1.4 The Second Generation Product: Self-Coupled ER Damper
1.4.3 Theoretical Model for the Self-Coupled ER Damper [107]
In order to analyze the damper property, the following denotations are adopted. The mass of piston is m; its radius is R and its weight is neglected. The driving force is a sinuate pressure and the function of pressure vs. frequency and time is Fp F sin Z t. The damping force generated by ER fluid is FR.
The direction that the piston moves down is the positive direction of axis X. When the piston stays at equilibrium position, its displacement is equal to zero.
The relevant coefficients of piezoelectric ceramic are d33 (piezoelectric constant) and capacitance Cpz. The elastic constant of spring is k. The piezo-ceramics and the spring in the damper are always the state being pressed even when the piston is at zero displacement, so the pre-pressure is set as F.
The viscosity coefficient of ER fluid is ȝ and its function as the external electric field E is
P D
P 0 AE . Where A and D are the experimental constants relevant to ER fluid materials andȝ0 is the magnitude ofȝ under zero electric field.
The gap between the inner and outer electrodes is d, and the height of electrodes is l. The capacitance induced by ER fluid is CER. The external pressure is expressed as
t F
FP sinZ . (1)
So the vibration equation of piston can be written as
x m kx F
FP R . (2)
By substitution, we get
x m kx F t
Fsin(Z ) R . (3)
The pressure applied upon the piezoelectric ceramic is
p
pz kx F
F . (4)
So the electric field between the electrodes can be derived as
) (
) (
) (
33 33
ER pz
p
ER pz pz pz
C C d
d F kx C
C d
d F C
d Q d E V
u
u
u u
u , (5)
where Qpz is the output charge of piezoelectric ceramic, V is the voltage between the electrodes and C is the whole capacitance of the circuit.
In this way, the viscosity coefficientPof ER fluid can be expressed as
D D P
P
P ằằ
ẳ º
ôô
ơ ê
u
u u
( )
)
( 33
0 0
ER pz
p
C C d
d F A kx
AE . (6)
The flowing velocity of ER fluid in damper cavity isx, so the damping force applied on the piston can be calculated as [108]
d x D
l F R
p
R u
u u u 3
4
12SP , (7)
whereDp is the average value of inner and outer electrodes. By substituting Equation (6) into Equation (7), we get
d x D
l R C
C d
d F A kx
F
ER p pz
p
R u
u u u
ằằ
ẳ º
ôô
ơ ê
u
u u
3
33 4
0 )
) (
) ( (
12
D
P
S . (8)
Moreover, substituting Equation (8) into Equation (3), we get
x m kx d x
D l R C
C d
d F A kx
t F
p ER
pz
p
ằằ
ẳ º
ôô
ơ ê
3
33 4
0 )
) (
) ( (
12 ) sin(
D
P S
Z . (9)
It is assumed that a sweeping sine excitation is applied on the piston rod of damper with a sweeping speed of 8 octave/min. Ȧ=2ʌ×1.054412 t . The relevant constants are set as Į=2;
F=500 N; à0=2.0 Pa•s; d33=350ì10-12 C/N; d=0.001 m; k=40000 N/m; R=0.020 m;
Dp=0.056 m;l=0.030 m;m=0.5 kg;Cpz=231×10-12F;CER=334×10-12F.
Firstly, setting A=0, it means that the output voltage of piezoelectric ceramic is absent, i.e. the electric field E=0. The calculated curve of piston displacement versus time is obtained in figure 14(a).
(a)
(b)
Figure 14. The simulation results for the vibration of the self-coupled ER damper (a) without the output voltage of piezo-ceramic and (b) with the output voltage of piezo-ceramic.
Secondly, setting A=10.0×10-12, it means that the output voltage of piezoelectric ceramic is present, i. e. E>Ec,Ec represents the critical electric field for making ER fluid active [109].
The calculated curve of piston displacement versus time is obtained in figure 14(b).
In figures 14(a), E=0, the displacement amplitude of piston is 0.005m at time of 0 second. The maximum of displacement amplitude is about 0.016m. In figure 15(b), E>Ec, the displacement amplitude decreases drastically. The maximum of displacement amplitude is about 0.004m or 0.007m. The decrement of the maximum of displacement amplitude is obvious. It means that the self-coupled ER damper has a manifest vibration suppression effect.
The driving force is set as a sinusoidal function, while the frequency of driving force is set as an exponential function of time, i.e., Ȧ=2ʌ×1.054412 t. With increase in time, the frequency also increases. By adopting the assumption, the decrements of displacement amplitude near resonant frequency, which is mainly concerned by us, can be showed clearly.
In figure 14(a)&(b), when time exceeds 70s, the frequency has reached very high, so the displacement amplitude trends to a small value.
In figures 13, when the output voltage of piezo-ceramics is connected onto the electrodes, the envelopes of FRF change drastically. It shows that the voltage generated from the piezo- ceramics has stimulated the ER fluid filled in the damping cavity and the working state of ER damper system changes manifestly. This point verifies that the method of using piezo- ceramics to stimulate ER damper is feasible in practice.
It can be seen form figure 13 that the envelope of FRF with the voltage of piezo-ceramics is generally lower than that of the ceramic-voltage-disconnected. Especially at the frequency range of 220-250 Hz, the decrement of FRF amplitude can be over 30% of the primitive one.
This means that the vibration suppression effect of self-coupled ER damper is manifest. This character is identical to that of conventional ER damper, which usually uses an external high voltage power supply.
It also can be seen from figure 14 that the resonant frequency of 198 Hz moves to 208 Hz or so when the voltage of piezo-ceramics is connected. The movement of resonance peak verifies that the ER fluid in the damper is stimulated and the vibration-state of damper is changed manifestly.
Compared with our previous adaptive ER dampers [104,105], The second-generation damper has been modified in (a) the number, arrangement style and electrically connecting style of piezo-ceramics; (b) load applying model; (c) hole-channel in the concentric cylinder electrode. These modifications improve not only the structural stability of the new damper, but the working reliability also. Compared with the first generation damper, the second- generation damper has a better suppression effect in FRF amplitude and a more manifest frequency movement of the resonant mode.
The theoretical model is an on-and-off state model. The off state represents that the voltage of piezoelectric ceramics is disconnected, while the on state is connected. The simulation result shows that the damper vibration changes accordingly from off state to on state, see figure 14(a)&(b), the amplitude of resonant peak decreases from 0.016m (off state) to 0.007m (on state). In damper experiment, see figure 13, it has been verified that the frequency response functions are varied from 6.5ms-2/N (voltage-disconnected state) to 2.2ms-2/N (voltage –connected state) at frequency of 220 Hz. In theoretical simulation, it is also shown that the displacement trends to small value when time exceeds 70s. The driving force, in which the frequency is set as an exponential function of time, leads to the result. We adopt the assumption in order to emphasize the on/off effect.
It is verified from the experimental results that the method of employing piezoelectric ceramics and ER fluid to form an automatic feedback control system is feasible. In addition,
the design method could be implemented extensively into many kinds of design for self- adaptive control system, such as acoustic insulation and control. The self-coupled control system based on the methodology is not only reduced in design and manufacture but also very convenient of operating in practical application.