3.3 Vibration Control Using Piezostack Mount
4.1.2 Design and Modeling
As mentioned, the hybrid active mount consists of active and passive elements connected in serial. The passive element is a rubber mount that is primarily used to support the mass (100 kg) and isolate the vibration from the base. The active elements are two piezostack actuators, which can exert actuating forces to control the vibra- tion for performance improvement. Moreover, an additional mass is adopted between the piezostack actuators and the rubber element as an inertial mass to increase the actuating force. The confi guration of the hybrid active mount is shown in Figure 4.1.
The base plate is fi xed to the vibrating base. A supported mass (100 kg) is loaded on the top plate that is fi xed to the bottom plate by a screw through the hole at the center of the rubber element. The mass of the piezostack actuators and that of the housing combined with the additional mass is considered as the inertial mass. The bottom plate is designed for fail-safe purpose; it protects the mount from being broken by a large force in the upward direction. In this confi guration, two high-stiffness piezostack actuators are serially connected with the relatively low-stiffness rubber. Therefore, the stiffness of the piezostack actuators does not much affect the total stiffness of the hybrid active mount.
It can easily be seen in the confi guration that actuating force of the mount is mainly generated by the inertial mass. In other words, the larger the inertial mass is, the larger the control force can be generated. However, there are some constraints
on dimensions of certain applications that limit the size of the mount for practical applications. The height and diameter of the mount are constrained to be smaller than 125 and 140 mm, respectively. (These constraints are translated from the equivalent constraints that limit the size of the designed mount not to exceed 10%
the size of the rubber mount 7E450 [17].) Due to these design constraints, the rubber element should be carefully designed so that its stiffness is large enough to support a 100 kg load.
In this test, the hybrid active mount is designed to improve the performance of the conventional rubber mount (Mount Code No. 7E450) [17], which is widely used for navy shipboard equipment. For this purpose, the characteristics of the rubber element should be satisfi ed based on this mount model for the possibility of replacement. On the other hand, the rubber element must be designed so that there is enough space for the adaptation of the piezostack actuators and the inertial mass. It is noted that the characteristics of the hybrid active mount in passive operation will not be much changed by the adaptation of the piezostack actuators. This is possible because the stiffness of the piezostack is much higher than that of the rubber element. According to the mount 7E450 specifi cations, the desired rubber element with a 100 kg mass should have a natural frequency in the range of 6–7.5 Hz, and the dynamic stiffness should be in the range of 142–222 N/mm. The damping ratio should be greater than 5% so that the vibration amplitude of the supported mass does not exceed 0.3 mm and is under 0.03 mm excitation at resonance. The defl ection at 100 kg load should be within the limits 6.6–11.2 mm, or the static dynamic stiffness is in the range of 87.6–148.6 N/mm. Finite element (FE) analysis was performed to investigate the deformation behavior and fatigue life of the rubber element. Figure 4.2 shows the FE model of the rubber element. The strain distribution of the designed rubber element under 100 kg static load in vertical direction is shown in Figure 4.3. The analysis result shows that the axial defl ection of the rubber element under 100 kg static load is about 10 mm, which is in the allowable range. From the FE analysis, principal design
Piezostack actuators
Top plate
1 2
Additional mass
Bottom plate
Piezostack housing
Base plate
Rubber element
FIGURE 4.1 Confi guration of the proposed hybrid active mount.
parameters of the rubber element are determined based on a trial-and-error method, and shown in Figure 4.4. The space underneath the rubber element is large enough for adapting two piezostack actuators and an inertial mass of 1.7 kg maximum.
After the design phase, the rubber element was manufactured, and experiments were then carried out to identify dynamic characteristics. Figure 4.5 shows the
Z Y X FIGURE 4.2 Finite element model of the ¼ rubber element.
Z Inc: 30
Time: 3.00e+001 2.731e + 000 2.388e + 000 2.046e + 000 1.703e + 000 1.360e + 000 1.018e + 000 6.749e – 001 3.323e – 001 –1.038e – 002 –3.530e – 001 –6.957e – 001
Y X
FIGURE 4.3 Strain distribution of the rubber element under 100 kg static load. (From Nguyen, V.Q. et al., Proc. Inst. Mech. Eng.: Part C, J. Mech. Eng. Sci., 223, 1327, 2009. With permission.)
photograph of the manufactured rubber element. The relation between defl ection and static load can be easily obtained by experiment; the results are shown in Figure 4.6, from which the static stiffness can be determined to be 97.5 N/mm.
The dynamic stiffness and the damping coeffi cient can be determined based on the experimental system transmissibility, whose magnitude is given by
2 m
dB 2 2 2
b
1 (2 ) ( ) 20 log 20 log
(1 ) (2 )
A r
T r
A r r
= = + ζ
− + ζ (4.1)
FIGURE 4.5 Photograph of the manufactured rubber element.
φ30 φ50
φ31 φ90 φ110
5713 9880
225
φ90 φ130
FIGURE 4.4 Principle design parameters of the rubber element (unit: mm).
where
TdB is the magnitude of the acceleration transmissibility in decibel (dB)
Ab and Am are the amplitudes of the base acceleration and the supported mass acceleration, respectively
ζ is the damping ratio of the rubber element
r is the ratio of the excitation frequency f to the natural frequency fn of the system (r = f / fn)
As the excitation frequency is equal to the natural frequency (r = 1), the damping ratio ζ can be easily obtained from Equation 4.1, which is reduced to
dB 0.5
0.5 10⎡ 0.1T 1⎤−
ζ = ⎣ − ⎦ (4.2)
Then the dynamic stiffness of the rubber element, kr, can be determined by
2 r (2 n)
k = πf M (4.3)
where M is equal to 100 kg. Figure 4.7 shows the measured acceleration transmis- sibility obtained from experiments. The natural frequency and the magnitude of the transmissibility are found to be 6.31 Hz and 17.5 dB, respectively. By utilizing Equations 4.2 and 4.3, the damping ratio and dynamic stiffness of the rubber ele- ment can be calculated by 6.7% and 157.4 N/mm, respectively. The corresponding damping coeffi cient is calculated by 537 N s/m. It is noted that these parameters vary in accordance with the excitation frequency as well as the ambient temperature or ageing. To cope with these variations, an SMC will be formulated to achieve the robustness of the control system, in which they are considered as uncertainties.
In this section, the hybrid active mount is designed to operate under the excita- tion levels defi ned in Figure 4.8, which is one of the military specifi cations for navy
0 2 4 6 8 10 12 14
0 200 400 600 800 1000 1200 1400
Experiment Simulation
Deflection (mm)
Load (N)
FIGURE 4.6 The load vs. defl ection characteristic of the rubber element. (From Nguyen, V.Q. et al., Proc. Inst. Mech. Eng.: Part C, J. Mech. Eng. Sci., 223, 1327, 2009. With permission.)
shipboard equipment [18]. The piezostack actuators adopted in the hybrid active mount must be carefully chosen so that they are able to improve the isolation performance in the frequency range of interest, 20–1000 Hz. For a proper selection of piezostack actuators, the electromechanical behavior of piezostacks is fi rst examined.
The constitutive equations for a piezoelectric material along the polarized direction can be written as
33 33
D= ε E+d T (4.4)
33
S d E 1T
= +c (4.5)
1000 100
1E–420 1E–3 0.01 0.03 0.1
Frequency (Hz)
Displacement amplitude (mm)
FIGURE 4.8 Specifi cation of base excitation. (From Nguyen, V.Q. et al., Proc. Inst. Mech.
Eng.: Part C, J. Mech. Eng. Sci., 223, 1327, 2009. With permission.)
1 10 100 1000
–80 –60 –40 –20 0 20 40
Frequency (Hz)
Magnitude (dB)
FIGURE 4.7 Transmissibility from base acceleration to mass acceleration. (From Nguyen, V.Q.
et al., Proc. Inst. Mech. Eng.: Part C, J. Mech. Eng. Sci., 223, 1327, 2009. With permission.)
where
D is the electrical displacement (C/m2) E is the electric fi eld (V/m)
T is the stress (Pa) S is the strain
ε33 is the dielectric constant under constant stress (F/m) d33 is the piezoelectric charge constant (C/N)
c is the elastic modulus at zero electric fi eld
For the piezostack actuator of length l subjected to an external preload force fex(t), Equation 4.5 can be developed as follows [9]:
ex 33 33
( ) ( )
( ) t V t
f t AT AcS Acd E Ac Acd
l l
= = − = δ − (4.6)
where A is the cross-sectional area of the piezoelectric element. By introducing the equivalent spring constant, kp = Ac/l, and the proportional constant, α = Acd33/l, Equation 4.6 can be rewritten as
ex( ) p ( ) ( ) or p( ) : ( ) p ( ) ex( )
f t = δ − αk t V t f t = αV t = δ −k t f t (4.7) where fp(t) is the force exerted by the piezostack actuator.
The mechanical model of the hybrid active mount supporting a 100 kg load can be described as a two-degree-of-freedom system shown in Figure 4.9. The governing equations of the vibration system are written as
( ) ( )
( )
( )
+ − + −
− − = −
+ − =
1 r 1 0 r 1 0
p 2 1 p
2 p 2 1 p
( ) ( ) ( ) ( ) ( )
2 ( ) ( ) 2 ( )
( ) 2 ( ) ( ) 2 ( )
my t b y t y t k y t y t
k y t y t f t
My t k y t y t f t
(4.8)
where
y1(t) and y2(t) are the displacement of the inertial mass and of the supported mass, respectively y0(t) is the displacement of the base (excitation) kr and br are the stiffness and damping coeffi -
cient of the rubber element, respectively kp is the stiffness of the piezostack actuators fp(t) is the force exerted by the piezostack
actuator
By defi ning the state vector x = [y1 y˙1 y2 y˙2]T, Equation 4.8 can be rewritten as the following state-space representation:
kr
m 2kp
y0
y2
y1
2fp
br
M
FIGURE 4.9 Mechanical model of the hybrid active mount system.
( ) ( ) p( ) ( ) ( ) ( )
t t f t d t
y t t
= + +
=
x Ax B F
Cx
(4.9)
where
y(t) is the system output, which is velocity of the supported mass M d(t):= br y˙0 (t) + kt y0(t) is the disturbance excited from the base The matrices A, B, C, and F are given by
⎡ ⎤
⎢− + − ⎥
⎢ ⎥
= ⎢ ⎥
⎢ − ⎥
⎢ ⎥
⎣ ⎦
r p r p
p p
0 1 0 0
( 2 ) 2 0
0 0 0 1
2 0 2 0
k k m b m k m
k M k M
A 兾 兾 兾
兾 兾
[ ]
⎡ ⎤ ⎡ ⎤
⎢− ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥
=⎢ ⎥ =⎢ ⎥ =
⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦
0 0
2 1
, , 0 0 0 1
0 0
2 0
m m
M
B 兾 F 兾 C
兾
(4.10)
The base excitation and control force are inputs, and the vibration of the mass is the output of the system. It can be seen that the vibration of the mass is the sum of the vibra- tion transmitted from the base excitation and the vibration transmitted from the control force. The mass acceleration levels under the exciting conditions in Figure 4.8 can be approximately determined by utilizing the acceleration transmissibility of the rubber element shown in Figure 4.7. To suppress these acceleration levels, the control force exerted by the piezostack actuators is required to be capable of gener- ating acceleration at the supported mass as large as the maximum acceleration that is transmitted from the base excitation. Hence, the control force can be reversely estimated from the output acceleration by employing the transmissibility from con- trol force to the mass acceleration that is evaluated by the simulation based on the model given in Equations 4.9 and 4.10. It is noted that the piezostack parameters are unknown while they need to be known for simulation. Therefore, some processes of actuator selection need some trials on commercially available piezostack actuators in the following steps: (1) choose a piezostack actuator, (2) perform simulation for transmissibility from control force to the mass acceleration, and (3) compute and check the required force for applicability. After undertaking these processes, two piezostack actuators (Piezomechanik PSt 350bp/16/70/25; Figure 4.10) are chosen.
They are of bipolar type and work with bipolar input voltage of ±350 V maximum.
To verify the performance of the piezostack actuators, experimental apparatus is set up as shown in Figure 4.11. A force transducer is used to measure the blocking force. In this experiment, the force transducer itself, which has a very high stiffness, is considered as a spring load, and a laser sensor is used to measure the displacement
at this load. Sinusoidal signals generated by the function generator are amplifi ed to activate the piezostack actuator. By measuring the actuating force and the dis- placement at various amplitudes of input voltage, the dynamic characteristics of the piezostack actuators can be identifi ed. Figure 4.12 shows experimental results of the dynamic characteristics of two piezostack actuators. The values of the spring constant, kp, and the proportional constant, α, of the piezostack actuators are experi- mentally determined by 48 N/μm and 3.1 N/V, respectively. The maximum blocking forces of two piezostacks are also checked: each piezostack can exert a maximum force of about 1030 N at the input voltage of 300 V. In addition, experimental results show that the two piezostack actuators have almost the same characteristics. Figure 4.13 shows the measured actuating force with respect to the frequency at 300 V. From the experimental results, the actuating force is about 4.5 N at 20 Hz, and over 1000 N at 1000 Hz. It is seen that as the frequency is increased, the force exerted by the piezostack actuators is signifi cantly increased due to the inertial mass.
FIGURE 4.10 Photograph of the piezostack actuator PSt 350bp/16/70/25.
Dynamic signal analyzer
Function generator High-voltage amplifier Force
transducer Laser sensor
Force Displacement
Piezostack actuator
Voltage
FIGURE 4.11 Experimental setup for the characteristic identifi cation of the piezostack actuator.
200 400 600 800 1000 0
200 400 600 800 1000 1200
Measured Curve fitting
Actuating force (N)
Frequency (Hz)
FIGURE 4.13 Actuating force exerted by the piezostack actuators. (From Nguyen, V.Q.
et al., Proc. Inst. Mech. Eng.: Part C, J. Mech. Eng. Sci., 223, 1327, 2009. With permission.) 0
400 800 1200
300 V 200 V 100 V
Displacement (μm)
Displacement (μm)
Blocking force (N)
(a)
0 5 10 15 20 25
0 5 10 15 20 25
0 400 800 1200
300 V 200 V 100 V
Blocking force (N)
(b)
FIGURE 4.12 Measured blocking force of the piezostack actuators. (a) Piezostack actuator
➀ and (b) Piezostack actuator ➁. (From Nguyen, V.Q. et al., Proc. Inst. Mech. Eng.: Part C, J. Mech. Eng. Sci., 223, 1327, 2009. With permission.)