be described by
8
< sin a ¼ cos p2 a ¼ rþqoq cos h
s
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðA3Þ
: cos a ¼ 1 sin2 a ¼ qo sin h
qs
Therefore, the radial velocity component and the circumferential velocity component can be obtained as
vsr ðr; hÞ ¼ vs ðr; hÞ cos a ¼ xs qo sin h
ðA4Þ
vsh ðr; hÞ ¼ vs ðr; hÞ sin a ¼ xs ðr þ qo cos hÞ
Fig. 20 Relationship between oil film thickness and pump
delivery pressure at transition zone.
(2) Second quadrant
Select an arbitrary point from the second quadrant of oxy
system, and the following relationships can be obtained
according to the trigonometric function
8
< sin a ¼ cos p2 a ¼ cos p p2 a ¼ rþqoq cos h
s
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
: cos a ¼ 1 sin2 a ¼ qo sin h
qs
ðA5Þ
It should be pointed out that the direction of the circumferential velocity component is negative. Based on Eq. (A5), the
radial component and the circumferential component of the
sliding velocity in the second quadrant are
vsr ðr; hÞ ¼ vs ðr; hÞ cos a ¼ xs qo sin h
ðA6Þ
vsh ðr; hÞ ¼ vs ðr; hÞ sin a ¼ xs ðr þ qo cos hÞ
Fig. 21 Relationship between oil film thickness and pump
swashplate angle at transition zone.
(3) Third quadrant
As shown in Fig. 4, the direction of the circumferential
velocity component is also negative
8
< sin a ¼ sinðp aÞ ¼ cos p a þ p2 ¼ rþqoq cos h
s
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðA7Þ
: cos a ¼ 1 sin2 a ¼ qo sin h
qs
964
vsr ðr; hÞ ¼ vs ðr; hÞ cos a ¼ xs qo sin h
vsh ðr; hÞ ¼ vs ðr; hÞ sin a ¼ xs ðr þ qo cos hÞ
C. SHI et al.
ðA8Þ
(4) Fourth quadrant
Similarly, the same conclusions can be obtained as follows:
8
< sin a ¼ sinðp aÞ ¼ cos p2 ðp aÞ ¼ rþqoq cos h
s
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
: cos a ¼ 1 sin2 a ¼ qo sin h
qs
vsr ðr; hÞ ¼ vs ðr; hÞ cos a ¼ xs qo sin h
vsh ðr; hÞ ¼ vs ðr; hÞ sin a ¼ xs ðr þ qo cos hÞ
ðA9Þ
ðA10Þ
In summary, considering the slipper spin speed xz , the
radial velocity component and the circumferential velocity
component of an arbitrary point ðr; hÞ under the slipper can
be described as
vsr ¼ vsr ðr; hÞ ¼ xs qo sin h
ðA11Þ
vsh ¼ vsh ðr; hÞ xz r ¼ ðxs xz Þr þ xs qo cos h
References
1. Chakraborty I, Mavris DN, Emeneth M, Schneegans A. A
methodology for vehicle and mission level comparison of More
Electric Aircraft subsystem solutions: application to the flight
control actuation system. Proc Inst Mech Eng, Part G: J Aerosp
Eng 2015;229(6):1088–102.
2. Shi C, Wang XJ, Wang SP, Wang J, Tomovic MM. Adaptive
decoupling synchronous control of dissimilar redundant actuation
system for large civil aircraft. Aerosp Sci Technol 2015;47:114–24.
3. Wang J, Wang SP, Wang XJ, Shi C, Tomovic MM. Active fault
tolerant control for vertical tail damaged aircraft with dissimilar
redundant actuation system. Chin J Aeronaut 2016;29(5):1313–25.
4. Shi C, Wang SP, Wang XJ, Wang J, Tomovic MM. Active faulttolerant control of dissimilar redundant actuation system based on
performance degradation reference models. J Franklin Inst
2017;354(2):1087–108.
5. Song Z, Jiao XY, Shang YX, Wang S, Hu WN. Design and
analysis of a direct load sensing electro-hydrostatic actuator. 2015
International conference on fluid power and mechatronics; Piscataway: IEEE Press; 2015. p. 624–7.
6. Shi C, Wang S, Wang X, Wang J, Tomovic MM. Active faulttolerant controller design for load sensing electro-hydrostatic
actuator with optimal reference model. 2016 35th Chinese control
conference (CCC); 2016 July 27–29; Chengdu, China. Piscataway:
IEEE Press; 2016. p. 6437–42.
7. Zhang J, Chao Q, Wang Q, Xu B, Chen Y, Li Y. Experimental
investigations of the slipper spin in an axial piston pump.
Measurement 2017;102:112–20.
8. Zhang J, Chao Q, Xu B. Analysis of the cylinder block tilting
inertia moment and its effect on the performance of high-speed
electro-hydrostatic actuator pumps of aircraft. Chin J Aeronaut
2018;31(1):169–77.
9. Han L, Wang S, Zhang C. A partial lubrication model between
valve plate and cylinder block in axial piston pumps. Proc Inst
Mech Eng, Part C: J Mech Eng Sci 2015;229(17):3201–17.
10. Xu B, Wang Q, Zhang J. Effect of case drain pressure on slipper/
swashplate pair within axial piston pump. J Zhejiang Univ – Sci A
2015;16(12):1001–14.
11. Xu B, Zhang JH, Yang HY. Investigation on structural optimization of anti-overturning slipper of axial piston pump. Sci China
Technol Sci 2012;55(11):3010–8.
12. Xu B, Zhang JH, Yang HY, Zhang B. Investigation on the radial
micro-motion about piston of axial piston pump. Chinese J Mech
Eng 2012;26(2):1–9.
13. Koc E, Hooke C, Li K. Slipper balance in axial piston pumps and
motors. J Tribol 1992;114(4):766–72.
14. Koc E, Hooke CJ. Investigation into the effects of orifice size,
offset and overclamp ratio on the lubrication of slipper bearings.
Tribol Int 1996;29(4):299–305.
15. Koc E, Hooke CJ. Considerations in the design of partially
hydrostatic slipper bearings. Tribol Int 1997;30(11):815–23.
16. Kazama T, Yamaguchi A. Application of a mixed lubrication
model for hydrostatic thrust bearings of hydraulic equipment. J
Tribol 1993;115(4):686–91.
17. Bergada JM, Watton J, Haynes JM. The hydrostatic/hydrodynamic behaviour of an axial piston pump slipper with multiple
lands. Meccanica 2010;45(4):585–602.
18. Bergada JM, Kumar S, Davies DL, Watton J. A complete analysis
of axial piston pump leakage and output flow ripples. Appl Math
Model 2012;36(4):1731–51.
19. Borghi M, Specchia E, Zardin B. Numerical analysis of the
dynamic behaviour of axial piston pumps and motors slipper
bearings. SAE Int J Passenger Car: Mech Syst 2009;2(1):1285–303.
20. Hesheng T, Yaobao Y, Jing L. Lubrication characteristics analysis
of slipper bearing in axial piston pump considering thermal effect.
Lubr Sci 2016;28(2):107–24.
21. Schenk A, Ivantysynova M. A transient thermoelastohydrodynamic lubrication model for the slipper/swashplate in axial piston
machines. J Tribol 2015;137(3):031701.
22. Pelosi M, Ivantysynova M. Heat transfer and thermal elastic
deformation analysis on the piston/cylinder interface of axial
piston machines. J Tribol 2012;134(4):041101.
23. Pelosi M, Ivantysynova M. A geometric multigrid solver for the
piston–cylinder interface of axial piston machines. Tribol Trans
2012;55(2):163–74.
24. Zecchi M, Ivantysynova M. Cylinder block/valve plate interface –
a novel approach to predict thermal surface loads. 8th IFK
international conference on fluid power; 2012. p. 285–98.
25. Zecchi M, Ivantysynova M. An investigation of the impact of
micro surface shaping on the cylinder block/valve plate inter-face
performance through a novel thermo-elasto-hydrodynamic model.
The 7th FPNI PhD symposium on fluid power; 2012. p. 589–610.
26. Balaras E. Modeling complex boundaries using an external force
field on fixed Cartesian grids in large-eddy simulations. Comput
Fluids 2004;33(3):375–404.
27. Xie FW, Hou YF, Yang P. Drive characteristics of viscous oil film
considering temperature effect. J Fluid Eng 2011;133(4):044502.
28. Beschorner K, Higgs CF, Lovell M. Solution of Reynolds
equation in polar coordinates applicable to nonsymmetric entrainment velocities. J Tribol 2009;131(3):034501.