Red Sea University
Faculty of Engineering
Department of Mechanical Engineering
HYDROSTATIC
LUBRICATION
Applied Fluid Flow
Moataz Abdelgadir Ali Abdelgadir
General considerations
 A hydrostatic system is
composed by two surfaces:
 one smooth,
 the other having one or more
pockets (or recesses),
 One can characterize two
regions:
 a first one where the film
thickness h is thin (AB and
CD)
 and a second one, composed
by the pockets where the film
thickness is larger (BC).
Surfaces
a) constant flow
b) constant pressure supply
Fluid supply
 The inlet of the external fluid is
located in the recesses.
 To feed the fluid inside the
bearing, two methods can be
used :


a constant flow ( valid for liquids
only) (a). a pump with a constant
output is used). When there are
many pockets, every bearing
recess can be supplied with a
single pump or using constant
output adjustment.
a constant pressure (b). In this
config., a hydraulic resistance is
located upstream of the pockets.
The restrictors commonly used
are capillary tubes and
diaphragms.
The fundamental questions
 Hydrostatic bearings have a
wide range of characteristics and
need to be carefully controlled
for optimum effect.
 The following questions
summarize the potential
problems that an engineer or
tribologist might confront.




how can these films be controlled
and produced when needed?
What are the practical applications of
this type of lubrication?
What are the critical design
parameters of hydrostatic bearings?
What is the bearing stiffness and
how can it be controlled?
Advantages and limits
 The main disadvantages are
their price and their complexity.
Nevertheless, very often, an
existing pressure source can be
used to supply the bearing.
 Hydrostatic bearings, particularly
those operating with liquids,
have many advantages:
 both surfaces are always
separated by a fluid film, even
when they are standing still,
which means, theoretically,
zero wear, and guarantees a
long life. The stick-slip
occurrence is avoided.
Advantages and limits …cont…


the pressure is distributed over
a large surface, so there is no
pressure peak.
since the load carrying capacity
is not related to surface motion,
the consequences of
machinery errors are less
important for liquids.
HYDROSTATIC BEARING
ANALYSIS
Circular hydrostatic pad bearing
 The analysis of hydrostatic bearings
is much simpler than the analysis of
hydrodynamic bearings. It is greatly
simplified by the condition that the
surfaces of these bearings are
parallel.
 Consider, a flat circular hydrostatic
pad bearing with a central recess as
shown
Circular h.p. bearing … cont…
 Where its parameters are:







pr is the recess pressure [Pa];
h is the lubricant film thickness
[m];
η is the lubricant dynamic
viscosity [Pa s];
R is the outer radius of the
bearing [m];
R0 is the radius of the recess [m];
Q is the lubricant flow [m3/s].
n is the speed of the bearing
[rev/s]
Pressure distribution

The pressure distribution can be
calculated by considering the
lubricant flow in a bearing. For a
bearing supplied with lubricant under
pressure, the flow rate given by:

the flow through the elemental ring at
radius ‘r’ is (circular bearing):

integration yields (h ≠ f(r)):
Pressure distribution …cont.
 Boundary conditions are:
p = 0 at r = R
 Substituting into equation (6.2),
yields the constant ‘C’:
 Hence the pressure distribution
for this type of bearing in terms
of lubricant flow, bearing
geometry and lubricant
viscosity is given by:
(1)
Lubricant Flow

By rearranging equation (1), the
lubricant flow, i.e. the minimum
amount of lubricant required from the
pump to maintain film thickness ‘h’ in
a bearing, is obtained:

Since at r = R0 , p = pr then:
(2)

By substituting Q (eq. 2), the pressure distribution (eq. 1) is expressed
only in terms of the recess pressure
and bearing geometry, i.e.:
(3)
Load Capacity
 The total load supported by the
bearing is obtained by integrating the pressure distribution
over the specific bearing area:
(4)
 From pressure distribution
shown, the expression for total
load is composed of two terms;
one related to the recess area
and the other to the bearing
load area.
(5)
Load Capacity … cont.
 Since the recess pressure is
constant (5) is reduced to:
(6)
Load Capacity … cont.

Substituting for pressure (eq. 3) into
eq. (6)

Integrating by parts and sub-stituting
gives

The expression for the total load that
the bearing can support is:
(7)

where: W is the bearing load
capacity [N].
Friction Torque
 The frictional resistance of a
rotating hydrostatic circular pad
bearing consists only of friction
torque (usually very small) can
be cal-culated from:(6)
 In a similar for load, the
expression for total torque has
two components; one related to
the recess area and the other
to the bearing load area:
Friction Torque …. Cont.

The frictional resistance of a rotating
hydrostatic circular pad bearing
consists only of friction torque
(usually very small) can be calculated from:(6)

the shear stress is:
(8)

The friction force in its differential
form ‘dF’, also has two components;
one for the recess and the other to
the bearing load (land) area,
(9)
Friction Torque… cont.

Substituting for U = 2πrn and Assuming
constant viscosity and velocity and
integrating yields: integration eq. (8)
the Friction torque is:

The friction power loss which is
transmitted through the operating
surfaces is calculated from:
 Hf = Tω = 2Tπn
 where:


ω is the bearing angular velocity, ω =
2πn, [rad/s];
Hf is the friction power loss in the
bearing [W].
(10)
OPTIMIZATION OF H. BEARING DESIGN

The parameters of a hydrostatic
bearing, such as bearing area,
recess area, lubricant flow rate, etc.,
can be varied to achieve:

either maximum stiffness,
maximum load capacity for a
given oil flow

or minimum pumping power.

The H. bearing is almost entirely
under external control

It is possible to regulate the
characteristics of such bearings to a
far greater extent than for those of
hydrodynamic bearings.
Non-dimensional parameters.
 Equations (2) & (10) of flow & Load
Capacity can be re-written in the
following forms for flat circular pads
(and flat square pads) in terms of nondimensional load and flow times a
non-dimensional scale factor
 where:


(11)
A is the_ total pad area [m2];
Ā and B are non-dimensional load and
flow coefficients defined as:
(12)
 Design coefficients for flat
circular pad bearings
Minimization of Power

From H.P. bearing Figure, if the
recess is made almost as large as
the bearing diameter, then supply
pressure is maintained over virtually
the entire area of the bearing. This
would ensure a higher load capacity
than with a smaller recess but with
the disadvantage of requiring a very
high rate of lubricant supply pumping
power.
 The total power required is the sum
of friction power (10) and the
pumping power
(13)
Minimization of Power … cont.

Pumping power ‘Hp’ is defined as the
product of the lubricant flow ‘Q’ and the
recess pressure ‘pr’,
(14)


The total power describes the rate at which
the friction and pumping energies are
converted into heat in the bearing.
The heat dissipation rate is the product of
mass flow rate, specific heat and
temperature, thus:
(15)
Q : the lubricant flow [m3/s];
ρ : the density of the lubricant [kg/m3];
σ : the specific heat of the lubricant [J/kgK];
ΔT : the temperature rise [°C].
Ratio of friction to pumping power ζ
 This ratio of friction power to pumping
power ‘ζ’ (i.e. ζ = Hf/Hp) is used as a
measure of the proportion of the
hydrodynamic effects to the
hydrostatic effects.
 If the bearing is not rotating then ζ = 0
(because Hf = 0)
 Bearings operating with ζ ≥ 1 are
considered as ‘high speed bearings’
and with ζ « 1 as ‘low speed bearings’
 When ζ ≥ 3 then the hydrostatic and
hydrodynamic effects on load are of
the same order
Low Speed Recessed Bearings
 In low speed bearings Hf ≈ 0, thus Ht ≈
Hp and ζ = 0. The bearing geometry
can be optimized so that at a given
total bearing area, film thickness,
viscosity and applied load the
pumping power is a minimum.
 This is achieved by calculating
pumping power to load ratio, i.e.:
High Speed Recessed Bearings
 If the bearing is forced to operate at
high speeds, the effects of viscous
shear due to the relative motion
between the surfaces, may become
significant.
 The ratio of friction power to pumping
power & the total power are expressed
as:
 Substituting and rearranging yields
High Speed Recessed …. cont.
 For any value of ‘ζ’ it is still necessary
to find a minimum value of the ‘H’
parameter to optimize the bearing
geometry for maximum load and
minimum power.
 Other parameters such as  and h can
also be optimized.
 For example, the lubricant viscosity
can be optimized by calculating power
losses and load capacity for a range of
viscosities, while maintaining all the
other parameters at required design
levels.
High Speed Recessed …. cont.
 The optimum clearance is obtained
when the power ratio ζ = 3.
 The bearing gives the optimum
performance when the power ratio ‘ζ’
is between 1 ≤ ζ ≤ 3
 The effect of optimization is relatively
small since the difference between the
most and least effective procedures is
only about 15% of total bearing power
consumption
CFD Analysis
H. W. (1)
 A
circular
hydrostatic
pad
is
supporting a load of W = 1000 N, and
the upper disk has rotational speed of
5000 RPM. The disk diameter is 200
mm, and the diameter of the circular
recess is 100 mm. The oil is SAE 10
at an operating temp. of 70ºC, having
a viscosity of  = 0.01 N-s/m2. The
efficiency of the hydraulic pump
system is 0.6 and that of the motor
and drive system is 0.9. Optimize the
clearance, h0, for minimum total power
consumption.
 Hint

Ignore friction loss at recess

Use excel to plot total power