Transcript Ohmic Switch Structural Dynamics

slide 1
An Experiment to Determine the Accuracy
of Squeeze-Film Damping Models in the
Free-Molecule Regime
Hartono (Anton) Sumali
1526 Applied Mechanics Development
Albuquerque, NM
Presented at
Purdue University
Birck Nanotechnology Center
April 6, 2007
Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company,
for the United States Department of Energy’s National Nuclear Security Administration
under contract DE-AC04-94AL85000.
Gas damping is important in MEMS.
slide 2
Motivation:
• Many micro/nano devices need high Q factor. Examples abound in
• MEMS switches need high speed (high Q).
• Resonant cantilever sensors need high responses.
• MEMS gyroscopes.
• MEMS accelerometers need controlled damping.
• Damping can reduce Q from several hundred thousands to several hundreds.
• Squeeze-film damping determines the dynamics of plates moving a few microns
above the substrate.
• Continuum models are not known to be valid in rarefied (free molecule) regime.
• Molecular-dynamics-based models for predicting squeezed-film damping give
different results.
• So which model should one use?
• Need experimental validation!
• Published experimental data were obtained for squeeze-film damping on flexible
structures. Have been used to validate theory derived for rigid structures.
Objective:
• Provide experimental validation to squeezed-film damping models for rigid plates.
Squeeze-film damping determines the
performance of MEMS.
Radio-frequency MEMS switch is designed to close
upon electrostatic actuation.
slide 3
Click below to animate.
Click below to animate.
In vacuum, it is very difficult to
close the switch without bouncing.
Click below to animate.
In atmospheric air, the right
actuation schemes can land the
plate softly.
Squeezed fluid damps oscillation.
slide 4
Oscillating
plate
z Click plate to animate.
a
Substrate
y
Gap thickness
Plate oscillates at fequency w.
e(t) = e0 cos(wt)
h
h
b
Gas gap
x
Time
The squeezed fluid between the plate and the substrate creates damping forces on the plate.
For sensors, rigid plate parallel to substrate, moving up and
down, is preferred over flexible plates. [H. Seidel et al. 1990
Sensors and Actuators, 21 312-315.]
Squeeze-film damping is important in the
design of MEMS/NEMS
slide 5
• Models are needed for predicting and designing performance of MEMS.
• Modeling tools are available, especially based on continuum mechanics.
Finite element models
Example
Reduced-order models
Example
c Andrews  0.42(ab) 2  / h3
http://www.comsol.com/showroom/gallery/1432.php
• High resolution.
• High fidelity.
• Simple.
• Gives clearer ideas of how parameters affect
damping.
• Well suited for design and optimization.
However,
Are models based on continuum mechanics valid for rarefied gas in the
free-molecule regime?
Knudsen number determines damping regime
slide 6
•Knudsen number Ks = 1.016 mean free path/(gap size).
•Mean free path    2RT
P mm
•Continuum models, eg.
R = universal gas constant,
T = temperature, K
mm=molecular mass, kg/mol
Ph2 2  p    p    z 
       
12  P  t  P  t  h 
Are based on spatial derivatives.
•Do they apply when the gap is not much larger than the mean free path?
Many researchers say no.
We need experimental data.
http://www.phas.ucalgary.ca/~annlisen/teaching/
Phys223/PHYS223-LECT34.html
Molecular models for free-space drag were
used to calculate squeeze-film damping.
slide 7
• The models below were derived for free-space drag, not squeeze-film damping.
• Have been used to explain data measured on squeeze-film damping.
Christian’s model
R.G. Christian 1966 Vacuum, 16 175
[S. Hutcherson and W. Ye 2004 J. Micromech. Microeng.
14 1726-1733.]
2m
cChristian  4
Ap
kT T
Kadar et al’s model
m
kT
T
A
p
= mass of one molecule, kg
= Boltzman’s constant, J/K
= Temperature, K
= Plate area, m2
= Pressure, Pa
Plate
No
c
substrate
Damping
z
m
k
Stiffness
Z. Kadar et al 1996 Sensors and Actuators A, 53 299-303
Improved on Christian’s model by replacing its Maxwell-Boltzman velocity distribution
with the Maxwellian stream distribution, resulting in
cKadar =  cChristian
Li et al’s model
B. Li et al 1999 Sensors and Actuators A, 77 191-194
Improved on Kadar et al’s model by taking into account the velocity of the moving
structure relative to the fluid, resulting in
1.5
cLi 
c
2 Christian
M
1  3 R0T u
These are “molecular-based’ models, not
molecular dynamic models.
Models for free-space drag underestimate
squeeze-film damping.
slide 8
Measured on squeeze-film
device [J.D. Zook et al 1992 Sensors
and Actuators A, 35 51-59]
Original free-space damping
model [R.G Christian 1966 Vacuum, 16 175]
Improved model [Z. Kadar et al 1996
Sensors and Actuators A, 53 299-303]
Improved model [B. Li et al 1999
Sensors and Actuators A, 77 191-194]
Figure in Li et al 1999 Sensors and
Actuators A, 77 191-194
Models gave higher Q (lower damping) than measured.
Squeeze-film damping is much larger than
free-space drag.
slide 9
Predicted with free-space drag:
Higher Q (lower damping).
Figure in C. Gui et al 1995 J.
Micromech. Microeng. 5 183-185.
Measured on squeeze-film damped
MEMS: Lower Q (higher damping).
Also concluded in R. Legtenberg and
H.A.C. Tilmans 1994 Sensors and
Actuators A 45 57-66.
Molecular models for free-space drag were
adapted to calculate squeeze-film damping.
slide 10
• The models below are adaptation of the free-space drag model to squeeze-film damping.
• Have been used to explain data measured on squeeze-film damping.
Bao et al’s model
Bao et al 2002 J. Micromech. Microeng. 12 341-346.
Adapts Christian’s model to squeeze-film damping (instead of free-space drag),
resulting in
cBao 
Lcircumference / h
16
Oscillating
plate
cChristian
Lcircumference = 2a+2b
h = Gap thickness
Hutcherson and Ye’s (HY) model
• Refines Bao’s model by taking into account the change of velocity of
molecules colliding with the plate and the substrate.
• True molecular model generated by molecular dynamic simulation.
• For the case considered in their paper,
cHY ≈ 2.2 cBao
h
Substrate
a
b
Gas gap
Squeeze-film damping models were closer to
measured data.
slide 11
Published molecular models still could not
explain published measured damping.
slide 12
Free-space damping model [R.G Christian 1966 Vacuum, 16 175]
Adapted to squeeze-film gap [M Bao et al
2002 J. Micromech. Microeng. 12 341-346.]
Data measured on
squeeze-film device [J.D.
Zook et al 1992 Sensors and
Actuators A, 35 51-59]
MD-simulation-based model
[S. Hutcherson and W. Ye 2004 J.
Micromech. Microeng. 14 1726-1733.]
• Hutcherson and Ye’s (HY) model
appears to be closest to Zook’s data.
• However, Zook’s data have a different
slope than all models.
• Zook’s geometry and test conditions
are not well known or modeled.
We need
• A model that is validated by measured data.
• Measurement data with better characterized test device and conditions.
“Molecular” models could not be validated with
available data. Try continuum models
• Forces on moving plate from gas layer
can be obtained from the linearized
Reynolds equation
Ph2 2  p    p    z 
       
12  P  t  P  t  h 
P
h

p
t
= ambient pressure, Pa
= gap size, m
= viscosity, Pa s
= pressure at (x,y), Pa
= time, s
slide 13
Assumptions:
1.Rigid plate
2.Small gap
3.Small displacement
4.Small pressure variation
5.Isothermal process
6.Small molecular mean free path
• Assumptions in the continuum models:
• Blech’s model
(7. On edges, pressure jumps
to ambient pressure.
8. Inertia of fluid neglected)
•Andrews et al.’s limit
(9. Zero Knudsen number limit of Blech’s)
•Veijola’s model
Free from 7 and 8.
Blech’s model was derived for continuum
regime, low Knudsen number.
slide 14
• Oscillating plate has stiffness and intrinsic (non-squeeze-film) damping.
Plate
Structural
stiffness
Solid damping
z(t) = e0 cos(wt)
mplate
ks
cs
cBlech
kBlech
~
m platez  cz  kz  f
cs  c Blech
k s  jw Blech
• Squeeze film causes an extra damping coefficient
• and an extra spring stiffness
e0 = amplitude, m
mplate= plate mass, kg
t
= time, s
z
= gap displacement, m
w = frequency, rad/s
Blech, J.J., 1983, “On Isothermal Squeeze Films”, Journal of Lubrication Technology, 105, p 615-620.
For low-squeeze numbers, Blech’s model
reduces to Andrew’s et al.’s model.
• Blech’s damping coefficient
slide 15
768 a 3b
m 2  n 2 a / b 
w   6 3   2 2 2 2
 h m,n odd m n m  n a / b 2 2  s 2 /  4
2
c
Blech
• Depends on the squeeze number

 a 
s  12   
 hm 
2
w 
 
P
• For low squeeze numbers, s <<2

a
b
h
P

c Andrews  0.42(ab) 2  / h3
w
= plate width, m
= plate length, m
= gap height, m
= ambient pressure, Pa
= viscosity, Pa s
= frequency, rad/s
• Sample applicable range:  = 1.82(10)-5Pa.s; a = 144m; h = 4.5m.
in atmosphere P=9.3(10)4Pa: s <1 for
w/2 < 70kHz
Andrews, M., Harris, I., Turner, G., 1993, “A comparison of squeeze-film theory with measurements
on a microstructure”, Sensors and Actuators A, 36, p 79-87.
Veijola’s model accounts for fluid inertia.
slide 16
• Taking into account rarefaction and the inertia of the gas flowing in and out of the gap,
Veijola (2004) modified Reynolds equation into
  rh
p    rh
p  rh

Qpr   
Qpr  
x  12
x  y  12
y  t
3
3
• If the gap oscillation is e(t) = e0 exp(jwt), then the damping force
complex amplitude is
Veij
F
w  
Qpr 

M
w   jw e0 w 
N
1
 
m 1, 3, n 1, 3, QprGmn  jwCmn
k 1, 3,...
Cmn 
Gmn 
 4 hmn2

= gap size, m
= pressure at (x,y), Pa
= time, s
= viscosity, Pa s
= density, kg/m3
a
b
e0
j
n
= width, m
= length, m
= amplitude, m
= √-1
= 1 for isothermal,
(= cp/cv for adiabatic).
= ambient pressure, Pa
= viscosity, Pa s
= frequency, rad/s
= gas mass density, kg/m3

r
64abn P
 h mn
768ab
6 3
2
P
m n 
 2  2 
b 
a
2
2
1  6K s
k 2 2 rh 2 1  10 K s  30 K s2
k 4 4
 jw
96
96 1  6 K s 
Knudsen number
Ks = 1.016 /h
 = mean free path, m

h
p
t

w
r
Veijola, T., 2004, “Compact models for squeezed-film dampers with inertial and rarefied gas effects”, Journal of
Micromechanics and Microengineering, 14, p 1109-1118.
Flexible test devices did not validate model
derived for rigid plates.
slide 17
• Blech’s model was
derived for rigid plates.
• Published test on
flexible cantilevers
gave lower damping.
C.C. Cheng and W. Fang 2005
Microsystem Technologies 11 104–110
We need a rigid plate oscillating
up and down while staying
parallel to the substrate.
A rigid plate gave correct validation
measurement for SFD models. But the data scatter is large.
slide 18
M. Andrews et al [1993 Sensors and Actuators A,
36 79-81] used a correct test device for
validating Blech’s model.
Oscillating
plate
Substrate
• Would be more obvious if damping were
in log scale, that
• scatter is large, especially at high squeeze
numbers (also Knudsen numbers).
A rigid-plate test device validated Veijola’s
model at low Knudsen numbers.But the data scatter is large.
-1
-2
10
6T
0.06T
-3
10
0.006T
-4
1
2
3
4
10
10
5
10
-1
1
2
3
4
-4
0
1
2
3
4
5
10
2
3
4
5
10
-1
-2
10
-3
10
10
1
10
-4
10 10 10 10 10
Squeeze number s
0
10 10 10 10 10
Squeeze number s
Damping ratio 
Damping ratio 
-3
10
10
10
-1
-2
-3
10
5
10
10
-2
10
-4
0
10 10 10 10 10
Squeeze number s
10
Damping ratio 
-3
-4
0
10 10 10 10 10
Squeeze number s
10
-2
10
Squeeze number
2
 a  w 
s  12     
h P
Blech
Andrews
Veijola (V)
V+NonSFD
Measured
10
Damping ratio 
60T
10
-1
10
640T
Damping ratio 
Damping ratio 
10
-1
0
1
2
3
4
5
10
-3
10
10
• Non-squeeze-film
damping accounts for
solid structural and
other unknown
damping.
• At high pressure,
nonSFD does not
contribute much.
• At near-vacuum:
• Gas damping is
negligible.
• Assume 90% of
damping is nonSFD.
-2
10
-4
10 10 10 10 10
Squeeze number s
slide 19
0
1
2
3
4
10 10 10 10 10
Squeeze number s
5
H. Sumali and D.S. Epp
2006 Proc. ASME IMECE.
10
• Damping being plotted in log scale, it is obvious that the scatter is
large, especially at high squeeze numbers (high Knudsen numbers).
Continuum models could not be validated with available
data. Try a better model:
slide 20
Gallis and Torczynski’s (GT) model Is a Direct Simulation Monte Carlo (DSMC) method.
Instead of the trivial boundary conditions at the plate edges, GT introduced
6 
 12U 
P  p   G  nˆ p    
1




G 
 G 
1
DSMC simulations were used to determine correlations for the gas-damping parameters

0.634  1.572   G 
1  0.537   G 

1  8.834   G 
1  5.118   G 
 
0.445  11.20   G 
G = gas film (gap) thickness. /G is modified Knudsen number
 = accommodation coefficient. (For this test device a = 1).
H. Sumali et al 2006 Proc. ASME IDETC to appear.
1  5.510   G 

2 


0   G 1
A rigid-plate test device validated GT model.
slide 21
• 2x3 array of plates
H. Sumali et al 2006 Proc. ASME
IDETC to appear.
• Again, the scatter is large, especially at low pressures (high Knudsen numbers).
• We need a better-controlled experiment to validate models at high Knudsen
numbers.
Present measurement was done on an
oscillating plate.
• Structure is electro-plated Au.
• Thickness around 5.7 m.
• Substrate is alumina.
slide 22
Folded-cantilever springs
Plate width
154.3 m
A = 29717(m)2
a = 154.3 m
Air gap between plate and substrate
Mean thickness = 4.1 m.
Anchored to substrate
• Assumed width a and length b,
where ab = true plate area.
b
a
The test structure can be modeled as SDOF.
slide 23
Plate
z(t) = e0 cos(wt)
mplate
Measured
c
k
Damping
Stiffness
zb
Substrate
• Equation of motion
mz  k zb  z   czb  z 
• Frequency response function (FRF) from base displacement to plate displacement:
Z w 
mw 2

1
Z b w   mw 2  jw cs  cgas   k s  k gas 
nonsqueezefilm
squeezefilm
damping
Need to compare computed damping factor c
with measured damping ratio .
slide 24
• Models predict damping factor c in the equation of motion
m platez  cz  kz  f ext (t )
• Measurement method gives damping ratio  in the equation of motion
z  2wn z  wn 2 z  fˆext (t )
• Quality factor Q is the inverse of damping ratio
  0 .5 / Q
• To compare prediction with measurement, use the relationship between c and 
• For Blech’s model,
 Blech 
• For Veijola’s model,
 Veij
cBlech
2m platewn


FVeij


 Re
 jw e 2 m w 
 d 0 plate n 
e0 = amplitude, m
hplate = plate thickness, m
j
= √-1
mplate = plate mass, kg
wn = natural frequency, rad/s
Scanning laser Doppler vibrometer enables
high temporal- and spatial resolutions.
slide 25
Click below to animate.
Laser Doppler vibrometry
principle
Scanning laser Doppler
vibrometry (scanning
LDV) gives spatial
vibration shapes.
http://www.polytec.com/int/158_6800.asp
Scanning LDV has been integrated with
a micro-otpical diagnostic instrument.
LDV can provide clear insight into dynamics
slide 26
Such work as
Click to animate.
can benefit a lot from LDV.
Click to animate.
Measurement uses LDV and vacuum chamber.
slide 27
• Substrate (base) was shaken
with piezoelectric actuator.
• Scanning Laser Doppler
Vibrometer (LDV) measures
velocities at base and at
several points on MEMS
under test.
Microscope
Die under test
Laser beam
PZT actuator
(shaker)
Vacuum
chamber
Oscillating plate is shaken through its supports.
slide 28
Air gap between plate
and substrate. Mean
thickness = 4.1 m.
1. Substrate is
shaken up and
down.
2. Plate moves up
and down.
3. Springs flex.
4. Air gap is compressed and expanded by
plate oscillation.
• LDV measured frequency response function (FRF) from base displacement to
plate displacement:
Z w 
mw 2
Transmissibility =

1
2






Zb w
 mw  jw cs  cgas  k s  k gas
FRFs are curve-fit to give natural frequency,
damping and mode shapes
• LDV measured transmissibility at 17
points on the plate and springs.
• Frequency response function (FRF) from
base displacement to gap displacement:
Z gap w 
w2

Z b w   w 2  jw 2w n  wn 2
= Transmissibility - 1
are curve-fit simultaneously using standard
Experimental Modal Analysis (EMA) process.
• Commercial EMA software gave natural frequency, damping, and mode shapes.
• Tests were repeated at different air pressures from atmospheric (640 Torr) to nearvacuum (<1 milliTorr).
slide 29
Experimental modal analysis gives natural
frequency, damping and mode shapes.
Measured deflection shape, first mode.
16910Hz. Up-and-down.
Higher modes are not considered.
Click below to animate.
27240Hz
Click below to animate.
Click above to animate.
slide 30
33050Hz
• Tests were repeated at different air pressures from atmospheric (640 Torr)
to near-vacuum (<1 milliTorr).
Damping from EMA agrees well with damping
from free decay.
1. Hilbert transform gives decay envelope.
slide 31
2. Exponential fit gives damping times
natural frequency.
3. Damping is constant with time.
4. For this case (P=3830milliTorr), both
experimental modal analysis
(frequency domain fit) and free decay
curve-fitting (time domain fit) give
damping ratio  = 0.0011
Non-squeeze-film damping is estimated from
zero-pressure asymptote.
slide 32
• At low pressures, Non-Squeeze-Film Damping
(NSFD) is the dominant damping.
• Linear-fit total measured damping at a few lowestpressures. The zero-pressure intercept is NSFD.
• To obtain squeeze-film damping from
measured total damping, subtract
NSFD from total measured damping.
Some models predict measured data very well.
slide 33
Conclusions:
slide 34
• On rigid plates with width ~150 m, oscillating
around 4.1 m above the substrate, squeezed air
film can cause large damping.
• Non-molecular models are not necessarily less
accurate than current molecular models.
• For the conditions tested here, in atmospheric air
the simplest model mentioned by Andrews et al.
is as good as any more sophisticated models.
c Andrews  0.42 A2  / h3
• In the high squeeze number regime (low pressures or high frequencies),
Veijola’s model appears to match experimental data accurately.
Acknowledgment
slide 35
The author thanks the following contributors:
• Chris Dyck and Bill Cowan’s team for providing
the test structures.
• Jim Redmond and Dan Rader for technical
guidance and programmatic support.
• David Epp for help with the modal analysis.
Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company,
for the United States Department of Energy’s National Nuclear Security Administration
under contract DE-AC04-94AL85000.