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International Journal of Mechanical Engineering and Technology (IJMET)
Volume 10, Issue 11, November 2019, pp. 3651, Article ID: IJMET_10_11_005
Available online at
ISSN Print: 09766340 and ISSN Online: 09766359
© IAEME Publication
VARIABLE STIFFNESS NONLINEAR
ISOLATOR: DESIGN, ANALYSIS AND
SIMULATION
T.D. Le
Department of Mechanical Engineering, Industrial University of Ho Chi Minh City
12 Nguyen Van Bao Street, Ho Chi Minh City, Viet Nam
Email: l
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ABSTRACT
It is difficult for a conventional linear isolator including an elastic element in
parallel with a damper to prevent the low frequency vibration band. Hence, this work
will design, analysis and simulate a variable stiffness nonlinear isolator (VSNI) with
air spring. The main feature of the VSNI is that the isolated object is supported by a
mechanism including wedgerollerair spring (named the main mechanismMM)
meanwhile the model of VSNI can obtain the lower resonance frequency and the
higher vibration attenuation than the equivalent linear isolation model (ELIM) but
guarantee the load bearing capacity through introducing a camrollerair spring
(named auxiliary mechanismAM). Because the pressure in air springs is the key
parameter which is used to adjust the stiffness of the MM and AM, the influences of
the pressure on the restoring force and the dynamic response of the VSNI are
presented. Furthermore, the effect of the isolated load on the isolation response of the
VSNI is investigated. For this purpose, the complex stiffness of the air spring will be
analyzed. Next, the motion equation of the system will be built. The numerical
simulation of the vibration transmissibility of the proposed model will be performed
through fourthorder RungeKutta algorithm.
Keywords: Nonlinear isolation, Air spring, Low frequency, Stiffness correction
Cite this Article: T.D. Le, Variable Stiffness Nonlinear Isolator: Design, Analysis and
Simulation. International Journal of Mechanical Engineering and Technology 10(11),
2019, pp. 3651.
1. INTRODUCTION
As commonly known, reducing the stiffness of a vibration isolator would produce the low
resonance frequency that extends the isolation band toward low frequency. However, this way
is difficult to carry out for a traditional linear isolator including a spring connecting with a
damper in parallel, because a reduction in stiffness will result in a large deflection and low
load capacity. This is also a major limitation of the traditional linear isolator for applying
widely in engineering practice such as vehicle suspension or protection of machinery,
equipment. Especially, high precision equipment because it is easily sensitive to external
Variable Stiffness Nonlinear Isolator: Design, Analysis and Simulation
37 editor@iaeme.com
vibration, shock for example instrumentation. Recently, the passive isolation method with low
static and high dynamic stiffness (LSHD) or quasizero stiffness (QZS) has been researched in
deep to improve the isolation effectiveness as well as to broaden the isolation range in
literature [1]. I. Kovacic et al. [2] analyzed the effects of the static force on the dynamic
response of the quasizerostiffness system and the stability of the steadystate response. An
isolation model using the negative stiffness structure for vehicle seat was analyzed and
simulated numerically by Le et al. [3], showing that this design model outperforms in
comparison with the equivalent linear model. Then, the experimental investigation of which
confirmed the theoretical model as shown in [4]. X.Ch. Huang et al. [5] studied the dynamic
response and stability of a highstatic and low dynamic stiffness isolator including an Euler
beam formed negative stiffness corrector paralleled with a conventional linear isolator. By
applying timedelayed active control strategy, the performance of the isolator with quasizero
stiffness would be improved as researched by X. Su et al. [6]. A novel dynamic model with
stablequasizero stiffness which was constructed by a positive stiffness component and a pair
of inclined linear springs providing negative stiffness was suggested by Hao et al. [7]. In
order to widen previous studies of the isolator with the characteristic of highstatic and low
dynamic stiffness, Shaw et al. [8] had indicated that simple changes in the shape of the force
displacement curve can have large effects on the amplitude and frequency of peak response,
and can even create unbounded response at the certain levels of excitation. An archetypal
dynamic model with quasizero stiffness which comprises a lumped mass denoting the
isolated object and a pair of the horizontal springs providing negative stiffness in parallel with
a vertical linear spring to bear the load was studied by Z. Hao et al. [9]. A multiDirection
QuasiZeroStiffness vibration isolator with timedelayed active control, which can be
realized excellent vibration isolation in three directions simultaneously was suggested and
analyzed by Xu et al. [10]. The effects of the equilibrium position on the dynamic response of
the isolation system using negative stiffness structure were analyzed by Le et al. [11]. In
addition, by integrating the linear mechanical spring and magnets, the highstaticlow
dynamic stiffness isolator was proposed and analyzed in [1213]. Besides, Q. Le et al. [14]
introduced a vibration isolator which is the combination between the magnetic spring and
rubber membranes to attain low natural frequency.
One of the main issues of the isolation method with the LSHD is the adjustment of the
stiffness according to the change of the isolated load. It can be realized by replacing the spring
or regulating the configurative parameters of the system. This adjustment or replacement may
cause difficulties for applying in practice. Hence, this paper will introduce an innovative
variable stiffness nonlinear isolator with air spring. The stiffness of the proposed model can
be adjusted via controlling the pressure in air springs so that VSNI can remain both the
desirable low stiffness at the wanted static equilibrium position and load capacity as well as
reduction in the static deformation. The proposed model can be used in the precision
fabrication field, instrumentation, etc. In addition, this isolation way can be employed in other
engineering fields such as the space antennae, satellites, isolation platforms, etc. The rest of
the paper is organized as follows. The configuration of the proposed system is presented in
section 2. Mechanicalpneumatic coupling model of the VSNI is analyzed in section 3.
Dynamic modeling of the VSNI is obtained and then the response simulation is carried out in
section 4.Finally, some conclusions are drawn in section 5.
2. CONFIGURATION OF THE VSNI
As shown in Fig. 1(a), the wedges (9) fixed on the vertical bars (8) along with rollers (3) work
as wedge mechanism. This mechanism combining with two air springs (1) is used to support
the load plate (4), which is named “main mechanismMM” indicated by dashedline
rectangle. Besides, the auxiliary mechanismAM plotted by the dotline rectangle, including
T.D. Le
38 editor@iaeme.com
semicircular surfaces (6) fixed on the vertical bars (8), rollers (5) and two air springs (2), is
introduced to modify the dynamic stiffness of the VSNI. Herein, the semicircular surface and
roller are considered as the cam mechanism. The load plate only moves in vertical direction
through the guide (7). During operation, the air springs are always compressed, it means that
the main mechanism offers positive stiffness, whereas, the auxiliary mechanism has the
vertically negative stiffness. The pressure in the air springs (1) can be adjusted to remain the
designed static equilibrium position when the weight of the isolated object is changed.
However, this adjustment may produce the negative or positive stiffness of the VSNI. To
remain low positive stiffness of the VSNI, the pressure of the springs (2) can be properly
regulated through a relationship between the pressure in the springs (1) and (2) which will be
discussed in next section. Fig. 1(b) presents the photograph of the VSNI. In order to reduce
the effect of the friction on the dynamic response, the linear bearings are installed in the
model.
Figure 1. (a) Physical model of the VSNI; (b) Photograph of the VSNI
3. MECHANICALPNEUMATIC COUPLING MODEL OF VSNI
3.1. Analysis of the Air Spring
Figure 2. Modeling of the air spring
Uncompressed
state
dEil, Til, Gil
dEol, Gol
Compressed
State
P,V,T
x
Fair
hd
hair
Dh
Variable Stiffness Nonlinear Isolator: Design, Analysis and Simulation
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The air spring model is considered as in Fig. 2. Without the heart change, the
thermodynamic equation in the air spring is described as following:
il ch ae oldE dE dE dE (1)
in which Eil and Eol are the air energies of input and output line, Ech is the air energy in
spring, Eae is the work of air expansion. These energies are given as following:
il p il il
ol p ol
ch v air v air
ae
dE C T G
dE C T G
dE C m dT C Tdm
dE PdV
(2)
where Cp and Cv are specific heat capacities at constant pressure and volume, respectively.
Til is the temperature of air at the inlet; mair, T and P are the mass, the temperature and
pressure of the air in the air spring, V is the volume of the air spring. Gil and Gol are mass low
rates at inlet and outlet.
From the ideal air equation, we have:
air gas
gas air
air
gas
PV m R T
PdV VdP R Tdm
m dT
R
(3)
Substituting Eq.(23) into Eq.(1), the air spring internal pressure equation is expressed as
below:
il il ol gas ol
n
P G RT G R T PV
V
(4)
where Rgas is the gas constant, n=Cp/Cv is the ratio of specific heat capacity.
Considering that the charging and discharging processes are not to occur, changing the
pressure in the air spring is given by:
P
P n V
V
(5)
The restoring force created by the air spring is obtained as following:
air atmF P P A (6)
Here in A is effective area of the air spring, Patm is the ambient pressure
By differentiating Eq.(6) versus the deformation of x, the stiffness of the air spring is
obtained as below:
airair atm
dF nAP dV dA
K P P
dx V dx dx
(7)
T.D. Le
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Let hd be the design height of the air spring. Adh, Vdh and Pdh are the effective area, volume
and pressure at the design height hd, respectively. Taking the linearization of the Eq. (7),
around this height, the stiffness of the air spring is approximated as following:
hd hd Vair dh atm A
hd
nA PdF dF
K P P
dx dh V
(8)
here, ;
d d
V A
h h h h
dV dA
dh dh
The linearized restoring force of the air spring can be rewritten as following:
air air dh atm dhF K h P P A D (9)
with air dh h h xD , hair is the height of the air spring at the uncompressed state, Pdh
and Adh are the pressure and the effectiveness area of the air spring at the design height,
respectively,
3.2. Restoring force and stiffness of the VSNI
In order to analyze the restoring force of the VSNI, the model of force acting on the wedge
and cam mechanisms is shown in Fig. 3. The dot line presents the initial position of the
system, meaning that at this position, air springs 1 and 2 are uncompressed. As the load plate
moves vertically down an amount of y, the result is that the both air springs are compressed
horizontally by an amount of x1 and x2 given by Eq. (10), respectively.
Figure 3. Modeling of force acting on the wedge and cam mechanisms
Ho
r
x2
y
x1
a
R
y
A
ir
s
p
ri
n
g
1
A
ir
s
p
ri
n
g
2
Fair1
Fair2
Variable Stiffness Nonlinear Isolator: Design, Analysis and Simulation
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1
2 2 2 2
2
tan (a)
(R r) ( ) (R r) (b)o o
x y
x H y H
a
(10)
in which R and r are the radii of the cam and roller, Ho is maximum displacement of the
load plate in vertical direction, a is the inclined angle of the wedge.
Consider a small vibration of the load plate around the design static equilibrium position
(shorten design positionDP) at which the load plate only is acted by the vertical restoring
force created by the air springs 1, the design height of the air springs 1 and 2 is:
1 1 0
2 2
2 2
tan (a)
R r (R r) (b)
d air
d air o
h h H
h h H
a
(11)
where subscripts „1‟ and „2‟ are designated for the air springs 1 and 2, respectively.
By using the Eq. (9), the restoring forces of the main mechanism FMM and the auxiliary
mechanism FAM are expressed as following:
12
1 12 tan 2 tan
h
MM air airF K y Fa a
D (12)
2
2 2
2 2
2 22 2
(R r)
2 1 2
(R r) (R r)
o h o
AM air o air
o o
H H y
F K H y F
H y H y
D
(13)
herein 11
h
airF
D
and 12
h
airF
D
are the forces of the air springs 1 and 2 at the initial position (y=0)
obtained as following:
1 1 1 1 1 1( )
h
air dh atm dh air air dF P P A K h h
D (14)
2 2 2 2 2 2( )
h
air dh atm dh air air dF P P A K h h
D (15)
in the analysis above,
2 2
1 1 2 2tan ; R r (R r)air d o air d oh h H h h Ha
By letting ou H y and introducing dimensionless parameters as below:
1
1 2
1 1 1
2 2
2
2 2
21 1
1
1 1
22 2
2 2
2
ˆ ˆ ˆˆ; ; ; ; ;
( ) ( ) (R r) (R r)
2
;B 2 ;
V
1 tan
ˆ1 1
C 2
(R r)
h
h d oMM MM
MM MM o
air air d
d V
d
atm d d
L d d atm
d
d d
d d
d d o
d
P HF F u
F F u H
K R r K R r P
nA
A
P A V
A
K nA P
A
V P
nA V
A A H
V
a
D
D
22
2
21 1
1
1 1
ˆ2 1 1
2
; ;
(R r)
1 tan
atm d o
atm d
L Ld d atm
d
d dh
P A H
P A
D
K KnA V P
A
V P
a
T.D. Le
42 editor@iaeme.com
Then, Eq. (1213) can be written in form of dimensionless as below:
1ˆ ˆ ˆˆ 2 tanhMM o MMF H u F aD (16)
2 2
2 2 2 2
ˆ ˆ1 1 ˆ1ˆ ˆ ˆ1 1
ˆ ˆ ˆ ˆ1 1 1 1
o o
AM
H H u
F B C u A u D
u u u u
(17)
The stiffness of the MM and AM can be attained by differentiating of Eq. (1617) versus
the dimensionless displacement uˆ , we obtain the dimensionless dynamic stiffness in vertical
direction as following:
Kˆ 1MM (18)
2 2 2 2
3 2 3 2
2 2
2 2 2 2
2 3 2 3
2 2
ˆ ˆˆ 1 1 ˆ 1
Kˆ A 1
ˆ ˆ1 1ˆ ˆ1 1
ˆ ˆˆ1 1 ˆ1
1
ˆ ˆ1 1ˆ ˆ1 1
o o
AM
o o
u H H u
D
u uu u
H u H u
B B C C
u uu u
(19)
Furthermore, to remain the design position, it is necessary to adjust the pressure in the air
springs 1 according to the change of the mass (M) of the isolated object as following
1
12 tan
dh atm
dh
Mg
P P
A a
(20)
in which g is gravity acceleration
3.3. Analysis of the restoring force
Considering the MM with the effectiveness area Adh1of the air springs 1 listed in table 1, Fig.
4 presents the relation between the isolated mass (M), the inclined angle (α) of the wedge and
pressure Pdh1 for which the VSNI would achieve the design equilibrium position. It can be
seen that in order to remain the DP, the inclined angle of the wedge or the pressure in the air
springs 1 should be increased or decreased in according with the rise or fall in the isolated
load, respectively. However, compared with the regulation of the inclined angle of the wedge,
the adjustment of the pressure in the air spring is easier to realize.
In addition, note that the load capacity of the isolated model is large for the low pressure
and the large inclined angle of the wedge. But in this case if the inclined angle is less than 10
o
,
to support the heavy load, the pressure in air springs of the MM must be very large that may
cause a difficulty for applying in practice. However, the case of the inclined angle of the
wedge larger than 45
o
results in the larger deformation of the air spring than the vertical
deformation of the load plate as given in Eq. (10a), which may cause a deformation exceeding
the limitation of the air spring.
Overall, it is necessary for the wedge to select its inclined angle to appropriate to a
specially practical application including the maximum deformation of the air spring,
maximum working pressure and the mass of the isolated object. For analysis purpose about
Variable Stiffness Nonlinear Isolator: Design, Analysis and Simulation
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the effects of the pressure in air springs on the stiffness of the MM and AM as well as the
dynamic response of the VSNI, the inclined angle of the wedge is chosen at value of 37
o
.
Figure 4. Relationship between the inclined angle α of the wedge, the mass M and pressure Pdh1
From Eq. (16), it is noted that the dimensionless restoring force of the main mechanism is
a linear function versus the displacement of the isolated object and is influenced by the
pressure in the air springs 1 at the design height. This is simulated in Fig. 5 for Pdh1=1.3, 1.5,
1.97, 2.5, 3.0, 3.5, 4.0 bar, the other parameters as listed in table 1. As observed, the slope of
the dimensionless restoring force of the ˆ
MMF which is defined as the ratio of the change in the
restoring force to the corresponding change in the displacement is the stiffness of the MM as
given by Eq. (18). The ˆ
MMF is varied according to the change of the pressure Pdh1 but the slope
of the curve of the ˆ
MMF is remained at the value of unity. This means that the pressure Pdh1
can be used to regulate the restoring force of the MM so that the system always obtains the
design position as there is a change in the isolated mass. In addition, the curve of the ˆ
MMF for
Pdh1=1.97 bar plotted by the solid line divides the plane ˆ ˆ,MMF y into two regions. In the
upper one, the ˆ
MMF is always positive, whilst, in the lower region, the
ˆ
MMF has negative value
when the vertical displacement begins from 0 to the specific position depending on the Pdh1 if
exceeding this position, ˆ
MMF will offer the positive value. For instance, Pdh1=1.3,
ˆ
MMF <0
within from 0 to 0.152 and out of this range ˆ
MMF >0. In order to guarantee the load bearing
function, the ˆ
MMF is always larger than zero in the vertical displacement region from 0 to
ˆ2 oH .
Therefore, the pressure Pdh1 in the air springs of the MM must be calculated to obtain the
positive restoring force of the ˆ
MMF in the displacement area within from 0 to
ˆ2 oH . In order to
guarantee this requirement, the restoring force of the MM at the initial position given in Eq.
(14) must be larger than zero.
P
d
h
1
(
b
a
r)
M
(Kg)
a (Deg
ree)
T.D. Le
44 editor@iaeme.com
Table 1 The parameters of simulation
Parameters Original values
R 60
r 20
1, 2,A A 0.11
1, 2,V V 0.0114
Adh1, Adh2 0.0105 m
2
Figure 5. Dimensionless restoring forces of the MM simulated by Eq.(16) for various values of Pdh1
(detailed notations of line types are seen in figure), another parameters given in table 1.
Fig. 6 shows the dimensionless restoring force curve ˆAMF of the auxiliary mechanism
through Eq. (17) for Pdh1=2.3 bar, the various values of the ratio () of the pressure Pdh2 to
Pdh1 (seeing details for notations of the types of lines and chosen values of in Figure), the
same other parameters as in Fig. 5, that are parameters given in table 1. It is interesting to
observe that at the design position, the ˆ
AMF
is always equal to zero. At the DP, the slope of the
curve of the ˆ
AMF exhibited by the dashed line is minus one for =1.53. As known, the slope of
ˆ
AMF is the equivalent the dimensionless stiffness of the auxiliary mechanism given by Eq.
(19), hence, with this value of the dimensionless stiffness of the AM is also equal to minus
one. If the value of exceeds 1.53, for instance, =1.8, the slope of the dimensionless force
curve of the AM denoted by the dasheddotdot line is smaller than 1, meanwhile, the slope
of which is quasizero for =0.42 (expressed by the solid line) and larger than zero for =0.35
(denoted by the dot line).
In addition, Fig. 7(a) and (b) present the dimensionless restoring force curves of the
auxiliary mechanism for various values of within 0.42<≤1.53. Herein, the chosen
parameters and the notations of the curves are given in detail in upperright panel of each
figures, the other parameters are the same as in Fig. 6. As observed, around the design
position, the restoring force of the auxiliary mechanism is decreased according to increasing
the vertical displacement, meaning that auxiliary mechanism offers the negative stiffness
around this position. However, with a smaller value of the AM will retain negative stiffness
0
0
0.8
0.2
Pdh1=1.97 bar
Pdh1=1.3 bar
Pdh1=1.5 bar
Pdh1=2.5 bar
Pdh1=3.0 bar
Pdh1=3.5 bar
Pdh1=4.0 bar
yˆ
ˆ M
M
F
0.152
0
ˆ2H
Variable Stiffness Nonlinear Isolator: Design, Analysis and Simulation
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over a smaller range of the vertical displacement as shown in Fig. 7(a). Furthermore, if the
value of Pdh2 is much greater than 0.42 for instance =1.4, 1.5, 1.53 the auxiliary mechanism
almost achieves the negative stiffness within 0 to ˆ2 oH as plotted in Fig. 7(b). In addition, the
larger the value of is, the lower the slope of the restoring force of AM is. Consequently, the
dynamic stiffness of the AM is decreased in accordance with the upturn in the pressure ratio.
Based on these analysis results, the dimensionless stiffness of the MM is always equal to
one regardless the change in the pressure of the air springs 1 but the stiffness of the AM is
always varied according to the change in the pressure of the air springs 2. This indicates that
the stiffness of the VSNI can be controlled to obtain the design position and the wanted low
stiffness at this position as the isolated load is varied. However, due to MM in parallel with
AM, the stiffness of the VSNI is linear combination of the stiffness of the MM and AM.
Reducing the dynamic stiffness of the AM would result in a decline of the dynamic stiffness
of the VSNI. Hence, to guarantee the load capacity, the dimensionless stiffness of the AM
must be larger than or equal to minus one. In the case of the dimensionless stiffness value of
AM being 1, the dynamic stiffness of the VSNI at the design position is equal to zero.
Figure 6. The displacementforce ˆ
AMF relation for the different values of (detailed annotation of
varied types of lines and selected values of are presented in upperright corner of the figure)
Figure 7. Restoring force curves of the auxiliary mechanism: (a) for =0.50, 0.55, 0.60, (b) for =1.40, 1.50,
1.53 (detailed annotations of various types of lines are presented in upperright corner of each figure)
0.4
0.0
0.4
=1.53
=0.42
=1.80
=0.35
Designed
position
yˆ
ˆ A
M
F
0
0
ˆ2H
0
ˆyˆ H
0.3
0.0
0.3
=1.53
=1.50
=1.40
yˆ
ˆ A
M
F
0
0
ˆ2H
(a) (b)
0.032
0.000
0.032
=0.50
=0.55
=0.60
yˆ
ˆ A
M
F
0
0
ˆ2H
Negative
stiffness region
Negative
stiffness region
T.D. Le
46 editor@iaeme.com
4. DYNAMIC MODELING AND RESPONSE SIMULATION OF THE
VSNI
As mentioned above, the restoring forces of the MM and AM as well as stiffness of which
depend on the pressure of the air springs 1 and 2, showing that the changes in the pressure
will inevitably affect on the dynamic response of the VSNI. Furthermore, the restoring force
of the AM offers strongly nonlinear characteristic. Hence, this section will express an accurate
dynamic model of the VSNI. Then, the complex dynamic response of which will be
performed through numerical simulation by using a fourthorder RungeKutta algorithm with
a fixed time step of 1/100 of the harmonic excitation period. The data of maximum amplitude
are sampled via using Poincaré sections [15].
4.1. Dynamic Modeling
Figure 8. Schematic diagram of VSNI. Specifically, the dashedline rectangle is presented for the
main mechanism. The dotline rectangle is representative for the auxiliary mechanism
In general, the VSNI is a one degree of freedom isolated model and shown in term of
simplified model as in Fig. 8. Herein, the isolated object (M) is supported by the main
mechanism with positive stiffness (KMM) denoted the dashedline rectangle which is
connected in parallel with the auxiliary mechanism offering the negative stiffness (KAM)
exhibited by the dotline rectangle. The stiffness of both mechanisms can be adjusted through
controlling the pressure of the air springs. Besides, C is the coefficient of damping to decay
the free vibration.
Consider a harmonic excitation Ze from the base as shown in Fig. 8 in which Z is the
absolute displacement of the isolated object.
Then, by applying the Lagrange equation, the dynamic modeling of the proposed system
is written as following:
24 tan MM AM e
e
Mu C u F F Mg Mz
z u z
a
(21)
M
KMM K AM
Z
Ze(t)
C
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4.2. Dynamic Response
a. The effect of the pressure ratio on the amplitudefrequency response
Consider the system given by Eq. (21), in which the pressure Pdh1 of the air springs 1 is 2.3
bar, the pressure ratio is varied from 0.42 to 1.53, other configurative parameters of the
VSNI are given in table 1. For these values, in order to vibrate around the design position the
load plate should support an isolated object having mass=102.86 Kg. The result is that the
vibration transmissibility curve of the model is plotted in Fig. 9 based on the integration of
Eq. (21) for the excitation amplitude of 10 mm and the excitation frequency swept from 0
to 50 rad/s. The detailed notations of the types of lines are presented in rightcorner of figure.
It confirms that when the value of is reduced from 1.48 to 0.75, meaning a corresponding
enhancement in the slope of the restoring force curve of the AM as shown in Fig. 6, the
resonance frequency and amplitude are increased. This result indicates that the VSNI will
offer a large isolation region corresponding to a great value of . Besides, the vibrated
attenuation of VSNI is upgraded according to the growth of .
Figure 9. Transmissibility of absolute displacement for M= 102.86, Pdh1=2.3 bar, various values of
(see more detailed notations for lines and selected values of in rightcorner of figure)
Especially, if =1.53, for which the slope of the restoring force curve of the AM at the
equilibrium position is 1 (seeing in Fig. 6), the isolated region and the vibration attenuation
rate of the VSNI denoted in Fig. 10(a) are larger than that of one considered in Fig. 9. In
addition, for this case, the peak amplitude of the VSNI is also reduced remarkably. But the
same excitation and simulation parameters as in Fig. 9 that are Pdh1, M, α, Ze, the isolated
effectiveness of the equivalent linear isolation model (ELIM) which is set by removing the
auxiliary mechanism from the VSNI in Fig. 1 is poorer than that of the VSNI as shown in Fig.
10(b). It can be observed that, the frequency for the vibration attenuation of the ELIM is
bigger than approximately 28 rad/s, whilst, the vibration attenuation of the VSNI having
=1.53 begins from 5 rad/s.
10 20 30 40 50
0
1
2
3
4
5
6
7
8
=1.42
=1.20
=0.98
=0.75
=1.48
De
cr
ea
se
o
f
Excitation frequency (rad/s)
Z
/
Z e

T.D. Le
48 editor@iaeme.com
Figure 10. The comparison of the vibration transmissibility of the VSNI having =1.53 (a) and ELIM (b).
b. The effect of the mass on the dynamic response
As analyzed above, it is evident that the advantages of the VSNI in the case of the accuracy of
the isolated mass, indicating that the VSNI achieves the design equilibrium position.
However, in fact, the weight of the isolated object can be changed or inaccurate. This can lead
to the effect on the isolation effectiveness of the VSNI. Hence, this subsection will be taken
account into the dynamic response when the VSNI is not to obtain the design equilibrium
position. By integrating Eq. (21) for the pressure ratio =1.31, M= 82.28 and 123.4 kg, other
parameters and the excitation are the same as in Fig. 9. The result is to obtain the vibration
transmissibility curve as shown in Fig. 11(a) for M=82.28 kg and in Fig. 11(b) for M=123.4
kg. It is interesting to observe that the frequency at which the vibration attenuation begins to
occur is approximately 17 rad/s for M=82.28 kg and 14 rad/s for M= 123.4 Kg. However, it
may exist a disadvantage that in isolated region, it appears a small area denoted by dashed
line ellipse called the amplified region in which the displacement of the isolated object can be
increased compared with the excitation.
This is verified through an excitation from the base with the amplitude of 10 mm, other
configurative parameters of the VSNI are remained as above. The first case, the isolated load
and the excitation frequency are 82.28 kg and 28 rad/s, respectively by which the time history
of the absolute displacement and the phase orbit are shown in Fig. 12(a) and 12(c),
meanwhile, Fig. 12(b) and 12(d) show the displacement of the isolated object and the phase
trajectory for the second case in which the former is at 123.4 kg and the later is at 23 rad/s. It
can be seen that both of cases of the excitation, the displacement of the isolated object is
bigger than the excitation. As observed, the amplitudes of the isolated object are
approximately 26 mm and 34 mm for the 1
st
and 2
nd
cases, respectively. Furthermor
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