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To be submitted to Journal of Automobile Engineering
DYNAMIC ANALYSIS AND CONTROL OF ACTIVE ENGINE MOUNT SYSTEM
by
Yong-Wook Lee and Chong-Won Lee
Center for Noise and Vibration Control (NOVIC)
Department of Mechanical Engineering
Korea Advanced Institute of Science and Technology
Science Town, Taejon 305-701, Korea
July 27, 2002
The number of figures : 15
The number of tables: 2
Postal Address
Professor Chong-Won Lee Address : Center for Noise and Vibration Control (NOVIC)
Department of Mechanical Engineering
Korea Advanced Institute of Science and Technology
Science Town, Taejon 305-701
South Korea
Phone : +82-42-869-3016
FAX : +82-42-869-8220
E-mail : [email protected]
Yong-Wook Lee
Address : NVH Team, Research & Development Division for
Hyundai Motor Company
772-1, Jangduk-Dong, Whasung-Si, Gyunggi-Do 445-706
South Korea
Phone : +82-31-369-6204
FAX : +82-31-369-6095
E-mail : [email protected] or [email protected]
Abstract
Dynamic characteristics of a prototype AEM, designed based on a hydraulic
engine mount, has been investigated and an adaptive controller for the AEM has been
designed. An equivalent mass-spring-damper AEM model is proposed, and the
transfer function that describes dynamic characteristics of the AEM is deduced from
mathematical analysis on the model. The damping coefficient of the model is derived
by considering nonlinear flow effect in the inertia track. Experiments confirmed that
the model precisely describes dynamic characteristics of the AEM. An adaptive
controller using the filtered-X LMS algorithm is designed to cancel the force
transmitted through the AEM. The stability of the LMS algorithm is guaranteed by
using the secondary path transfer function derived based on the dynamic model of the
AEM. The performance test in the laboratory shows that the AEM system is capable
of significantly reducing the force transmitted through the AEM.
Key Words
active engine mount (AEM), modeling, filtered-X LMS algorithm, stability
NOMENCLATURE A : transfer function for passive characteristics of active engine mount
dA : area of decoupler
eA : equivalent cross-sectional area of upper chamber
pA : area of the magnetic pole
tA : cross-sectional area of inertia track B : transfer function for active characteristics of active engine mount C : transfer function for electro-magnetic actuator: current vs. displacement c : damping coefficient for the flow in inertia track D : secondary path transfer function D : estimated secondary path transfer function
hD : hydraulic diameter of inertia track F : force from electro-magnetic actuator
1F : transmitted force due to engine vibration
2F : transmitted force due to actuator vibration
eF : engine excitation force
TF : total force transmitted to chassis f : friction factor for flow in inertia track G : transfer function for electro-magnetic actuator: voltage vs. displacement
0g : nominal gap of electro-magnetic actuator I : electric current in the coil of electro-magnetic actuator cI , : command current input to electro-magnetic actuator ci
0I : bias electric current in the coil of electro-magnetic actuator i : control current for electro-magnetic actuator
bK : bulge stiffness of main rubber
cK : gain of current amplifier
dK : derivative gain of the feedback controller
iK : current stiffness of electro-magnetic actuator
pK : proportional gain of the feedback controller
rK : main stiffness of main rubber
yK : position stiffness of electro-magnetic actuator
εK : compliance of lower chamber
eL : minor head loss at inlet and outlet of inertia track
tL : length of inertia track l : coordinate along inertia track m : mass of fluid in inertia track
em : mass of engine
rm : mass of the runner in electro-magnetic actuator N : number of coil turns P : pressure in fluid 1P : pressure in upper chamber
2P : pressure in lower chamber R : reference signal for filtered-X LMS algorithm
eR : Reynolds number u : flow velocity in inertia track
cV : control signal input to current amplifier
kW : coefficient vector for adaptive controller X , x : displacement of engine
dX , : displacement of actuator dx
ex : equivalent deformation of bulge stiffness
tX , : displacement of fluid in inertia track txY , : position of the runner in electro-magnetic actuator yZ : vector of filtered reference signal µ : convergence coefficient of filtered-X LMS algorithm
0µ : absolute permeability ρ : density of fluid
cτ : low-pass filtering time constant of current amplifier
1. INTRODUCTION
Automobiles are getting more important as being not only a transportation
means but also a part of modern life. Especially, passenger cars are considered as a
part of space for everyday life so that comfort is one of the most important criteria in
the market. As a result, the noise, vibration and harshness (NVH) characteristics are
seriously taken into account in designing an automobile. On the other hand, there is a
strong requirement on fuel-efficient vehicles: customers want to reduce the
expenditure on their cars, and governments want to control the air pollution from cars.
To improve fuel efficiency, makers are trying to improve engine performance and cut
down the weight of automobiles, but in general the weight reduction is harmful to the
NVH characteristics [1,2]. Hence, active noise/vibration control technologies are
extensively developed to resolve this problem.
Engines are one of the most important sources of noise and vibration in
automobiles, so the isolation of engine vibration is critical to the improvement of
NVH characteristics. Engine mounts have to meet two contradictory functional
requirements: effective vibration isolation and firm engine support [3]. The primary
way to cut off paths of noise and vibration from the engine is to use soft mounts.
However, engine mounts must also constrain or control engine excursions caused by
rough roads, firing in cylinders, wheel torque reactions, etc. To limit engine motions,
engine mounts should be stiff and heavily damped. These conflicting demands on
engine mounts have prompted automotive industries to search for a new engine
mounting method.
Hydraulic engine mounts [4-6] have been promising alternatives to conventional
rubber mounts due to their capability of creating frequency dependent damping, but
they had limitations due to their pre-determined dynamic characteristics. To improve
the characteristics of hydraulic mounts, a number of adaptive engine mounts were
proposed [7,8] but they still had the limitation that they could only reduce the
vibration to some extent but not cancel it out. Recently, AEMs [9-15] appeared, which
were equipped with high-power, high-speed actuators to generate secondary vibration
so that it destructively interferes with the primary vibration from the engine.
However, most of the previous work on the AEMs concentrated on control schemes
so there is little material that thoroughly analyzes dynamic characteristics of AEMs.
In this paper, we present a dynamic model of a prototype AEM and the
influence of the model on the stability of adaptive control algorithm. The dynamic
behavior of the AEM is analyzed by using an equivalent mass-spring-damper model,
where the damping coefficient is derived by considering nonlinear flow effect in the
inertia track. An adaptive controller is designed by using the filtered-X LMS
algorithm [16-21] to cancel out disturbance forces from engines. The dynamic model
of the AEM was experimentally verified, and the AEM showed good performance in
canceling the force transmitted to the base of the AEM.
2. STRUCTURE DESIGN AND MODELING OF AEM
Two conditions are usually imposed in designing an AEM: the AEM should
work as a passive mount under abnormal situations such as actuator, controller and/or
sensor malfunctioning, and the power consumption of the AEM should be reasonably
low so that the power to operate the AEM can be supplied in the automobile [10]. To
reduce power consumption, the engine weight should be supported by a passive spring
element, and to guarantee the performance as a passive mount, another spring element
should be inserted between the engine and actuator so that the spring can alleviate
shocks due to the malfunctioning of AEM components. Figure 1 illustrates these
design concepts, introducing two springs: one supports the engine weight and the
other connects the engine and the actuator.
Figure 2 shows the structure and equivalent model of a hydraulic engine mount
[5,6]. Hydraulic engine mounts are intended to introduce stiffness and damping that
are dependent to the vibration amplitude and frequency so that the engine mounts
would have different stiffness and damping for different operating conditions. The
hydraulic engine mount is comprised of a main rubber, two fluid chambers, an inertia
track, and a decoupler as shown in Fig. 2(a), and Fig. 2(b) is its equivalent model. The
main rubber bears engine load in two different ways: one is due to the vertical
deflection of the engine and the other is due to the volumetric change in the upper
chamber, hence it is modeled via two stiffness elements Kr and . The main
stiffness element,
Kb
Kr , models the reaction due to the vertical deflection so that it
supports the static as well as dynamic load, and the bulge stiffness element, ,
models the reaction due to volumetric change in the upper chamber so that it supports
the dynamic load only. The engine vibration forces the fluid in the upper chamber to
flow between the upper and lower chambers through the inertia track, where the flow
Kb
velocity in the track is much faster than that in the two chambers because of the small
cross-sectional area of the inertia track. Hence there exist considerable inertia and
damping which were modeled as and c . m
t
Remarkable similarities exist between the conceptual model of AEM in Fig. 1
and the equivalent mass-spring-damper model of hydraulic engine mounts in Fig.
2(b). Both have two spring elements, and one spring in each model supports the
engine weight. Moreover, the second spring in Fig. 2(b) is not directly connected to
the chassis but to the decoupler, just as the second spring in Fig. 1 is connected to the
actuator. These facts strongly imply that the structure in Fig. 1 can be realized by
installing an actuator in the lower chamber of the structure in Fig. 2 and connecting it
to the decoupler as shown in Fig. 3.
Figure 3(a) shows the structure of the AEM, which is an amalgamation of the
basic structure of the hydraulic engine mount and an actuator system. But, due to the
actuator, the role of the decoupler is changed to a piston so that it transmits the force
from the actuator to the engine and chassis through the upper chamber. The role of the
inertia track is also changed: in hydraulic engine mounts, the inertia track generates
frequency dependent stiffness and damping [6], but in the active engine mount, it just
relieves the static pressure in the upper chamber. Accordingly, the model for
describing the dynamic behavior in the upper chamber is changed from the link with
clearance as in Fig. 2(b) to the piston-cylinder structure as in Fig. 3(b). In Fig. 3(b),
is the equivalent cross-sectional area of the upper chamber (see Appendix A),
is the decoupler area and is the cross-sectional area of the inertia track. The
compliance of the lower chamber is modeled as . The compliance of the lower
chamber is usually much smaller than
eA
dA A
εK
rK or so it is frequently neglected in
many hydraulic engine mount models, but we included it for better accuracy. The
bK
equivalent model of the AEM in Fig. 3(b) shows that there are two paths for force
transmission in the AEM: one is through the main stiffness element Kr and the other
is through the bulge stiffness element and the actuator. The force transmitted
through each path is designated as and , respectively.
bK
Fe
1F
(txe
2F
)(t=&&
)t
)t(1F)(tT +=
)1F xKr=
P
)t2F PAe=
{ (xe)(tP Kb
Figure 4 shows free body diagram of the engine-AEM system of which equation
of motion is given as
)() tFm T− (1)
where is mass of the engine, is displacement of the engine, is
engine excitation force coming from gas pressure in cylinders and reciprocating parts
of the engine [22], and
em (x )(tFe
)(tFT is the force transmitted to the chassis that is the sum of
and , i.e. )(1 tF )(2 tF
.)(2 tFF (2)
Since the main stiffness transmits the engine vibration to the chassis, is given
as
)(1 tF
)(( tt . (3)
The actuator vibration affects the pressure t( ) in the upper chamber that exerts
force on the chassis as
)(( t . (4)
Since the bulge stiffness element deforms in proportion to the pressure in the upper
chamber, we have
})() txtAe −= (5)
where is the deformation of the bulge stiffness element. The pressure also
forces the fluid in the inertia track to flow, which is expressed as
)(txe
)()()()( txKtxctxmtPA tttt ε++= &&& (6)
where is the displacement of the fluid in the inertia track. Here, the damping
coefficient is a variable depending on the flow velocity , and detailed
analysis on the equivalent damping coefficient and mass will be given in the next
section. The fluid can be considered as incompressible, thus the continuity equation
becomes
)(txt
c )(txt&
)()()( txAtxAtxA ddttee =+ . (7)
From Eqs. (2) to (7) expressed in Laplace domain, we can express the transmitted
force in terms of the engine and actuator displacements as
)()()()()( sXsBsXsAsF dT += (8)
where
( )( ) bte
ber
KAKcsmsAKcsmsKAKsA 222
22)(
+++
+++=
ε
ε (8-a)
( )( ) bte
bde
KAKcsmsAKcsmsKAAsB 222
2)(
+++
++−=
ε
ε (8-b)
and )(sFT , and are the Laplace transforms of )(sX )(sX d )(tFT , and
, respectively.
)(tx
)(txd
By substituting in Eq. (8) into Eq. (1) in Laplace domain we get the
mathematical model that describes the transmitted force in terms of engine excitation
force and actuator motion as
)(sX
.)()(
)()()(
)()( 2
2
2 sXsAsm
sBsmsFsAsm
sAsF de
ee
eT
++
+= (9)
3. MODELING OF FLOW IN THE INERTIA TRACK
Figure 5 shows the flow between the upper and lower chambers through the
inertia track. The fluid in the inertia track is forced to flow between the upper and
lower chambers due to the pressure difference in those chambers. The flow is
expressed by the momentum equation as [23, 24]
uuDf
tu
lP
h 2ρ⋅+
∂∂
ρ=∂∂
− (10)
where P is the pressure in the inertia track, is the coordinate along the inertia
track, is the average flow speed at a cross-section in the inertia track, is the
friction factor, and is the density of the fluid. Provided that and
, Eq. (10) can be rewritten as
l
u
),tl
f
(tuρ )u =
(PP =
uuD
LLfuLPPh
ett 221
ρ⋅
++ρ=− & (11)
where and are the pressures in upper and lower chambers, respectively, and
is the length of the inertia track, and is the equivalent length for minor head
loss at inlet and outlet. Here, the friction factor is the function of Reynolds
number given as [24]
1P 2P
tL Le
f
( ){ }
>
<<×+−
<
=
.42003164.0
420023001031.123002961
230064
25.0
52
ee
eee
ee
RR
RRR
RR
f (12)
For , the second term in the right hand side of Eq. (11) can be
approximated by using the describing function as
tXx tt ω= sin
( ) ( )tbX
tXbttXxxuu
t
ttttωω=
ωω≅ωωω==
coscoscoscos 22&&
(13)
where
( )
.38
coscos20
2
π=
ωωπω
= ∫ωπ dtttb
Substituting Eq. (13) into Eq. (11) we obtain
tth
ettt xX
DLLfxLP &&& ω
π⋅
ρ⋅
++ρ=∆
38
2. (14)
Finally, multiplying to Eq. (14) by At , we get
tt
ttth
ettttt
xcxm
xXAD
LLfxLAPA
&&&
&&&
+≡
ωρ+
⋅π
+ρ=∆34
(15)
where
tt LAm ρ= (15-a)
ωρ+
⋅π
= tth
et XAD
LLfc34 . (15-b)
These are the equivalent mass and damping coefficient for the flow in the inertia track
described in Eq. (6).
4. DESIGN OF ELECTRO-MAGNETIC ACTUATOR
The engine excitation force is a resultant of gas pressure in cylinders and inertia
forces generated in moving parts such as pistons or crankshafts. This mechanism was
fully analyzed in [24], but to calculate the engine excitation force, we have to know
detailed data on the engine: mass, rotational inertias and dimensions of moving parts,
and gas pressure in cylinders, etc. However, what is needed in designing an AEM is
the outline, or the maximum values, of the required force, stroke, and dynamic range,
but the exact description of them. Hence, if we can estimate the maximum force and
stroke by simple measurements, then we don’t have to go through detailed analysis on
the engine.
The force transmitted to the chassis can be simply estimated, without detailed
data on engine, by multiplying the amplitude of the engine vibration with the stiffness
of the engine mount. The amplitude of the engine vibration is largest in idling state,
and it gets smaller as the rotating speed gets higher. Hence, the transmitted force is
largest in idling state. The amplitude of the engine vibration in idling state was
measured to be 0.22mm in the test vehicle, and the force transmitted to the chassis
was calculated to be about 70N. The actuator force affects the pressure in the upper
chamber and, in turn, the pressurized fluid exerts force to the chassis and engine as
described in Eq. (4). This process amplifies the actuator force by the ratio of A Ae d ,
which is about 1.5 for typical commercial hydraulic engine mounts. Hence the
minimum force requirement on the actuator to control the transmitted force of 70N is
about 50N.
The operating frequency range of AEM was selected as 20-50Hz which
corresponds to the firing frequency for engine speed of 600-1500rpm for 4-cylinder 4-
cycle in-line engines of which the typical idling speed is about 750rpm.
The actuator stroke required to isolate engine vibration can be derived from Eq.
(9) by letting the transmitted force FT equal to zero. In this formulation, we have to
know the engine excitation force, which is difficult to get both analytically and
experimentally: it is hard to directly measure the excitation force acting on the mass
center of the engine, and it is a burden to get inertias and dimensions of the moving
parts necessary to calculate the excitation force. However, if we derive the actuator
stroke from Eq. (8), we can use engine displacement which can be easily measured.
By letting the transmitted force FT in Eq. (8) equal to zero, we get
( ) )s(XKcsmsAA
AKAA
KK)s(X
)s(B)s(A)s(X
de
tr
d
e
b
rd
+++
+=−=
ε2
21 . (16)
Figure 6 shows that the actuator stroke should be 0.7mm or larger over the frequency
range of 20-50Hz in order to cancel the engine vibration of 0.22mm in amplitude. The
typical size of hydraulic engine mounts is about 100mm in diameter and 80mm in
height, so the size of the actuator should be smaller than this.
In short, the actuator should be able to produce force larger than 50N and stroke
larger than 0.7mm over 20-50Hz, and it should be smaller than 100mm in diameter
and 80mm in height.
Based on these specifications, we compared the characteristics of stacked piezo-
actuators, hydraulic actuators, electro-dynamic actuators and electro-magnetic
actuators. Among the four types of actuators, electro-magnetic actuators showed the
best characteristics: stacked piezo-actuators have very limited stroke; hydraulic
actuators are usually large and expensive; electro-dynamic actuators could not
produce sufficient force with reasonable size.
Figure 7(a) shows the schematic of an electro-magnet, of which the magnetic
force is given as
20
220
4g
AINF pµ= (17)
Here, is the absolute permeability µ0 ( )mAWb ⋅×π −7104 , N is the number of
coil turns, I is the current in the coil, is the area of magnetic pole face, and
corresponds to the maximum stroke which is pre-determined from the
specifications. Because the magnets can produce attractive force only, the electro-
magnetic actuator is composed of a pair of electro-magnets to produce bi-directional
motions as shown in Fig. 7(b). The net force acting on the runner is the difference
between the forces from the two electro-magnets given as
Ap
g0
( )
( )( )
( )20
20
20
20
20
20
44 yg
iIAN
yg
iIANF pp
+
−µ−
−
+µ= (18)
where is the offset current, is the control current, and is the displacement
of the runner from its nominal position. We can linearize Eq. (18) by using the Taylor
series expansion as
I0 i y
( ) ( )
yKiK
yyFi
iFFyiF
yi
yiyi
+≡
∂∂
+∂∂
+≈==== 0,00,0
0,0, (19)
where and are the current and position stiffnesses, respectively, given as Ki Ky
KN A I
gip
=µ0
20
02 and K
N A I
gyp
=µ0
202
03 .
The equation of motion for the runner in Fig. 6(b) then becomes
m y F K i K yr i&& y= = + (20)
or
iKyKym iyr =−&& (21)
In this equation, the stiffness term has negative sign, implying that the system is
unstable. This instability can be compensated by applying the proportional-derivative
(PD) feedback control as
( )yKyKi dp &+−= . (22)
This control scheme modifies Eq. (21) as
( ) 0=−++ yKKKyKKym ypidir &&& . (23)
Hence, we can make the system stable by selecting proper and values. If
we add command term in Eq. (22) such as
pK dK
( ) cdp iyKyKi ++−= & (22)
then Eq. (21) becomes
( ) ciypidir iKyKKKyKKym =−++ &&& (23)
or we can get the transfer function as
.
2 )()()()(
ypidir
i
c KKKsKKsmK
sIsYsC
−++== (24)
We used a current amplifier that drives electric current according to control signal in
voltage of which transfer function is
s
KSVSI
c
c
c
cτ+
=1)(
)( (25)
where is the control signal in voltage, is the gain of the amplifier, and cV cK cτ
is the time constant for low pass filtering determined by the gain of the amplifier and
inductance of the coil. The transfer function of the electro-magnetic actuator is given
as
{ ( )} .1)()()()( 2 sKKKsKKsm
KKsVsYsG
cypidir
ic
c τ+−++== (26)
The electro-magnetic actuator was designed to produce 100N of force and
1.0mm of stroke over the frequency range of 0-60Hz. The parameters of the elctro-
magnetic actuator are listed in Table 1, and Fig. 8 shows that the upper limit of the
actuator bandwidth is higher than 60Hz.
5. DESIGN OF ADATPIVE CONTROLLER
The control schemes for active vibration isolation can be classified into two
categories: feedback and feed forward controllers. Between them, the feed forward
controllers are generally accepted to be more advantageous in active vibration control.
However, the feed forward controllers require precise modeling of dynamic
characteristics of the systems to be controlled, which may vary from system to system
and change according to the aging of the system. Adaptive nature is introduced to feed
forward controllers to resolve this problem because it enables the feed forward
controllers to adjust themselves according to such variations. The filtered-X LMS
algorithm [16] is widely adopted for its simple structure and good performance.
The system model to be provided to the adaptive controller can be derived from
Eq. (9) and Eq. (26). Since the actuator displacement Y in Eq. (26) is the same as
in Eq. (9), we can substitute Eq. (26) into Eq. (9) to get dX
.)()()()()(
)()()( 2
2
2 sVsAsmsGsBsmsF
sAsmsAsF c
e
ee
eT
++
+= (27)
From this equation, we can see that the secondary path transfer function to be used in
filtered-X LMS algorithm should be
)()()()( 2
2
sAsmsGsBsmsD
e
e
+= (28)
Accordingly, we can obtain the controller update formula as
ZWW )(21 nFTkk µ+=+ (29)
where is the controller weight vector, kW µ is the step size, )(nFT is sampled
data of the transmitted force, and is the vector of the sampled reference signal Z
filtered through the transfer function , the discrete-time domain expression for
. Figure 9 shows the block diagram of the AEM system using the filtered-X
LMS algorithm, where is the estimated and is the reference signal
vector filtered through .
)(zD
)(sD
)(sD
)(ˆ zD
)(ˆ zD
)(zD Z
Note that the transfer function in Eq. (28) becomes very small at very
low frequencies. This implies that the AEM is ineffective in controlling low frequency
disturbances.
6. EXPERIMENTS
Prior to investigating the vibration isolation performance of the AEM, we tried
to verify the analytical model in Eq. (8) by experiment. If Eq. (8) is verified, then Eq.
(9) is derived by a natural consequence. Equation (8) has no coupling term between
the engine vibration X s( ) and the actuator vibration , we can get the transfer
functions and
X sd ( )
A s( ) B s( ) separately: by exciting the free top side of the
AEM and measuring the force at the fixed bottom while the actuator runner is fixed,
and
A s( )
B s( )
)t
by exciting the actuator and measuring the force while the top and bottom
sides of the AEM are fixed. Figure 10 depicts the test setup, where the AEM is
installed upside down in a commercial hydraulic shaker. During the model
verification tests, a proximity probe and a force transducer measured and )t(x
(FT , respectively, and is measured from the runner position feedback
system by a built-in proximity probe. Figures 11 and 12 compare the experimental
and analytical results of and
)t(xd
A s( ) B s)( , respectively. The parameters in Eq. (8)
were measured from the prototype AEM and listed in Table 2. Note that the analytical
results based on Eqs. (8-a) and (8-b) agree well with the experimental results,
confirming that the model in Eq. (8) accurately describes the dynamic behavior of the
AEM. Figure 11 shows the typical behavior of a hydraulic engine mount because the
AEM was designed to function as a hydraulic engine mount when the actuator is out
of order. In Fig. 12, the small dynamic stiffness below 10Hz implies that the static
pressure acting on the actuator is very small. This allows the actuator to control the
dynamic loads only as intended in designing the structure.
The vibration isolation performance of the AEM incorporated with the filtered-
X LMS algorithm is experimentally investigated in the laboratory. The algorithm is
programmed on a TMS320C30 digital signal processor and implemented to the AEM.
In the experiment, the secondary path transfer function is estimated as an FIR
filter, which is designated as in Fig. 9. The adaptive controller and the
secondary path transfer function have 150 taps each, and the sampling
frequency is 4kHz which is considered fast enough. The step size
)(zD
)(ˆ zD kW
)(ˆ zD
µ in Eq. (29) is set
to be 0.001. Figure 13 shows the laboratory test setup. The function generator
produces harmonic signal which is fed to the exciter to exert disturbance force on the
AEM. Meanwhile, the square wave signal, having the same frequency as the harmonic
signal, is generated to simulate the tachometer signal from the engine. A mass block,
instead of the engine, is attached on the top side of the AEM to imitate the real
engine-mounting system. The force transducer at the bottom of the AEM measures the
transmitted force and feeds it to the DSP. An additional force transducer is inserted on
the topside of the mass block to monitor the excitation force. The DSP computes the
control command according to the filtered-X LMS algorithm by using the tachometer
and force signals. The PD controller stabilizes the actuator and the current amplifier
drives electric current to the actuator to produce control force. Figure 14 shows the
typical performance of the AEM system with the excitation of 25Hz which
corresponds to the typical excitation frequency of the 4-cylinder 4-cycle in-line engine
at idling state. Note that the transmitted force was almost completely eliminated.
Figure 15 shows the effective dynamic stiffness of the AEM over the frequency range
of 10 to 70Hz. Owing to the control effort, the dynamic stiffness was significantly
reduced over the frequency range of 15 to 60Hz. The slight increase in the dynamic
stiffness beyond 60Hz is due to the limitation of the actuator bandwidth, and the
stiffness increase below 15Hz is due to the low frequency characteristics of the
secondary path transfer function commented at the end of section 5.
7. CONCLUSIONS
An equivalent mass-spring-damper model was proposed to describe the
dynamic characteristics of the AEM. The mathematical analysis on the equivalent
model, accounting for the nonlinear flow effect in damping, provided the transfer
function of the AEM that agrees well with the experimental results. The requirements
on the actuator for force, stroke, bandwidth, and size were deduced from engine
vibration characteristics as well as the dynamic model of the AEM, and the electro-
magnetic type actuator was designed to fulfill the requirements. The adaptive
controller using the filtered-X LMS algorithm was employed to cancel out the
disturbance force due to the engine vibration. The stability of the LMS algorithm was
guaranteed by deriving proper secondary path transfer function that take into account
the influence of the control force on the engine vibration. The laboratory experiments
confirmed that the AEM combined with the adaptive controller is able to significantly
reduce the vibration transmission.
ACKNOWLEDGEMENT
The authors are grateful for the support from Hyundai Motor Co., Ltd., Hyundai
Electronics Industries Co., Ltd. and Pyonghwa Industrial Co., Ltd. during the
production and testing of the AEM prototype.
REFERENCES
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idle isolation. SAE paper 850976, 1985.
2. Hata, H. and Tanaka, H. Experimental method to derive optimum engine mount
system for ilde shake. SAE paper 870961, 1987.
3. Choi, S.H. et al. Performance analysis of an engine mount featuring ER fluid and
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APPENDIX A. Equivalent cross-sectional area of upper chamber
Figure A.1 shows the shape of the upper chamber. The volume of the fluid
contained in the main rubber is
( )2221
213
rrrrHV ++π
= . (A-1)
When the main rubber is vertically deformed, then volume change of the upper
chamber is given as
( ) ( )( )( ).
3
332221
21
2221
21
2221
21
rrrrx
rrrrxHrrrrHV
++π
=
++−π
−++π
=∆ (A-2)
Hence, the equivalent piston area is defined as the volume change due to vertical
deflection divided by the vertical deflection itself as
( )2221
213
rrrrxVAe ++
π=
∆≡ . (A-3)
r1
r2
x
H
Figure A.1 Volume change of upper chamber due to vertical deflection.
List of Tables
Table 1. Parameters of electro-magnetic actuator.
Table 2. Parameters of prototype AEM.
List of Figures
Figure 1. Desired architecture of active engine mount.
Figure 2. Hydraulic engine mount: (a) structure; (b) equivalent model.
Figure 3. Active engine mount: (a) structure; (b) equivalent model.
Figure 4. Free body diagram of AEM system.
Figure 5. Flow between upper and lower chambers through inertia track.
Figure 6. Required actuator stroke for suppressing engine vibration of 0.22mm.
Figure 7. Structure of electro-magnetic actuator: (a) schematic diagram of an
electromagnet; (b) dual electro-magnet structure.
Figure 8. Bandwidth of the electro-magnetic actuator.
Figure 9. Block diagram of filtered-X LMS algorithm.
Figure 10. Test setup for model verification.
Figure 11. Passive transfer function of active engine mount, A(s).
Figure 12. Active transfer function of active engine mount, B(s).
Figure 13. Laboratory test setup for vibration isolation performance.
Figure 14. Typical laboratory performance result of AEM system at 25Hz.
Figure 15.Vibration isolation performance of AEM.
Table 1. Parameters of electro-magnetic actuator.
Parameter Value
coil turns, N turns480
offset current, 0I A.51
pole face area, pA 2216mm
nominal gap (or maximum stroke), 0g mm.01
Table 2. Parameters of prototype AEM.
Parameter Value
main rubber stiffness, rK m/N. 3104127 ×
bulge stiffness of main rubber, bK m/N. 3106313 ×
compliance of lower chamber, εK mN /0.2
the equivalent cross-sectional area of the upper chamber, eA
24123mm
decoupler area, dA 21662mm
cross-sectional area of inertia track, tA 250mm
fluid mass in inertia track, m g5.12
damping coefficient in inertia track, c msN ⋅08.0
Engine
Fig
Weight supporting spring
Actuato
Chassis
ure 1. Desired architecture of
Shock alleviating spring
r
active engine mount.
Main Rubber
Upper Chamber
Lower Chamber
Engine
Decoupler
Inertia Track
(a)
Engine
Kr
KbA At e:
m
c
(b)
Figure 2. Hydraulic engine mount: (a) structure; (b) equivalent model.
Main Rubber
Upper Chamber
Engine
Piston (decoupler)
Inertia Track
Actuator
Bellow
Lower Chamber
(a)
xdAd
Engine, me
Kr
Kb
AtAem
Kε
F1 F2
xe xt
x
c
eF
(b)
Figure 3. Active engine mount: (a) structure; (b) equivalent model.
Engine, em
AEM
TF
TF
TF
eF
TF
x
Figure 4. Free body diagram of AEM system.
1P 2P
l
Inertia track
)(tu
),( tlPUpper
chamber Lower
chamber
tL
Figure 5. Flow between upper and lower chambers through inertia track.
0 20 40 60 80 1000.0
0.3
0.6
0.9
1.2
1.5
Act
uato
r stro
ke (m
m)
Frequency (Hz)
Figure 6. Required actuator stroke for suppressing engine vibration of 0.22mm.
F g0
Current, I
N-turns
Pole face, Ap
(a)
Magnet Core
Coil ( I i0 + )
Runner
y
Coil ( I i0 − )
(b)
Figure 7. Structure of electro-magnetic actuator: (a) schematic diagram of an electromagnet; (b) dual electromagnet structure.
1 10-24
-23
-22
-21
-20
-19
-18
-17
-16
100
3dB
Mag
nitu
de (d
B)
Frequency (Hz)
Figure 8. Bandwidth of the electro-magnetic actuator.
+
Engine-AEM
+
)(nR
)t(FT
)(tFe
D/AA/D
)()(
2 sAsmsA
e +
)(ˆ zD
A/D
W
)n(FT
×
Correlated signal (tachometer)
ZWW )(21 nFTkk µ+=+
Z
)(tR
)(sD
Figure 9. Block diagram of filtered-X LMS algorithm.
crossbar
force transducer
proximity sensor
hydraulic exciter
Figure 10. Test setup for model verification.
1 10 100104
105
106
107
experiment simulation
Stiff
ness
(N/m
)
Frequency (Hz)
(a) stiffness
1 10
0
40
80
120
160
100
Phas
e (d
eg.)
Frequency (Hz)
(b) phase
Figure 11. Passive transfer function of active engine mount, A(s).
1 10103
104
105
106
107
100
experiment simulation
Stiff
ness
(N/m
)
Frequency (Hz)
(a) stiffness
1 10 100-200
-160
-120
-80
-40
0
Phas
e (d
eg.)
Frequency (Hz)
(b) phase
Figure 12. Active transfer function of active engine mount, B(s).
proximity sensor
excitation force
runner position
control current
PC with DSP exciter
tachometer
PD controller
power amp.
mass Block
force transducer
AEM
function generator
(a) schematic diagram of experimental setup
force transducer
mass block
force transducer
Figure 1
runner position sensor
(b) close-up view of AEM
3. Laboratory test setup for vibration isolation
performance.
0.00 0.05 0.10 0.15 0.20-80
-60
-40
-20
0
20
40
60
80
Time (sec.)
Tran
smitt
ed fo
rce
(N)
controlled uncontrolled
Figure 14. Typical laboratory performance result of AEM system at 25Hz.
10 20 30 40 50 60 70-100
0
100
200
300
400
500
Frequency (Hz)
Stiff
ness
(kN
/m)
controlled uncontrolled
(a) stiffness
10 20 30 40 50 60 70
0.0
0.2
0.4
0.6
0.8
Frequency (Hz)
Am
plitu
de (m
m)
controlled uncontrolled
(b) vibration amplitude
Figure 15.Vibration isolation performance of AEM.
