Sensitivity Analysis of Design Parameters on Fluidlastic
Isolators Performance
J H Deng and Q Y Cheng
*
Science and Technology on Rotorcraft Aeromechanics Laboratory, China Helicopter
Research and Development Institute, Jingdezhen 333001, China
Corresponding author and e-mail: Q Y Cheng, qy_cheng@163.com
Abstract. Control of vibration in helicopters has always been a complex and challenging task.
The fluidlastic isolators become more and more widely used because the fluids are non-toxic,
non-corrosive, nonflammable, and compatible with most elastomers and adhesives. In the
field of the fluidlastic isolators design, the selection of design parameters is very important to
obtain efficient vibration-suppressed. Aiming at getting sensitivity of property of fluidlastic
isolator to design parameters, a dynamic equation is set up based on the theory of dynamics.
The orthogonal experimental method is used to analyze the parametric sensitivity of the
design parameters on the property of fluidlastic isolator. Two control indexes for design are
taken as the experimental indexes, and five parameters influencing the property of the isolator
are taken as the experimental factors. Arranged for the tests based on the orthogonal
experiment table, 2 indexes 6 factors orthogonal experiment is carried out. Range analysis is
adopted to study the sensitivity. The results show that for the combustion efficiency of
dynamic stiffness of fluidlastic isolator, the order of significance levels in turn decreases with
η, K
1
, c , L, ρ and K
2
respectively. For the combustion efficiency of dynamic stiffness of
fluidlastic isolator, the order of significance levels in turn decreases with η, K
1
, c, K
2,
ρ and L
respectively.
1. Introduction
Helicopter vibration is a critical aspect of helicopter design and a major reason for extended lead time
during the aircraft development phase. Control of vibration in helicopters has always been a complex
and challenging task. Increasing demands for expanding the flight envelop of helicopters, such as nap
of earth flying, high speed, high maneuvers, coupled with the need to improve system reliability and
reduce maintenance costs has resulted in more stringent vibration specifications.
Various methods have been applied to vibration control in the engineering field [1-5].
Traditionally, passive isolators and dampers are used to attenuate mechanical vibrations. The
traditional approach to passive vibration isolation is to install relatively soft springs or elastomeric
isolators to provide a primary low natural frequency [6-9]. These isolators would also incorporate
sufficient damping to control resonant response. Soft systems with primary natural frequencies well
below the N/rev exciting frequency are required to achieve isolation. Such systems result in large
relative motion between the pylon and the airframe due to static loads. Natural frequencies low
enough to isolate N/rev vibration would have static (1G) deflections up to 0.50 inches. Since flight
64
Deng, J. and Cheng, Q.
Sensitivity Analysis of Design Parameters on Fluidlastic Isolators Performance.
In Proceedings of the International Workshop on Materials, Chemistry and Engineering (IWMCE 2018), pages 64-70
ISBN: 978-989-758-346-9
Copyright © 2018 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
controls and power transmission drive shafts cross this interface it is advantageous to keep the
relative motion as small as practical. An effective method for isolating the N/rev vibration that did
not allow large relative motions between the pylon and airframe was needed.
There are many design parameters such as the density of fluid, area coefficient (R
1
), and rubber
performance that can affect the property of fluidlastic isolator [10-12]. Each of these parameters is
important considerations in the design of a fluidlastic isolator and affects property of fluidlastic
isolator to a greater or lesser degree. A key ingredient in developing reliable and efficient procedures
for design optimization of fluidlastic isolator is sensitivity analysis.
The paper discusses the sensitivity of property of fluidlastic isolator to design parameters such as
the density of fluid, rubber performance. These parameters are important consideration in the design
of a fluidlastic isolator and affects performance to a greater or lesser degree. The dynamic equation is
set up based on the theory of dynamics. The dynamic analysis is carried out. The effects of design
parameters on property of fluidlastic are studied by orthogonal experimental method.
2. Mathematical model
Figure 1 and Figure 2 show the cross-section and the mechanical model of Fluidlastic Isolator.
Figure 1. Cross-section of Isolator.
Figure 2. Mechanical model of Isolator.
According to the Figure 1 and Figure 2, the equations of motions (EOM) for the fluidlastic tuned
isolator can be expressed as:
d
xKxcxxmfxxiKxM
202021111
1
(1)
wd
fxcxKfxxiKxM
0221122
1
(2)
0020020
)( fAAAxKAAxcxxm
pdu
(3)
Where, c is the Viscosity coefficient of liquid, ηis Loss coefficient of rubber, f is the load that the
M
2
imposed on the lever mechanism. The A
p
, R
1
and R
2
is defined as:
Sensitivity Analysis of Design Parameters on Fluidlastic Isolators Performance
65
dup
AAAA /
0
(4)
(5)
d
d
x
x
A
A
R
0
0
2
(6)
When the vibration load is a harmonic excitation, the load and displacement can be written as
follows:
tj
tj
tj
w
eXx
eXx
Fef
22
11
(7)
The Transmissibility of vibration can be written as (8):
2
21
2
21
22
2
2
1
2
12
2
2
22
2
21
2
2
2
1
2
12
2
211
2
2
1
1
11
1
RiKRMRRjcRKRmR
RiKRRjcRKRRRm
X
X
T
(8)
When
,,,
11
cX
are all equal to zero, the frequency of undamped isolation system can be
obtained:
2
1
2
211
2
12
2
21
1
mRRR
RKRK
iso
(9)
The amplitude of vibration can be written as:
2
2
2
1
22
2
2
1
2
12
2
21
2
1
11
2
RjcRRRmRKRiK
FRR
X
(10)
From the equation (7), the dynamic stiffness of isolation system can be obtained:
2
2
2
2
1
22
2
2
1
2
12
2
21
2
11
R
RjcRRRmRKRiK
X
F
K
D
(11)
According to the static equation, the static stiffness of isolation system can be also obtained:
2
2
2
1
21
R
R
KKK
(12)
3. Parameter sensitivity analysis
Table 1 gives the design parameters of fluidlastic including the density of fluid, viscosity coefficient
of fluid, tuning port lengh, stiffness of rubber, loss coefficient of fluid, stiffness of (K
2
). The
boundary values of K
1
, K
2
, ρ, L, c and η are set at 5e6 to 1.5e7, 1e4 to 8e4, 1e3 to 5e3, 0.1 to 0.3,
0.05 to 5 and 0.05 to 5.
IWMCE 2018 - International Workshop on Materials, Chemistry and Engineering
66
Table 1. Design parameters of fluidlastic.
level
K1
K2
ρ
L
c
η
1
5e6
1e4
1e3
0.1
0.05
0.05
2
8e6
3e4
1.5e3
0.12
0.1
0.1
3
1e7
4e4
1.8e3
0.2
0.5
0.5
The sensitivity analysis of design parameters on property of fluidlastic are studied through
orthogonal experimental method. Five relevant factors including the density of fluid, viscosity
coefficient of fluid, tuning port length, stiffness of rubber, loss coefficient of fluid, stiffness of (K
2
)
are studied. The orthogonal table L
25
(5
6
) is designed and show in Table 2 based on five investigation
factors and five corresponding levels.
Table 2. Design of orthogonal table L
25
(5
6
) and analysis results.
No
K
1
K
2
ρ
L
c
η
T
K
D
/23.87
1
5e6
1e4
1e3
0.1
0.05
0.05
0.122
90.106
2
5e6
3e4
1.5e3
0.12
0.1
0.1
0.050
62.935
3
5e6
4 e4
1.8e3
0.2
0.5
0.5
0.143
138.419
4
5e6
5 e4
3 e3
0.25
1
1
0.342
253.485
5
5e6
8 e4
5 e3
0.3
5
5
0.532
474.724
6
8e6
1e4
1.5e3
0.2
1
5
0.205
743.486
7
8e6
3e4
1.8e3
0.25
5
0.05
0.199
146.667
8
8e6
4 e4
3 e3
0.3
0.05
0.1
0.316
156.815
9
8e6
5 e4
5 e3
0.1
0.1
0.5
0.173
969.695
10
8e6
8 e4
1e3
0.12
0.5
1
0.462
2504.493
11
1e7
1e4
1.8e3
0.3
0.1
1
0.345
3161.499
12
1e7
3e4
3 e3
0.1
0.5
5
0.279
1857.246
13
1e7
4 e4
5 e3
0.12
1
0.05
0.123
103.191
14
1e7
5 e4
1e3
0.2
5
0.1
0.392
596.747
15
1e7
8 e4
1.5e3
0.25
0.05
0.5
0.218
2251.392
16
1.2e7
1e4
3 e3
0.12
5
0.5
0.473
2930.703
17
1.2e7
3e4
5 e3
0.2
0.05
1
0.413
5674.165
18
1.2e7
4 e4
1e3
0.25
0.1
5
0.442
3353.537
19
1.2e7
5 e4
1.5e3
0.3
0.5
0.05
0.100
399.778
20
1.2e7
8 e4
1.8e3
0.1
1
0.1
0.478
944.904
21
1.5e7
1e4
5e3
0.25
0.5
0.1
0.335
551.430
22
1.5e7
3e4
1e3
0.3
1
0.5
0.613
5074.972
23
1.5e7
4 e4
1.5e3
0.1
5
1
0.949
10439.771
24
1.5e7
5 e4
1.8e3
0.12
0.05
5
0.743
5570.818
25
1.5e7
8 e4
3 e3
0.2
0.1
0.05
0.083
606.392
The range analysis is applied to clarify the sensitivity of design parameters on the property of
fluidlastic isolator including dynamic stiffness and transmissibility of vibration. First, the average
value of each experimental index of factors at five levels is calculated. Next, the range of each factor
of at five levels is also calculated. The influence coefficient of these six factors can be confirmed by
comparison with the value of range. The bigger the range is, the more the influence will be.
4. Results and discussion
A civil helicopter is chosen for this study. The N/rev frequency of this helicopter is close to 25Hz.
High loads and very small motions of high stiffness isolators are even more challenging.
Effectiveness of a tuned isolator in a system can be estimated by measuring the dynamic stiffness
Sensitivity Analysis of Design Parameters on Fluidlastic Isolators Performance
67
(K
D
) over the frequency range of interest. The K
D
value at the tuned (N/rev) frequency is a good
indicator of the effectiveness of the isolator in the system.
Fluidlastic isolators must also be designed to handle a specific range of input motions. The
dynamic stiffness versus frequency curve is shown in Figure 3. The dynamic stiffness varied with the
frequency. The dynamic stiffness varies from nominally 1.03E7 N/m statically (f = 0 Hz) to
approximately 1.91E6 N/m at 25 Hz. Effective dynamic stiffness at 25Hz is less than 20% of the
static stiffness. So, the “rigid” isolator can still provide effective performance. However, the static
stiffness is relatively high. It is advantageous to keep the relative motion between the pylon and
airframe as small as practical.
Figure 3. Fluid isolator stiffness versus frequency.
The range analysis is applied to clarify the sensitivity of design parameters on the property of
fluidlastic isolator including dynamic stiffness and transmissibility of vibration.
According to Table 2, it is showed that the design parameters have obvious influence on the
performance of isolator. The value of dynamic stiffness and transmissibility of vibration varies
widely at different assembly of design parameters. The result of transmissibility is in the range 0.05
to 0.949. Dynamic analysis has shown that fluidlastic isolator can reduce the vibration effectively.
The integration of these parameters is very important when selecting parameters for fluidlastic
isolators design.
The range analysis of sensitivity of design parameters on dynamic stiffness is shown in Table 3.
Table 3 summarizes the range analysis of the effect of different factor on the dynamic stiffness.
According to the results of constricting the R values of different factors, the significant sequence of
all the investigated influencing factors of design parameters was in turn. For the combustion
efficiency of dynamic stiffness of fluidlastic isolator, the order of significance levels in turn
decreases with η, K
1
, c, L, ρ and K
2
respectively.
IWMCE 2018 - International Workshop on Materials, Chemistry and Engineering
68
Table 3. Range analysis of influencing factors for T.
level
K1
K2
ρ
L
c
η
1
0.093
0.093
0.100
0.110
0.106
0.036
2
0.082
0.090
0.089
0.093
0.062
0.092
3
0.077
0.119
0.096
0.076
0.079
0.088
4
0.106
0.094
0.114
0.120
0.119
0.184
5
0.149
0.111
0.125
0.130
0.165
0.129
R
0.104
0.055
0.057
0.085
0.144
0.203
Sensitivity: η> K
1
> c >L>ρ>K
2
The range analysis of sensitivity of design parameters on transmissibility of vibration is shown in
Table 4. Table 4 summarizes the range analysis of the effect of different factor on the transmissibility
of vibration. According to the results of constricting the R values of different factors, the significance
sequence of all the investigated influencing factors of design parameters was in turn. For the
combustion efficiency of dynamic stiffness of fluidlastic isolator, the order of significance levels in
turn decreases on η, K
1
, c, K
2
, ρ and L respectively.
Table 4. Range analysis of influencing factors for K
D.
level
K1
K2
ρ
L
C
η
1
204
1495
2324
2860
2749
269
2
904
2563
2779
2234
1631
463
3
1594
2838
1992
1552
1090
2273
4
2661
1558
1161
1311
1424
4407
5
4449
1356
1555
1854
2918
2400
R
6369
2954
2418
2340
3484
6386
Sensitivity: η>K
1
>c>K
2
>ρ>L
Referring to the Table 2, Table 3 and Table 4, it should be noted that the property of fluidlastic
isolators can be affected by design parameters obviously. But the property of rubber (loss coefficient
of rubber and stiffness of rubber) is, by far, the most significant design driver.
However, six parameters are analyzed only in this paper. The intercoupling among different
parameters has not been considered. In order to obtain more accurate results, more parameters such
as tuning port length, area coefficient should be considered in the next study. The interrelationship
among these parameters will be also analyzed.
5. Conclusions
The dynamic equation is set up based on the theory of dynamics. The orthogonal experimental
method is used to analyze the parametric sensitivity of the design parameters on the property of
fluidlastic isolator. Findings are listed below:
(1) The fluidlastic isolator can reduce the vibration effectively. The integration of these
parameters is very important when selecting parameter for fluidlastic isolators design.
(2) The property of fluidlastic isolators can be affected by design parameters obviously. But the
property of rubber (loss coefficient of rubber and stiffness of rubber is, by far, the most significant
design driver.
(3) The order of significance levels in turn decreases with η, K
1
, c, L, ρ and K
2
respectively. For
the combustion efficiency of dynamic stiffness of fluidlastic isolator, the order of significance levels
in turn decreases with η, K
1
, c, K
2
, ρ and L respectively.
Sensitivity Analysis of Design Parameters on Fluidlastic Isolators Performance
69
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70