ABSTRACT. Micromachined directional microphones (sound source localization sensor) mimicking an auditory organ of Ormia. Ochracea have been intensively ...
W3P.046
NOVEL THEORETICAL DESIGN AND FABRICATION TEST OF BIOMIMICRY DIRECTIONAL MICROPHONE Shigeru Ando1* , Toru Kurihara1 , Kentaro Watanabe1 , Yoshiki Yamanishi2 , and Takahiko Ooasa2 1 Dept. Information Physics and Computing, University of Tokyo, Japan 2 Tokyo Electron AT Limited, Hyogo, Japan ABSTRACT Micromachined directional microphones (sound source localization sensor) mimicking an auditory organ of Ormia Ochracea have been intensively studied based on centersupported, marginally free 1-D or 2-D diaphragm structures. Based on an exact mathematical formulation, a Gaussian weight function on the diaphragm is shown to be inevitable in the transduction of 1st and 2nd moments. An analysis was performed to obtain the weight distribution assuming an arbitrary thickness distribution of a shell-structured diaphragm. The best result was a marginally-supported faceto-face cantilever structure. We fabricated this structure using SOI wafer and RIE. The experimental results show a strong dependence of the vibration pattern on both zenith and azimuth angles, and a greatly enhanced sensitivity to inclined sound pressure.
KEYWORDS sound source localization, Ormia Ochracea, weighted integral method
(a) Ormia ochracea
mechanical coupling
pivot
air-filled chamber auditory nerve
(c) model of auditory organ
Figure 1: External anatomy of (a) Ormia ochracea and (b) after removal of the head [1]. The left and right tympanum is connected to generate common/different mode vibrations for an inclined input of sound wave.
inclined sound pressure (2D)
inclined sound pressure
The parasitoid fly Ormia Ochracea shows a remarkable ability to detect the direction of sound even though its ears are in very close to each other, where the interaural differences in intensity and time are extremely small (Fig.1). It is based on the measurements of 0th and 1st order moments generated by an inclined sound pressure distribution. Micromachined microphones mimicking this structure are very attractive, and have been intensively studied [1]–[6] using center-supported, marginally free 1-D or 2-D diaphragms (Fig.2). But they have not obtained sufficient success because of their low sensitivity, low accuracy, and narrow bandwidth. In this paper, based on an exact mathematical formulation[7, 9], we showed the former interpretation of this structure is wrong, and obtain a novel one which can provide 1) enhanced sensitivity to inclined sound pressure, 2) wide-bandwidth and frequency-invariant localization, and 3) close resemblance with Ormia ochracea’s organ.
(a) 1-D type
(b) Gimbal supported 2-D type
Figure 2: Old types of biomimicry directional microphone.
Then the wave field f (r, t) from the source satisfies a partial differential equation (PDE) ∇f (r, t) =
n ˙ f (r, t). c
(1)
By means of an arbitrary weight function w(r), the PDE is identically converted into an integral form as[7, 9] n ∇f (r, t)− f˙(r, t) = 0 ∀ r ∈ Γ c n (∇f (r, t)− f˙(r, t))w(r)dr = 0 ∀ w(r). (2) ↔ c Γ
EXACT THEORETICAL DESIGN Exact theory of sound source localization Let the spatial coordinate be r = (x, y, z)，the unit direction vector of a source be n, the sound velocity be c.
tympanal membrane
auditory apodome
bulba acustica
INTRODUCTION
978-1-4244-4193-8/09/$25.00 ©2009 IEEE
(b) tympanic membranes
Let us consider here the Gauss function w(r) = e−α|r | whose gradient has a form of 1st order moment such as
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Transducers 2009, Denver, CO, USA, June 21-25, 2009
(∂/∂x)w(r) = −2αxe−α|r | = −2αxw(r). Then the integral form is rewritten as 2
gx (t) gy (t)
n ˙ = 0, + g(t) c
z a( x) -X
Γ
w(r)f (r, t)dr
gx,y (t) ≡ −2α
Γ
0
x
x
(a) diaphragm with thickness distribution z
where n = (nx , ny ) are unknown source direction, and g(t) ≡
X p( x)
(3)
z( x)
FL -X
(x, y)w(r)f (r, t)dr
are the Gauss-weighted 0th and 1st order moments of the pressure distribution. Therefore, Eq.(3) can provide an exact solution of the source direction regardless of the measurement conditions such as the sound direction and a spectrum of the sound source. Uniqueness of Gaussian weight function So that the integral form (2) is expressed by 0th and 1st order moments only, the weight function w(r) must satisfy uniformly a vector differential equation ∇w(r) = −2αrw(r), where α is an arbitrary positive constant. Dividing both side by w(r) where w(r) = 0, we obtain ∇w(r) = ∇ log w(r) = −2αr, w(r) and the general solution of it as w(r) = Ae−α|r | , 2
where A is an arbitrary real constant, and w(r) = 0 everywhere. This concludes the Gaussian function is the unique weight distribution. Gaussian weight by thickness distribution The exactness is only when the weight function is the Gauss function. Therefore, we considered a method to generate it in a natural way. It is a non-uniform stiffness or thickness of the diaphragm. Fig.3 shows the geometrical model of a diaphragm plate. Both sides are connected to rigid walls (as the Ormia Ochracea’s ear). At the center, the inclination angle and the deviation from the stationary plane due to an anti-symmetric pressure component and a symmetric component, respectively, are measured. The transfer coefficients from a point load to these measurements define the weight distribution. Fig.4 shows two typical cases; (a) is a marginally thick design, and (b) is a marginally thin one (former types). In (b), the weight distribution has a completely different shape
F -x
FR x
0
F
x
X
(b) anti-symmetric 2 point loads
Figure 3: Geometrical model of non-uniform, elasticallysupported diaphragm excited by symmetric loads and antisymmetric loads.
from Gauss. Contrarily in (a), it is very close to Gauss. Although the sides are stiffly connected, the inclination angle of (a) at the center is far larger than (b). This shows the directional sensitivity is enhanced. In comparison with the Ormia Ochracea’s ear shown in Fig.1(b), the marginally thick membrane will be seen as the corrugated part of the tympanum, which is considered formerly as only a screen to avoid a pressure leakage.
PRACTICAL DESIGN AND FABRICATION Thin-plate-with-slit (TPWS) structure Thin plate structure of crystal silicon with an equal thickness has various advantages such as an ideally uniform elasticity, high Q value of the resonance, easiness of fabrication, etc. To obtain a practical design based on this structure shown in Fig.5 (a). A 2W × 2W (W = 5mm for the experiments) square thin area was built from an SOI wafer by a backside wet-etching. The thin area with thickness a was cut like an “X” shape to create four triangular cantilevers supported by the frame (thick marginal area). In the center, they are connected with each other with 2L × 2L (L = 1.25mm for the experiments) square region. The connections are with four fine beams with width w and length d in which the torsional force is ignorable. Each triangle zone receives the sound pressure and bends like a cantilever. The connecting points between the beams and the square region are shifted inside toward a 2l × 2l central square of the square region. The structure is made from an SOI wafer using a wet etching and RIE. The common mode bents of four face-to-face cantilevers generate parallel out-of-plane vibration of the square region. The x- and y-differential mode bents of them generate xand y-inclination vibrations of the central square region. They are magnified by the shifted connecting points of the beams in proportion to the ratio l/L. The resonant modes of
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(b) marginally thin design (old type) Figure 4: Calculation results of weight distribution for two typical cases. In the marginally thick design (a), the distribution is very close to Gauss, whereas in the marginally thin design (b) which corresponds to the former 1-D of 2-D gimbal-supported design, it is far from the Gaussian shape.
the proposed structure obtained by FEM analysis are shown in Fig.6. Weight distribution of TPWS structure Fig.7 shows the weight distribution of the thin-platewith-slit structure of a device for the fabrication test. Although the geometrical parameters has not been optimized, the curve tracks well the Gaussian distribution. Frequency characteristics of TPWS structure The components that determine the fundamental resonant modes of this structure are: 1) spring constant of the triangular cantilevers, 2) spring constant of the connecting beams, and 3) mass and moment of inertia of the central square. 1) Spring constant of triangular cantilever — Let the thickness, width, and length of a square cantilever be a, 2W , and W , respectively. The spring constant at the tip side of the cantilever is expressed as k = 3E ·
a3 (2W ) 12
·
Figure 5: A novel design of biomimicry directional microphone and its vibration modes. Four triangular cantilevers are connected by a small central square plate.
2) Spring constant of connecting beam — Let the width and length be b and d. When both connecting points are kept parallel, it is expressed as kB =
k=
(5)
kB kC . kB + kC
(6)
The motion equation and the resonance frequency of the parallel vibration mode are then expressed as 2
4ρaL z¨ = 4kz, ωp =
k . ρaL2
√ 4ρaL4 ¨ l θ = 4kl2 θ, ωr = 3ωp · , 3 L
(4)
(7)
For the inclination vibration modes, we obtain
1 = , W3 2W 2
Ea3 . 2W 2
Ea3 b . d3
3) Vibration modes of the square region — Let the density be ρ. The structure is simply modeled by a spring-supported plate shown by Fig.8, where the mass is 4ρaL2 , and each spring is a series of the connecting beam and the cantilever, hence, the spring constant is
Ea3
where E is the Young’s modulus. For a triangular cantilever, FEM simulation shows kC 0.548 ·
(c) photograph
(8)
where θ is the angle of plate from the baseline. The ratio of both frequencies can be adjusted by the ratio l/L, i.e., the deepness of connecting √ points inside the square region. They coincide when l = L/ 3 0.5774L.
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imaging area
(b) x-differential mode
(a) common mode
amplitude
π phase -π
180deg
135deg
90deg
45deg
Figure 9: Measurement of vibration mode excited by an incident sound from various direction using a white-light vibrometry[8]. (c) y-differential mode
(d) mixed differential mode
Figure 6: Various mode of vibration excited by an incident sound from (a) frontal, (b) x-lateral, (c) y-lateral, and (d) π/4-lateral direction.
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Figure 7: Weight distribution of the proposed structure for the common mode (a) and the differential mode (b) excitations of incident sound.
Figure 10: Vibration amplitude distribution when the zenith angle was changed from 40 degree to 80 degree.
the next step, will easily realize the directional microphone.
EXPERIMENTS Fig.9 shows results of white-light vibrometry[8] when sound is incident from various directions. Each image show vibrating amplitude (left) and phase (right). Linear black zone in amplitude shows the vibration nodes, and inversion of phase there shows the differential vibration mode is excited strongly. Lateral offset of the vibration nodes indicates the common vibration mode is also excited. Fig.10 shows the lateral shift of vibration amplitude according to the zenith angle of incident sound. Both results show the direction of sound can be captured well by this structure. Implementation of probes for both the vibration modes, in
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l k
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Figure 8: A model for the vibration analysis of square region.
REFERENCES [1] R. Miles, D. Robert, and R. Hoy, J. Acoust Soc. Am., vol.98, no.6, pp.3059-3070, 1995. [2] R. Miles, D. Robert, and R. Hoy, J. Comp. Physiology A, vol.179, no.1, pp.29-44, 1996. [3] N. Ono et al. Proc. Transducers’03, Boston, 2003. [4] N. Ono, A. Saito, and S. Ando, “Bio-mimicry sound source localization with gimbal diaphragm,” Trans. IEEJ, vol.123-E, no.3, pp.92-97, 2003. [5] N. Ono et al. Proc. Transducers’05, Seoul, 2005. [6] Cui, W. et al. MEMS 2006, Istanbul, 2006. [7] S. Ando and N. Ono, 4th Joint Meeting ASA and ASJ, Honolulu, 2006. [8] S. Sato wt al. Proc. 25th Sensor Symposium, pp.700703, 2008. [9] S. Ando and T. Nara, “An exact direct method of sinusoidal parameter estimation derived from finite Fourier integral of differential equation,” IEEE Trans. Signal Processing (accepted).
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