Dynamic Modeling of SFD Pressure Measurement System GU Zhiping , HE Xingsuo

Physical and Numerical Simulation of Geotechnical Engineering
7th Issue, Jun. 2012
Dynamic Modeling of SFD Pressure Measurement System
GU Zhiping1, 2, HE Xingsuo1, FENG Ningsheng3
1．Dept. of Engineering Mechanics, Northwestern Polytechnical University, 710072, P.R.China
2．Dept. of Civil Engineering, Xian Technological University, 710032, P.R.China,
3．School of Mechanical and Manufacturing Engineering, Univ. of NSW, 2052, AUSTRALIA
[email protected]
ABSTRACT: To study the dynamic behaviors of the squeeze film dampers (SFD), it is vital to know
the oil film pressures in the dampers. Considering the outer race length restriction, a system consisting
of capillaries and mass-springs at the other large piston ends is proposed to measure the pressures. By
relating the mass of the liquid in the capillaries as an additional mass to the piston mass at the end, a
simplified model of a mass-(viscous) damping-stiffness system for the measurement device is
established, and a new approach on indirect measurement of oil film pressure is provided and proven
with the aid of the similar theory and theorem of kinetic energy.
KEYWORDS: Squeeze-Film, Dynamic Modeling, Measurement System, Vibration, Theorem of
kinetic energy
1 INTRODUCTION
A major problem faced by designers and manufacturers
of high-speed rotating machinery is the vibration and
dynamic loads caused by rotor unbalance and self-excited
whirl. High shaft speeds and the use of long slender shafts
in modern multipool turbo engines have made the problem
more acute. Squeeze film dampers, which are probably the
most significant development of the last two decades
benefiting high-speed rotor dynamics, are well-known
devices used to control vibration characteristics of
rotor-bearing systems [1][2]. Experience has shown that the
problems described can be alleviated to a large extent by
the installation of one or more SFDs in a rotor-bearing
system. In its simplest form, the SFD is an annular
fluid-filled space surrounding the outer race of a
rolling-element bearing. The outer race of the
rolling-element bearing is fitted with a radial clearance in
the bearing support housing, in which radial and orbital
motion of the rolling-element bearing outer race is
permitted, but rotation is prevented by some type of
mechanical restraint. The annular clearance thus formed is
kept filled with oil, and hydrodynamic forces are generated
by the motion of the bearing under the influence of
unbalance or other excitation. The squeeze film acts as a
nonlinear spring and damper system which, if properly
designed, can significantly reduce the dynamic loads,
eliminate resonance, and suppress dynamic instability [3][4].
Although the SFD is an inherently stable machine element,
it is nevertheless important to emphasize here that in spite
of its inherent stable feature, its operation at certain
conditions or parameters may give rise to undesirable
non-synchronous vibration.
In order to investigate deeply the mechanism or reasons
for SFDs to alleviate the problems, the radial and axial
oil-film pressures in SFDs need to be measured. Because
© ST. PLUM-BLOSSOM PRESS PTY LTD
the oil-film pressures vary along axial distribution, several
measurement spots along axial direction on the
rolling-element bearing outer race should be used. Due to
the outer race length restriction; however, the pressure
sensors are not installed directly at those measurement
spots. Instead, the pressure measurements are made through
motions of pistons connected to the SFD via capillaries.
Therefore, this paper studies dynamic modeling of a
squeeze film damper oil film pressure measurement system,
analyses the dynamic response characters of the system,
and investigates the effect of the pressure conduction liquid
in capillaries on the measuring pressures in order to achieve
indirect measurements.
2 MODEL OF THE SQUEEZE FILM DAMPER
PRESSURE MEASUREMENT SYSTEM
Figure1 is a schematic of a squeeze film damper oil film
pressure measurement system, which consists of three parts:
(a) squeeze film damper; (b) capillary; and (c) mass-springs
at the other large piston end. Since measurement of the
mass block motion is very convenient, this paper tries to
find the pressure in the squeeze film damper by measuring
motions of the mass block at the other large piston end. To
achieve this, the measurement system needs to be modeled
so that the liquid mass in the capillary can be transformed
to the mass block at the large piston end under the
condition that the system responses and the other
conditions remain unchanged. The liquid in the capillary
can then be considered to transmit pressures only.
According to the theorem of kinetic energy of differential
form [5] ， the following equation can be written in
connection with the system:
dT  We + Wi
(1)
Dynamic Modeling of SFD Pressure Measurement System
DOI: 10.5503/J.PNSGE.2012.07.021
Squeeze Film Damper
(Block)
Capillary
P
r
Pistono end
b
(c)
Pump
Oil
(b)
(a)
Disp.
probe
Figure 1 Schematic of a squeeze film damper oil film pressure measurement system
Where
and
T
Wi
is the entire kinetic energy of system;
dTm  P * (t ) A2 dx  kxdx
We
are respectively the work of all external and
In the above equations,
internal forces acting on the system shown as Fig.1. When
the motion of the liquid in capillary is a lamina flow,
interaction forces between any two layers are action and
reaction, so that
coordinate of the mass block;
capillary wall. Since the
incompressible, one can obtain
flow state of liquid is in capillary, one always gets
*
0
is a tangent stress on
liquid
in
capillary
is
(7)
A1ds  A2 dx
From eqns. (5)-(7), the differential kinetic energy of the
total measurement system may be given by
A
(8)
dT1  d (Tl  Tm )  P(t ) A2 dx  kxdx   0Dl 2 dx
(2)
No matter what flow states of liquid in capillary, for an
infinitesimal displacement ds the work done by the
external forces is
*
(3)
Wel = P(t ) A1 ds  P (t ) A1 ds   0D lds
A1
where T1 is the total kinetic energy of the primary
system.
Let the system shown as Fig.2 be a simplifying system of
the primary measurement system shown as Fig.1. In the
simplifying system, the liquid in capillary is massless
because the liquid mass has been transformed to the mass
block at the other large piston end. Except the liquid mass,
all the boundary conditions of two systems are completely
same, such as damping of capillary wall, motion of mass
block, etc.
With regard to the mass block, for an infinitesimal
displacement dx corresponding to ds the work done
by the external forces is
Wem = P* (t ) A2 dx  kxdx
are respectively
is the length of capillary; P (t ) and P (t ) are
respectively oil pressures in the squeeze film damper and
the other large piston end; x (t ) is displacement
0. Therefore, no matter what
Wi  0
A2
l
in capillary is a turbulent flow, the speed distribution of the
liquid in capillary can be considered as even distribution [6].
Thus, the total liquid in the capillary can be considered as a
Wi =
and
the cross sections of capillary and large piston end; D is
a diameter of capillary; k is stiffness coefficient of spring;
Wi  0 . When the motion of the liquid
rigid body, so that
A1
(6)
(4)
According to the equations (1), (2), (3), the differential
kinetic energy of the liquid in capillary is
(5)
dTl  P(t ) A1 ds  P * (t ) A1 ds   0Dlds
According to the equations (1), (2), (4), the differential
kinetic energy of the mass block at the other large piston
end is
Figure 2 Schematic of simplifying measurement system
dTm  F * (t ) A2 dx  kxdx
For the simplifying system, similar equations can be
written
(9)
dTl  0  P(t ) A1ds  F * (t ) A1ds   0Dlds
(10)
*
In eqns. (9) and (10), F (t ) expresses liquid
pressures at the other large piston end shown as Fig. 2.
105
Physical and Numerical Simulation of Geotechnical Engineering
7th Issue, Jun. 2012
Adding eqn. (9) to (10) and using the relationship in eqn. (7)
result in
A
(11)
dT2  d (Tl  Tm )  P(t ) A2 dx  kxdx   0Dl 2 dx
A1
capillary, the
m
can be calculated respectively:
2.1 Lamina flow state of liquid in capillary
When viscous liquid flows in a capillary under laminar
flow state, the liquid speed distribution, shown as Fig. 3, is
expressed by: [6]
where T2 expresses the total kinetic energy of the
simplifying system.
Comparing eqn.(8) with (11), it is known that the kinetic
energies of the above two system are equal, that is
  2r  2 
v  v max 1    
  D  
T1  T2
(12)
In order to decide completely the simplifying system, the
additional mass m shown as Fig.2 has to be obtained.
According to three different motion states of the liquid in
(13)
Figure 3 Liquid speed distribution under lamina flow
Thus, the kinetic energy of the liquid in capillary under the
lamina flow state is
D/2
1
1
(14)
Tl     2  r  dr  l  v 2    A1  l  v m2 a x
2
6
0
where

is the liquid density.
Let
va
be the average speed of the liquid in capillary.
liquid speed distribution can be considered as uniform:
v  va  vmax
T1 
and
Thus, the additional mass of the mass block under turbulent
flow state is:
D/2
(16)
0
m 
From eqns.(13) and (16), one can get:
vm a x 2va
A22
l
A1
(22)
(17)
2.3 Transitional flow state of liquid in capillary
According to eqns. (14), (15), (16) and (17), the total
kinetic energy of primary measurement system can be
obtained:
T1  Tm  Tl 
(21)
1
1 2 1 A22
2
mx 
lx 2 = (m  m) x
2
2
2 A1
(15)
 2  r  dr  v  A1va
1 2 1 A22
mx 
lx 2
2
2 A1
From eqns.(12) and (21), one has:
Since the liquid is incompressible, one has:
A1va  A2 x
(20)
Therefore, the total kinetic energy of primary measurement
system is:
1 2 2 A22
mx 
lx 2
2
3 A1
Common transitional flow state can be considered as the
one between the two extreme lamina flow and turbulent
flow. As a result, the additional mass of the mass block
under transitional flow state should be:
(18)
From eqns.(12) and (18), one can get:
m  h
2
2
1 2 2A
1
mx 
lx 2  (m  m) x 2
2
3 A1
2
1 h 
4
3
(23)
In general, the additional mass of mass block under
various flow states is
The additional mass of the mass block under lamina flow
state of the liquid in capillary is expressed as:
4 A22
m 
l
3 A1
A22
l
A1
m  h
(19)
A22
l
A1
1 h 
4
3
(24)
After obtaining the additional mass m , the simplified
system shown as Fig.2 is determined. The liquid in
capillary is considered as massless and only transmitting
pressures. The effect of liquid mass has been transformed
2.2 Turbulent flow state of liquid in capillary
When liquid in capillary is under turbulent flow state, the
106
Dynamic Modeling of SFD Pressure Measurement System
DOI: 10.5503/J.PNSGE.2012.07.021
to the mass block. Thus, the measurement system has been
greatly simplified.
From eqn.(30), it can be seen that for any flow state, the
pressures in squeeze film damper can be obtained by
measuring the motions of mass block and the pressures at
the other large piston end.
When viscous liquid in a capillary is in lamina flow state,
we can further simplify eqn.(30). According to Ref. “[6]”:
8
(31)
0 
va
D
where  is the viscosity. From eqns.(15) and (31), one
has:
3 DYNAMIC MODELING OF MEASUREMENT
SYSTEM
Once the liquid in capillary is modeled, the dynamic
model of the measurement system is easily established.
With regard to the simplifying measurement system,
according to Newton’s second law one has:
(25)
F * (t ) A2  kx  (m  m) x
P(t ) A1  F (t ) A1   0Dl  0
*
0 
(26)
2


D
*
 P(t )  F (t )  Dl 0
2
 

 
Where C  8l  A2 
f
 
(28)
2
(34)
 A1 
From eqn.(33), it is seen that when the liquid in capillary
is under a lamina flow state, the pressures in squeeze film
damper P (t ) can be obtained by measuring only the
motions of mass block.
(29)
From eqns. (27) and (29), one can get:
m  mx  kx  a1 P(t )  a2 F * (t )
(32)
From eqns.(27), (28) and (32), the relationship between
the motion of system and the pressure in squeeze film
damper under the lamina flow state becomes
(33)
(m  m) x  C f x  kx  A2 P(t )
From eqns. (25) and (26), the relationship of the motion
of system, the shear stress on capillary wall and the
pressure in squeeze film damper is:
(27)
(m  m) x  C 0  kx  A2 P(t )
where
C  D l A2 / A1
According to Ref. “[5]” and “[6]”:
8A2
x
DA1
(30)
where
a1  A2 
a2 
DC
4l
DC
4l
,
a1P(t)+a2F*(t)
k
Figure 4
model for
flow state
Analytical
general liquid
.
Cf
Figure 5 Analytical model under liquid lamina flow state
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Dynamic Modeling of SFD Pressure Measurement System
DOI: 10.5503/J.PNSGE.2012.07.021
from eqn.(38):
Figure 4 shows the dynamic model of the measurement
system in general flow conditions, while Fig. 5 shows the
model for lamina flow state. Both the dynamic models are
quite simple, so will be the analysis of the dynamic
characteristics of the measurement systems.
k
m  m

(40)
From eqn.(39), one gets the magnifying coefficient:
k (m  m)
(41)
Cf
From eqn.(41), it is seen that the magnifying coefficient
increases with increase of the additional mass, but reduces
 max 
4 DYNAMIC CHARACTERISTIC ANALYSIS OF
THE MEASUREMENT SYSTEM
with increase of the coefficient
From eqn.(30) and the dynamic model shown as Fig.4,
when the liquid in capillary is under any flow state, the
resonance frequency  of measurement system is:
k
k
(35)


m  m
hA2 l
m
2
magnifying coefficient
A1
; when liquid flow speed is high and liquid
3 A1
is under turbulent flow state, h  1 , m = A22 l
. Since
A2 , 
l
and
A squeeze film damper oil film pressure measurement
system is simplified into a linear system with a mass,
spring and damper, so that the analysis of the dynamic
characteristics of the measurement system is greatly
simplified.
By investigating dynamic modeling of the squeeze film
damper oil film pressure measurement system, a new
approach on indirect measurement oil film pressure is
provided. For viscous liquid in capillary under general flow
state, both motions of mass block and pressures at the other
large piston end need to be measured in order to obtain the
pressures in the squeeze oil film damper. When viscous
liquid in a capillary is under lamina flow state, the
pressures in the squeeze film damper can conveniently be
obtained only by measuring the motions of mass block.
Finally, when the frequency of working pressures in the
squeeze film damper approaches the resonance frequency
of measurement system, sizable amplitude motions are
generated so that the measurement system becomes invalid.
Therefore, in designing the measurement system, the
resonance frequency of measurement system should be
avoided to approach the frequency of working pressure in
the squeeze film damper by selecting the system
parameters.
A1 , but decreases with increases
. When the frequency of working
pressures P (t ) in the squeeze film damper approach the
resonance frequency  of system, sizable amplitude
motion will be generated so that the measurement system
becomes invalid. Therefore, in designing the measurement
system, the resonance frequency  should be avoided to
approach the frequency of working pressure P (t ) .This can
be done by selecting system parameters, such as m,
k , A1 , A2 , l and  .
When the liquid in a capillary is under lamina flow state,
the motion equation of measurement system can be further
simplified from eqn.(30) into eqn.(33), corresponding to the
model of measurement system shown as Fig.5. It is a linear
system with a mass, spring and damper. Eqn.(33) can be
changed further into the following equation
x +2 px + p 2 x  X 0 pP(t )
(36)
Where

Cf
2 k (m  m)
k
, X  A2 p
0
k
m  m
,p
(37)
According to Ref.[14], the resonance frequency of system
is:
  1  2 2
k
m  m
REFERENCES
(38)
[1].
Now, the maximum magnifying coefficient of amplitude is:
 max 
1
2 1  
(39)
[2].
2
If damping coefficient

is very small ( 
decreases with increase of
5 CONCLUSIONS
flow states, from lamina flow to turbulent flow, the
additional mass decreases with the increase of liquid flow
speed, and the resonance frequency increases with the
increase of liquid flow speed. From eqn. (35), it is seen that
the resonance frequency  increases with increase of the
of
 max
A1
the lamina flow and turbulent flow are the two extreme
cross section of capillary
m
the flow speed of liquid in capillary, so is the amplitude of
mass block.
In addition, from eqn.(36), it is obvious that the pressures
P (t ) in the squeeze film damper can be obtained by
measuring only the motions of mass block at the other large
piston end. This is of extreme importance because an
indirect measurement of oil film pressure can be achieved
with the aid of eqn.(36). Of course, the resonance of
measurement system should be avoided.
3
2
. Because
decreases from lamina flow to turbulent flow, the
According to the above discussion, when liquid flow
speed is low and liquid is under lamina flow state, h  4 ,
m  4 A2 l
Cf
 1 ), one has
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