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Fig. 4. Amplitude-frequency characteristics of end point of the cantilever
structure, obtained in the vicinity of the fundamental frequency
of the PMPG in the presence of squeeze-film damping for a
constant air-film thickness (h 0 = 50 μm) at different levels
of ambient pressure p 0: 100 Pa, 1 kPa, 5 kPa, 10 kPa,
25 kPa, 100 kPa (curves from top to bottom).
Fig. 5. Amplitude-frequency characteristics of end point of the cantilever
structure, obtained in the vicinity of the fundamental frequency of the
PMPG in the presence of squeeze-film damping for a constant
ambient pressure (p 0 = 100 kPa) at different air-film thickness h 0:
50 μm, 75 μm, 100 μm, 150 μm (curves from bottom to top).
The topmost curve (green) is a frequency
response with damping excluded.
moves towards this nearby rigid surface with a thin air-film
in-between. Thus, if the bottom face (boundary) of the proof
mass is located relatively close to some stationary ground
surface, then during transverse motion of the mass its fairly
small displacement in normal direction would compress (or
pull back) a significant amount of air out of (or into) the
narrow gap. However, the viscosity of the air will limit the
flow rate along the gap, and thus the pressure will be
increased inside the gap and act against the structure. The
squeezed air-film between the mass and ground surface will
likely to have a significant effect on PMPG dynamic
behavior due to the induced counter-reactive pressure force
that is exerted on the vibrating cantilever structure [5].
Nonlinear compressible isothermal Reynolds equation is
11-13 May 2011, Aix-en-Provence, France
usually used for modeling of squeeze-film damping
occurring in micro-scale [6]:
∂ ⎛ 3 ∂P
⎞ ∂ ⎛ 3 ∂P
⎞ ⎛ ∂P
∂h
⎞
⎜ Ph ⎟ +
⎜ Ph
⎟ 12μ=
eff ⎜h
+ P ⎟ , (1)
∂x
⎝ ∂ ⎠ ∂yx
⎝ ∂
⎠ ⎝ ∂t
∂t
⎠
μ
eff =μ . (2)
1.159
⎛
0PL
⎞
1+
9.638
⎜ a
⎟
⎝ hp 00 ⎠
Here the total pressure in the gap P and the gap thickness h
are functions of time and position (x, y). μ is the dynamic
viscosity of the gas, μ eff is the effective viscosity coefficient,
which is used to account for gas rarefaction effects (a model
of T. Veijola [7] is used here; it is adopted by the COMSOL
as one of the optional approaches), p 0 is the initial (ambient)
pressure in the gap, L 0 is the mean free path of air particles
at atmospheric pressure P a , and h 0 is the initial air-film
thickness. For the P a = 101325 Pa, L 0 ≈ 65 nm. Total
pressure in the gap is equal to P = p 0 + Δp, where Δp is an
additional film pressure (variation) due to the squeezed airfilm
effect.
“Film Damping Application” mode, which uses (1) and
(2), was added to the piezoelectrical model in order to
simulate frequency and time responses of the PMPG taking
into account the effect of squeeze-film damping (a
linearized version of (1) is used for harmonic analysis).
B. Numerical Analysis
Numerical study of the developed PMPG finite element
model commenced from the determination of the natural
frequencies and the associated vibration mode shapes (Fig.
2). Performed modal analysis indicates that the fundamental
frequency of the PMPG is equal to 184 Hz. Fig. 2 illustrates
the first four mode shapes: a) the 1 st out-of-plane flexural
mode, b) the 1 st torsional mode, c) the 2 nd torsional mode, d)
the 2 nd out-of-plane flexural mode. This analysis also
provided results on distribution of air pressure forces in the
gap when the structure is vibrating in its flexural and
torsional resonant modes. Pressure mode shapes in Fig. 3
reveal obvious coupling between structural displacements of
the structure and pressure distribution in the gap. For
example, in the 1 st torsional mode (Fig. 2(b)), the upward
motion of left side of the proof mass corresponds to a
concave profile in the respective region of pressure mode
shape (Fig. 3(b)), which indicates the reduction of pressure
in this part of the gap (i.e. decompression effect). And, in
contrast, the downward motion of right side corresponds to
a convex pressure profile – zone of increased pressure with
respect to atmospheric (i.e. compression effect).
The aim of the subsequent numerical experiments was to
determine influence of viscous air damping on dynamical
behavior of the PMPG and its generated voltage. The
simulations were performed with zero structural damping.
The model was subjected to sinusoidal kinematic excitation
by applying vertical acceleration through body load that is
equal to F z =aρ in each subdomain, where a=Ng (N=0.1,
g=9.81 m/s 2 ) and ρ is density of the corresponding material
(Si or PZT-5A).
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