The transmission spectrum of SPL shows a typical resonance effect and, intuitively, it is relevant to vibrations in the sandwiched ER layer (plastic electrodes and ER fluid) in the presence of electric field. The thin ER layer is excited by the vibrating air in the parabolic reflector and also vibrates forcedly. The vibrating ER layer can be approximately modeled as
a piston sound-radiator with diameter 2a smaller than the side length of the square panel, see inset (a) of Fig. 21. The sound pressure at the point P, which is at the central-vertical axis and with a distance of z from the piston center, can be expressed as [133]
[ ( ) ]
2 2
2 0 0 sin ( ) 2
i t k R z a
p c u k R z e
Z S
U , (1)
whereU0is the air density, co the sound speed in air, k the wave vector, R the distance from P to the rim of piston, and ua the velocity amplitude of piston surface along z. In the case of long wavelength or low frequency, i.e. ka<<1, so k(R-z)<ka<<1, we has the approximation of
sin ( ) ( )
2 2
k R z |k R z . By substitution we get
[ ( ) ]
2 2
i t k R z
p p em Z S
, (2)
denoting the sound pressure amplitude pm U0 0c u k R za ( ).ua is a key factor to determine pm, here, we set up a reduced vibration model to calculate it.
50 100 150 200 250
0 10 20 30 40
50 E=0 kV/mm0.17kV/mm 0.33kV/mm 0.67kV/mm 1.00kV/mm
M o d u le o f Z2 A 2 /F 0
Frequency (Hz)
FIG. 21. Simulation results of Z2 A F2 0 as a function of frequency for different electric field E. The ER panel can be approximately modeled as a piston sound-radiator in inset (a) and the transmitted sound pressure amplitude pm at point P is proportional to Z2 A F2 0. The two pieces of plastic film and ER fluid can be approximately modeled as a bi-freedom system in inset (b) and the effect of electric field on ER fluid can be reflected from the elastic element keand viscous one ce.
As depicted in the inset (b) of Fig. 21, the upper plastic film and the lower one can be modeled as two mass elements connecting on ground via elastic elements k1 and k2, and viscous element c1 and c2 respectively. The sandwiched ER fluid is treated as an elastic
element ke and viscous one ce in the so-called Voigt model for viscoelastic materials and the mass of ER fluid (me) is fixed on the lower plastic film. A force F excites the lower plastic film due to the vibrating air beneath. F is set asF e0 i tZ , where F0 is the amplitude of F and Z
the sound frequency. Moreover, we set displacements of the lower and the upper plastic film as x1 A e1 i tZ , x2 A e2 i tZ , respectively, where A1,A2 represent the displacement amplitude and can be complex numbers. The vibration equations for the two pieces of plastic film can be written as
0 1 1 1 1 2 1 1 1 2 0
0 2 2 2 2 1 2 2 2 1
( ) ( ) ( )
( ) ( ) 0
i t
e e e
e e
m m x k x k x x c x c x x F e m x k x k x x c x c x x
Z
®
¯
, (3) wherem0 is the mass of lower or upper plastic film. So A2 can be derived as
2 2 1
2 2 0 2 1 0 1
2
2 2 0
0
( )[ ( ) ]
( ) e e e
e e
k k m i c i c k m m i c
A k i c m
F k i c
Z Z Z Z Z
Z Z
Z
ê º
ô ằ
ơ ẳ
. (4)
The sound is directly radiated from the surface of the upper plastic film, so the velocity of the upper plastic film is of primary importance and it can be calculated as
2 2 2 i t
v x i A e Z Z with the module of Z A2 . By substituting it into the expression of pm, we get pm U Z0 2 A R z2 ( ). In the equivalent vibration model, ke and ce for ER materials can be approximately expressed as a simple function of storage modulus G’ and loss modulus G”, i.e., ke S G r d c 2/ , and ce S G r cc 2/( Z d ), where r is the radius of piston and d the thickness of ER fluid. G’and G”can be measured in an experiment setup with one vertically oscillating electrode and other unmoved electrode [135]. Here, we use Liu et al’s [135]
experimental data in simulations. G’=0, 675, 1330, 2250, 2700Pa and G”=50, 400, 470, 720, 755Pa, when the electric field E is 0, 0.17, 0.33, 0.67, 1.00kV/mm, respectively. The data is obtained at small sinusoidal oscillation strain (~10-5) so that ER fluid works in linear regime.
The other constant parameters are set as: k1=k2=15000N/m, c1=c2=5Paãs, r=33mm, d=1mm, m0=6×10-3kg, and me=2.8×10-2kg. Figure 21 plots the simulation results of Z2 A2 /F0, which can represents pm, as a function of sound frequency. The sound radiation peaks about 120Hz are exhibited with the presence of electric field due to the reinforced vibration occurring in the plastic film of ER layer. In addition, both the amplitude and the frequency of the sound radiation peak increase with increase in E as a result of the variation of the electric- field-dependent storage and loss modulus. The simulation qualitatively agrees with the experiment plot in Fig. 16(a). Strictly, the presented model could be somewhat crude since many details are simplified. For example, the vibration of the four-side-fixed ER layer may be far more complex and the storage and loss modulus vary with frequency. These may account for the simulation deviation in higher frequencies. In essence, the simple model presents a clear physical picture of sound transmission process that explains the characteristics observed in experiments very well.
Within the band of 80-150Hz, the flexible thin ER fluid layer exhibits manifest tunable features. Sound pressure level and phase angle can all be adjusted by varying the electric field strength. The corresponding wavelengths of the responding frequencies of 80-150Hz are far larger than the size of the ER layer. These tunable characteristics provide a methodology of designing phononic crystals with tunable acoustic band gap in low frequency domain. For example, these ER layers could be employed as inclusion layers in layered one-dimensional phononic crystals or applied in the of unit cell composite as a design element in multi- dimensional phononic crystals. The composite inclusions or unit cells with tunable characteristics might achieve the active control of acoustic band gap. Besides, the flexible thin ER layers own can be utilized to develop controllable acoustic devices, such as sound amplifiers and phase tuners.
Summary
The tunable and quick rheological responses to external electric field of ER fluid have attracted highly attentions due to the potential use in both conventional and intelligent devices. Since the discovery of ER fluids, many developments in the mechanisms, materials and applications have made. We discuss some new advances in design and preparation of ER materials based on two routes including molecular & crystal structure design and nanocomposite & hybrid design. Especially, in order to achieve the design about optimal physical and chemical properties of ER materials, some advanced preparation techniques, such as self-assembly, nanocomposite, and so on, are also used. These new design and preparation ways not only extend ER materials but also bring merit for high-performances.
On the other hand, based on the idea of intelligent materials and systems, the ER fluid is employed to act as actuator for vibration control and the piezoelectric ceramic is used as vibration responder and exciter for the solidification of ER fluids. The first generation ER/piezo damper realized the adaptive control and the manifest vibration suppression effect have been observed in experiments. In the second generation ER self-coupled damper, many new designs have been adopted. These modifications improved the structural stability and the working reliability of the ER damper. A flexible sandwiched ER composite layer is designed and fabricated and the sound transmission behaviors of the ER layer have been investigated carefully. The transmitted sound pressure level and the phase can be modulated by the external electric field. ER fluids can be used in constructing tunable acoustic devices.
Acknowledgements
This work was supported by the National Natural Science Foundation of China for Distinguished Young Scholars(No.50025207), the National Natural Science Foundation of China (No. 59832090, 50272054) and the ‘863’ Foundation of China ( No. 2001AA327130).
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