Force Multiplier in a Microelectromechanical
Silicon Oscillating Accelerometer
by
Nathan St. Michel
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
Masters of Science in Mechanical Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
-May 2000.
@ Nathan St. Mi-hel, MM. Allrights reserved.
The author hereby grants to MIT permission to reproduce and
distribute publicly paper and electronic copies of this thesis document
in whole or in part.
A u th or ........................................................
......
Department of Mechanical Engineering
e, May 22, 2000
a
Certified by.........................
David L. Trumper
Associate Professor
Thesis Supervisor
C ertified by .....................
.....
Marc
$'. Weinberg
Draper Laboratory
e-,.-.2iiesis Supervisor
Accepted by .....................
Dr. Ain A. Sonin
Chairman, Department Committee on Graduate Students
MASSACHUSETTSISTITUTE
OF TECHNOLOGY
SFP 2 0 2000
LIBRARIES
Force Multiplier in a Microelectromechanical Silicon
Oscillating Accelerometer
by
Nathan St. Michel
Submitted to the Department of Mechanical Engineering
on May 22, 2000, in partial fulfillment of the
requirements for the degree of
Masters of Science in Mechanical Engineering
Abstract
This thesis describes the third generation design of a silicon oscillating accelerometer
(SOA.) The device is a micromachined vibrating beam accelerometer consisting of
two tuning fork oscillators attached to a large proof mass. When accelerated along
the input axis, the proof mass exerts a force on the oscillators, causing their natural
frequencies to shift in opposing directions. Taking the difference of these frequencies
gives a measurement of the applied acceleration.
A significant feature in the new design is a force multiplying lever arm which
connects the oscillators to the proof mass. With this addition, the surface area of the
proof mass is reduced by a factor of four, while the scale factor and axial resonant
frequency of the previous design is retained. All vibration modes of the device are
identified, and stiffness nonlinearity in the oscillators is investigated. A method to
compensate for thermally induced acceleration errors is also introduced. Testing of
the fabricated device is shown to agree with model predictions.
Thesis Supervisor: David L. Trumper
Title: Associate Professor
Thesis Supervisor: Marc S. Weinberg
Title: Draper Laboratory
2
Acknowledgments
This thesis was prepared at The Charles Stark Draper Laboratory, Inc., under Draper
Laboratory IR& D/CSR project 15059.
Publication of this thesis does not constitute approval by Draper or the sponsoring
agency of the findings or conclusions contained herein. It is published for the exchange
and stimulation of ideas.
Permission is hereby granted by the Author to the Massachusetts Institute of
Technology to reproduce any or all of this thesis.
A uthor .............
................
Nathan St. Michel
I am very grateful to all the people at Draper and MIT who have helped me to
complete this project. In particular, I am thankful to Dr. Marc Weinberg and Prof.
David Trumper for the enormous amount of guidance and instruction that they have
given me. Without their influence, this thesis would have never happened.
I am also thankful to all of the Draper staff, including Ralph Hopkins and the
rest of the SOA team, who have helped bring this project to completion. Thanks to
David Nokes and Greg Kirkos for their invaluable instruction and technical expertise.
Thanks also to Bernie Antkowiak and Jim Bickford for their outstanding analysis
work and help. I appreciate the help of Joe Miola and Rich Elliott, without whom
there would be no test data to report. Jeff Borenstein, Lance Niles, Nicole Gerrish,
and the rest of the fabrication team have been key to the success of the project, as
have Paul Ward and the electronics team. Thanks to Rob White, Phil Rose, Dave
Hom, Steve Daley, Paul Magnussen, and everyone else in the micromechanical test
lab who were always willing to help out and always kept things interesting.
Finally, I am very grateful for the support of my family and friends, who were
always patient with me and full of encouragement. I feel blessed that God has placed
you all in my life.
3
Contents
1
Introduction
10
2
Background
14
3
2.1
Device Description .......
............................
14
2.2
Basic Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
2.3
Stiffness Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.3.1
General Formulation . . . . . . . . . . . . . . . . . . . . . . .
18
2.3.2
Error Predictions . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.4
Thermal Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.5
Capacitive Drive System . . . . . . . . . . . . . . . . . . . . . . . . .
22
Design and Analysis
25
3.1
Previous Work
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
3.2
Thermal Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
3.3
Beam Models using ANSYS
. . . . . . . . . . . . . . . . . . . . . . .
33
3.4
Detailed Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
3.4.1
Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . .
38
3.4.2
Thermal Modeling
. . . . . . . . . . . . . . . . . . . . . . . .
42
3.4.3
Nonlinear Analysis . . . . . . . . . . . . . . . . . . . . . . . .
45
4 Fabrication and Testing
49
4.1
Fabrication Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
4.2
Scale Factor and Drive Frequency Testing
54
4
. . . . . . . . . . . . . . .
4.3
Q Testing
4.4
. . . . . . . . . . . . . . . . . . . . . . . . .
58
Temperature Sensitivity
. . . . . . . . . . . . . . . . . . . . . . . . .
59
4.5
Axial Frequency.....
. . . . . . . . . . . . . . . . . . . . . . . . .
62
4.6
Stiffness Nonlinearity
. . . . . . . . . . . . . . . . . . . . . . . . .
63
........
5 Conclusions and Recommendations
65
A Predicting Stiffness Nonlinearity
68
B SOA Vibrational Modes
73
5
List of Figures
1-1
Completed SOA design . . . . . . . . . . . . . . . . . . . . . . . . . .
12
2-1
Schematic diagram of SOA-2 (not drawn to scale) . . . . . . . . . . .
15
2-2
Simple model of flexure and oscillating mass. . . . . . . . . . . . . . .
17
2-3
Portion of capacitive comb drive . . . . . . . . . . . . . . . . . . . . .
23
3-1
Concept drawing of force multiplying lever arm connecting the proof
m ass to the oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
3-2
Plot of oscillator reaction forces vs. input force
. . . . . . . . . . . .
28
3-3
Spring model of SOA used in thermal analysis . . . . . . . . . . . . .
29
3-4
Concept drawing of oscillator and force multiplier with anchor beam
. . . . . . . . . . . . . . . . .
33
3-5
Beam model of SOA created in ANSYS . . . . . . . . . . . . . . . . .
34
3-6
Finite element model of one half of the SOA . . . . . . . . . . . . . .
37
3-7
The contribution of base beam width to proof mass axial frequency,
repositioned for thermal compensation
drive frequency, and total scale factor (frequency difference of two oscillators.) Lever arm width for this analysis is 125 pm.
3-8
. . . . . . . .
39
The contribution of lever arm width to proof mass axial frequency and
total scale factor (frequency difference of two oscillators.) Base beam
3-9
width for this analysis is 45 pm. . . . . . . . . . . . . . . . . . . . . .
40
. . . . . . . . . . . . . . . . . . . . . . . . . . .
43
. . . . . . . . . . . . . . . . . . . . . . .
44
Drive mode for SOA
3-10 Out-of-plane mode for SOA
3-11 Finite element model of oscillator and force multiplier used in nonlinear
analysis
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
46
3-12 Predicted nonlinear stiffness curve for flexure displacement . . . . . .4 47
4-1
SOA with rotationally symmetric configuration
. . . . . .
50
4-2
SOA with mirror configuration . . . . . . . . . . . . . . . .
50
4-3
Profile- of flexure beam . . . . . . . . . . . . . . . . . . . .
52
4-4
Profile of comb finger . . . . . . . . . . . . . . . . . . . . .
53
4-5
Frequency response of an SOA . . . . . . . . . . . . . . . .
55
4-6
Scale factor vs. drive frequency for twelve SOA units. Values calculated
4-7
using FEA are also shown. . . . . . . . . . . . . . . . . . .
56
. . . . . . . . .
58
. . . . . .
60
Oscillator position signal during ringdown
4-8 Thermal sensitivity data for one SOA oscillator
4-9
Thermal sensitivity data of the SOA. Here, the frequency difference
has been taken to show the overall effect of temperature on the device.
61
4-10 PSD of amplitude modulated drive signal showing the 3.4 kHz axial
. . . . . . . . . . . . . . . .
62
4-11 Plot of measured and calculated nonlinear stiffness curves .
63
A-i Modified spring model of SOA oscillator and force multiplier . . . . .
69
B-i Proof mass axial mode . . . . . . . . . . . . . . . . . . . . . . . . . .
74
"Hula" mode in which flexures resonate in phase with one another . .
75
frequency of the proof mass.
B-2
B-3 Rotational mode in which the proof mass rotates about its center point 76
B-4 Drive mode for SOA
. . . . . . . . . . . . . . . . . . . . . . . . . . .
77
B-5 Cross-axis mode, similar to axial mode in Figure B-1, but along the
axis perpendicular to the input axis . . . . . . . . . . . . . . . . . . .
78
B-6 Another in-plane mode that resembles a "tail wagging" motion of the
oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
B-7 First out-of-plane mode. This mode is symmetric to the other half of
the device.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
B-8 Second out-of-plane mode. This mode is antisymmetric to the other
half of the device.
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
81
B-9 Third out-of-plane mode. This mode is symmetric to the other half of
the device.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
B-10 Fourth out-of-plane mode. This mode is symmetric to the other half
of the device.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
B-11 Fifth out-of-plane mode. This mode is antisymmetric to the other half
of the device.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
B-12 Sixth out-of-plane mode. This mode is symmetric to the other half of
the device. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
B-13 Seventh out-of-plane mode. This mode is symmetric to the other half
of the device.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
B-14 First out of plane mode. This mode is symmetric to the other half of
the device. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
B-15 First out of plane mode. This mode is antisymmetric to the other half
of the device.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
88
List of Tables
3.1
Modal frequencies of SOA with force multiplier. All in-plane modes
were calculated using the full FEA model, and out-of-plane modes were
calculated using the half FEA model. . . . . . . . . . . . . . . . . . .
9
42
Chapter 1
Introduction
In recent years micromechanical inertial sensors have been gaining popularity in a
number of applications. The advantage to using these devices is their compact size,
low weight, low cost, and high reliability. These benefits are gained by the use of
semiconductor fabrication technology, which has the potential to produce large quantities of micromachined devices that are cheaper and physically less complex than
their conventional counterparts. With these benefits, micromechanical sensors have
the potential to become wide-spread not only in military applications, but in a variety of commercial uses as well. In addition to military applications including "smart"
munitions and missile guidance, these sensors have a virtually endless number of other
uses including automobile systems, factory automation systems, bio-medical devices,
and many other areas where inexpensive, robust sensors are required [3].
A variety of micromechanical inertial sensors have been designed and fabricated
from such materials as quartz and silicon. These include gyroscopes using tuning
fork resonators that are sensitive to coriolis acceleration [11], [10] and vibrating beam
accelerometers [8], [1].
Draper Laboratory is currently developing a version of a
vibrating beam accelerometer called the Silicon Oscillating Accelerometer (SOA).
This device consists of a large proof mass and two mechanical oscillators which are
attached to it. The proof mass and oscillators are attached via anchors to a Hoya
glass substrate below.
The oscillators are each formed out of two flexure beams
which are driven at resonance using a capacitive comb drive system. When the proof
10
mass experiences acceleration along the input axis of the device it exerts opposing
tensile and compressive forces on the oscillators, causing the frequency of one to
increase and the frequency of the other to decrease. By taking the difference of these
frequencies, a measure of acceleration is obtained. The change in differential frequency
per unit acceleration is known as the device's scale factor, and the unloaded natural
frequency of an oscillator is referred to as the drive or bias frequency. Chapter 2 gives
a more in-depth description of this device, along with the basic equations governing
its operation.
Two previous designs of this device have accomplished much by first proving the
feasibility of the device and then refining its operation by increasing its sensitivity and
developing a means for eliminating errors caused by temperature sensitivity. The first
design, SOA-1, which was developed in Kevin Gibbons' Master's thesis [7], produced
a device measuring 2,400 ,um square, with a scale factor of 4.0 Hz/g, a temperature
sensitivity of 1.03 Hz/ 0 C (41 ppm/SC), and an oscillator resonant frequency of 25.1
kHz. The second design, SOA-2, created at Draper Laboratory, produced a device
with a scale factor of 100 Hz/g, a temperature sensitivity of approximately 1.6 Hz/ 0 C
(75 ppm/ 0 C), and an oscillator resonant frequency of 21 kHz. This dramatic improvement in scale factor was accomplished by altering the configuration of the oscillators
and increasing the size of the proof mass to 1 cm square. Unfortunately, the large
size of the proof mass has also presented a number of manufacturing difficulties in
the microfabrication process.
In the present project, we have developed new SOA design with performance
similar to the previous SOA-2 design, but with one quarter the surface area. In
addition to maintaining the previous scale factor and temperature sensitivity, it is
also specified that the resonant frequency of the proof mass along the input axis be
above 4 kHz. This requirement is set so that externally applied vibrations, which are
specified to exist below this frequency, will not cause false acceleration measurements.
To meet these objectives, a design was created that incorporated the use of a lever
arm to produce a factor of four force magnification [15]. This mechanism, attached to
a modified oscillator, allowed the required proof mass size to be reduced while leaving
11
LOW MAG PHOTO OF THE DEVICE.
Figure 1-1: Completed SOA design
the scale factor constant. Other requirements were met by altering beam dimensions
and anchor placement. The completed device is shown in Figure 1-1.
Work during a previous thesis [13] provided the groundwork for this design. This
work, which is discussed in Section 3.1, involved the creation of a Matlab script
that allowed a network of one-dimensional beams to be constructed and analyzed for
displacements and reaction forces. With this script, a number of early designs were
analyzed, and requirements on necessary beam dimensions were found for the device
to function as desired.
Following this work, a more complex finite element model, described in Section
3.3, was created that again used one-dimensional beams for the oscillator and force
multiplier, but also included the proof mass. This model was used to both verify the
results from the previous analysis and to perform a preliminary calculation of the
SOA's expected thermal sensitivity.
Using a simple beam and spring model, described in detail in Section 3.2, the
12
behavior of the new SOA under thermal expansion was analyzed, and an optimal
placement for the lever anchor point was found.
With this information, a finite
element half model of the proposed instrument was created, and modal analysis was
performed to find the device's scale factor and natural modes. This analysis was later
verified with a full FEA model. After final optimization of the beam dimensions, a
thermal analysis was performed. A nonlinear analysis on the oscillator was also carried
out to predict the cubic stiffness nonlinearity of the flexures during oscillation. These
analyses are described in section 3.4. Unique contributions compared to previous
work [13] include creation of a full finite element device model and meeting a number
of design objectives such as temperature sensitivity, scale factor, drive frequency, and
mode placement.
The predicted scale factor of the new SOA design was 91 Hz/g, and the expected
drive frequency was 20.9 kHz. The axial frequency was calculated to be 4.1 kHz,
and the closest natural mode to the drive mode was at 89% of the drive frequency.
Thermal modeling predicted that the device would have a temperature sensitivity of
0.87 ppm/ 0 C, and nonlinear analysis predicted a cubic stiffness nonlinearity that was
four times greater than the previous SOA design.
Devices which were fabricated in the Draper Laboratory micro-fabrication facility
possessed thinner flexures and comb teeth than designed. Test results from these
devices, given in Chapter 4, show that this decrease in mass and stiffness of the
beams caused the average natural frequency of the devices to drop to 16.6 kHz and
the average scale factor to increase to 197 Hz/g. Cubic stiffness nonlinearity measurements were found to be close to predicted values. Temperature tests revealed
a 196 ppm/ 0 C sensitivity of each oscillator, but common mode effects reduced the
sensitivity of the overall instrument to 6.6 ppm/0 C. Further modeling of the device
shows that this difference between expected and measured values of temperature sensitivity occurred because materials underneath the glass substrate were neglected in
the initial modeling stage.
13
Chapter 2
Background
2.1
Device Description
As discussed in
[7],
the SOA is a micromachined device formed from single crystal
silicon using techniques common to semiconductor fabrication.
It is a monolithic
structure that consists of two oscillators attached to a large proof mass. A schematic
representation is shown in Figure 2-1.
Each of the oscillators is made up of two
smaller masses, suspended by flexures, which are driven at resonance through an
electrostatic comb-drive system. Acceleration is sensed by reading the change in
resonant frequencies of the two oscillators due to loads placed on them by the proof
mass.
The proof mass is attached to the glass substrate via anchor beams, which are
machined out of the proof mass. These structures are flexible in the direction of the
input axis (which is the Y-direction in Figure 2-1,) but are rigid along the other axes.
In this way only motion along the input axis is allowed.
The oscillators each consist of two oscillating masses, which are placed in the
center of a flexure. The flexures are attached at either end to base beams, which run
between the flexures. The base beams are then attached to either an anchor at one
end of the oscillator, or to the large proof mass at the other end. This configuration
is sometimes referred to as a "tuning fork" design.
The SOA is operated by driving each of the small masses in an oscillator at
14
-axis
Silicon Proof Mass
Y-axis (Input Axis)
Anchor Point
/
A
X-axis
Base Beam
A
/
'1
VI
F~jih
4AV(
Jimi--
W
C1+_
0
[-.L-7
aff6/i
go
I
-
LL
I
--
iI
7
I
Oscillating Mass
/
Flexure
Oscillating
Mass
I
Line of
Line
of
Symmetry
'/7
Resonator
II
Glass Substrate
Glass Substrate
resonance (referred to as the drive or bias frequency,) 180' out of phase with the other,
in the direction of the X-axis of Figure 2-1. Driving the oscillating masses out of phase
with one another serves to cancel out reaction forces caused by each individual mass'
motion. Excitation of the oscillators is accomplished using a capacitive comb-drive
system, which is described in Section 2.5.
When the SOA is accelerated along the input axis, the proof mass exerts an
equal force on each of the two oscillators. Since the oscillators are arranged in an
opposing orientation, one of the oscillators is placed in tension, while the other is
placed in compression. These forces, which are transmitted equally to the flexures in
the oscillators, cause the resonant frequency of the oscillator in tension to increase
and that of the oscillator in compression to decrease proportional to the amount of
acceleration applied to the device. Taking the difference of these frequencies provides
a measure of the applied acceleration. The amount of differential change in oscillator
frequency due to acceleration is known as the scale factor of the device.
Since other factors, such as variations in device temperature or oscillator drive
amplitude, affect each oscillator's frequency in the same way, most of the errors caused
by such sources are rejected by taking the frequency difference. Slight manufacturing
differences between the oscillators, however, cause small differences in each oscillator's
sensitivity to these inputs. This situation prevents the common mode error rejection
from being total. For this reason, a great deal of effort must be placed into reducing
these types of error sources.
2.2
Basic Operation
This section describes some of the basic equations governing the operation of the
SOA. The flexures and masses that comprise each oscillator can be modeled as shown
in Figure 2-2. In this model, the base beams are assumed to be much stiffer than
the flexures. Each flexure can then be modeled as two cantilever beams with guided
ends, with each attached to the mass, m, at the center of the flexure. Detailed finite
element analysis in Section 3.4 confirms the validity of this model.
16
P
Base Beam
Flexure
Oscillation
ase Beam
Figure 2-2: Simple model of flexure and oscillating mass.
The transverse deflection of a cantilever beam with guided ends is given in Table
10 of [19] as
X =
kP 1
2 tan
U) -- ki
(2.1)
2
where W is the transverse load, P is the axial compressive load (applied to the
flexures by the proof mass), 1 is the length of the beam , E is the Young's Modulus
of Elasticity, I is the area moment of inertia of the flexure, and k = (P/EI)1. The
stiffness, K, of the beam is obtained by
K
W
kP
_
X
2tan
()-
k
i
k
(2.2)
A linear approximation of the stiffness as a function of P can be made by substituting for k and expanding about P = 0. Dropping all second and higher order terms
yields
K ~5
12EI
17
6P
(2.3)
Since each flexure is made up of two of these beams, placed end-to-end, the total
linearized stiffness for one flexure is then
24E1
12P
3
5
K(oscillator =-
51
13
(2.4)
24
The oscillating mass and flexure system of figure 2-2 can now be modeled as a
second order mass-spring system. The natural frequency is then
K(oscilator
m
24EI
=3
12P
=5
m
(2.5)
Equation 2.5 demonstrates how the stiffness of the flexure, and thus the natural
frequency of the oscillator decreases when the flexure is placed in compression and
increases when it is placed in tension. This is the main effect we wish to utilize
in order to measure the acceleration applied to the device. Other effects, however,
also contribute to shifts in natural frequency. These error sources, which include
cubic stiffness effects and temperature sensitivity, are introduced in the following two
sections.
2.3
2.3.1
Stiffness Nonlinearity
General Formulation
An important property of the SOA flexures that affects the oscillators' resonant frequencies is stiffness nonlinearity. This condition describes spring softening or stiffening as displacement amplitude increases. This change in stiffness acts to increase
or decrease the resonant frequency of the oscillator as the amplitude of oscillation
increases. In this case, the springs stiffen with amplitude, and thus the resonant
frequency is increased.
To observe these effects, we first assume that the spring forces caused by a flexure's
18
transverse motion can be described by
(2.6)
F = k1 x + k 2 x 2 + k 3x 3 .
The undamped equation of motion for such an oscillator is then
m, + k 1 x + k 2 x 2 + k3 x 3 = 0.
(2.7)
If we assume a sinusoidal motion of the flexure of
x(t) = X sin (wt)
(2.8)
then substituting into 2.7 yields
-w2X sin(wt) + ki X sin(wt) + k2 X2 sin2(Wt) + k3 X 3 sin3 (Wt) = 0
m
ki
M-
in
w
2
'
m
X sin(wt) +
1 k2
-X
2m
m
cos(2wt)] +
[m-
2
X3
M
(2.9)
- sin(wt)
4
-
I sin(3wt)
4
0.
(2.10)
Since this motion is occurring at resonance, the higher frequency terms involving
2w and 3w can be neglected, as can the bias term. This leaves the equation
ki
m
3
_)2 +
k3X2
4m
)X
= ki
3 k 3 X2
sin(wt) = 0.
(2.11)
Solving for w2 gives
2
Expanding 2.12 about X
=
m
4m
(2.12)
0 gives an expression for the natural frequency as a
function of amplitude
wa =
-+
m
3
ink 3
8
ki m
19
(2.13)
Retaining the first two terms yields
I
1+
=
3
X).
(2.14)
The change in frequency with amplitude is then
-w - 3k3X2
8 ki
wn
(2.15)
Nayfeh and Mook [14] provide a similar derivation of this result, as well as other
methods for finding a more accurate solution to this problem.
2.3.2
Error Predictions
Since the SOA uses shifts in the oscillators' resonant frequencies to measure acceleration, any changes that occur because of the flexure stiffness nonlinearity will become
sources of error. If the oscillators were able to be driven at a constant amplitude,
both oscillator frequencies would shift by the same amount, and this error would be
canceled by taking the difference of the two frequencies. However, since small variations in the drive amplitude exist, the stiffness nonlinearity will contribute to bias
uncertainty.
If we assume that instead of 2.8, the periodic motion of the flexures includes small
amplitude variations
X(t) = (X + AX) sin(wt)
(2.16)
then equation 2.15 becomes
/dw 3k 3
Aw - -(X+
8 ki
Wn
AX) 2 .
(2.17)
Bias uncertainty is caused by frequency variations about a nominal, stiffnessshifted frequency. This variation is the difference between 2.17 and 2.15
(ZAw) 0
Wn
3 k3
k(X + AX)2
8 k,
20
k
8ki X2
(2.18)
(Aw)o = 3 k Wn [2X (AX) + AX2
8 ki
(2.19)
where (Aw). is the change in frequency (due to AX) about a nominal shifted frequency (due to X).
If we now assume that the amplitude of oscillation can be controlled to some level
A =
- ,and if we allow for the common mode effect to provide some error reduction,
C, then equation 2.19 becomes
3 k3
2
(Aw) 0 = 3k*nX
AC
(2.20)
where the higher order terms have been ignored, and C < 1. Equation 2.20 shows
that, while improving amplitude control of the oscillators helps to decrease bias uncertainty, the largest factor in reducing this error is lowering the amplitude at which
the oscillators are driven.
2.4
Thermal Sensitivity
Temperature sensitivity in the SOA is caused by two factors which act against each
other. The first of these factors involves the bonding of the silicon device to the
Hoya glass substrate. The coefficient of thermal expansion (CTE) for Hoya glass
(a,,= 2.533 x 10-6 /oC) [9] is less than that of silicon (a, = 2.75 x 10-6 /oC) [6].
Because of the orientation of the oscillators and the placement of the oscillators'
anchor points, an increase in device temperature causes the flexures to be placed
in tension. This tension causes an increase in their natural frequency, according to
equation 2.5. Section 3.2 provides an analysis of this effect.
The second way in which temperature affects the natural frequency of the oscillators is by lowering Young's modulus of silicon as temperature increases. Equation
2.5 shows that the unloaded natural frequency of an oscillator depends on the square
root of E.
Wf
24EI
3
Un
21
m13
.
(2.21)
The change in the oscillators' natural frequencies as a result of the change in the
Young's modulus of Silicon can be found by differentiating this expression. This
gives
613 m I
EI 1M11
Aw,
AE =
(2.22)
The fractional change in resonant frequency with the Young's modulus is then obtained by dividing equation 2.22 by equation 2.21
=Aw
I
w,,
E
2 (E
(2.23)
As with the stiffness nonlinearity effect, if both oscillators responded in exactly
the same way to temperature, differencing their frequencies would cancel out this
effect. In actual devices, however, small manufacturing differences between the two
oscillators cause slight variations in each oscillator's temperature sensitivity. To correct for this condition, the SOA can be designed so that the two temperature effects
cancel each other in each oscillator.
The Young's modulus of silicon decreases with increasing temperature by
-50 ppm/ 0 C
AEE
AT-
__
(-50 x 10-6 /oC) [12]. This change corresponds to a downward shift
in the SOA's resonant frequency of -25 ppm/ 0 C. In section 3.2, an SOA design with
close to zero temperature sensitivity is created by positioning the oscillator anchor
points so that the CTE mismatch effect raises the natural frequency by this amount.
2.5
Capacitive Drive System
The flexures in the SOA are excited at resonance through the use of intermeshed
comb capacitor drives. Figure 2-3 shows a view of a portion of the comb drive. Each
oscillating mass has two of these capacitors - one to drive the oscillation of the flexure,
and the other to sense the flexure's displacement. A discussion of this type of drive
capacitive system can be found in [16].
22
Stator
Comb Motion
Oscillating
Mass
Figure 2-3: Portion of capacitive comb drive
The capacitance between two charged parallel plates is
Av
C =
(2.24)
9
where
co =permittivity of free space
A = area of comb finger overlap
g = gap between comb fingers
v = fringing coefficient (v ~ 1.5 for SOA combs).
Each comb capacitor consists of N intermeshed fingers, which create 2N parallel plate
capacitors. Referring to Figure 2-3, the total capacitance is then
C=2N
23
WV
9
(2.25)
where I is the comb engagement, and w is the depth of the finger. During operation,
engagement changes with the motion of the rotor comb, x, so that 1 = 1. - x, where
lis the equilibrium value of comb overlap.
These capacitors drive the motion of the flexure by using electrostatic forcing. For
a capacitor, the force acting between the two sides is
F = -V2 0 0
(2.26)
Ox
2
where V is the voltage across the capacitor, and 2 is the change in capacitance with
the transverse motion of the oscillating mass. For the capacitors described above,
DC
Ox
= 2N
cowii
.
g
(2.27)
Note that since the force depends on the voltage squared, applying a zero-bias sinusoidal voltage at the resonant frequency of the flexures will not create any force at
that frequency. To remedy this problem, a bias voltage is added to the voltage drive
signal [18].
While the flexure is driven at resonance by one comb capacitor on the oscillating
mass, the other comb measures the flexure's displacement. This is accomplished by
again utilizing the change in capacitance that corresponds to the motion of the comb
fingers. The current caused by a change in capacitance of the combs is
I =
-(CV)
dt
DV
DC
C
+V
.(2.28)
at
at
By setting the voltage across the capacitor to a constant level, half of this expression is set to zero so that
I = VD C
at
(2.29)
A voltage readout of position can then be obtained by passing this current into a
preamplifier configured as an integrator [18].
24
Chapter 3
Design and Analysis
The first SOA design, SOA-1, created by Kevin Gibbons in his Master's thesis [7], had
a slightly different configuration from that depicted in Figure 2-1, with two flexures
attached to each oscillating mass. The proof mass was approximately 2.6 mm square
and was 12 pm thick. This design had an oscillator frequency of 27 kHz, a scale factor
of 4 Hz/g, and a temperature sensitivity of 1.03 Hz/ 0 C (38 ppm/SC).
To improve scale factor and decrease temperature sensitivity, David Nokes, of
Draper Laboratory, created a second design, the SOA-2, with a layout similar to
Figure 2-1. In this design, the scale factor was increased to 100 Hz/g by increasing
the size of the proof mass to 1 cm square. This increase in mass provided a larger
force per unit acceleration to be transmitted to the flexures, thereby increasing the
oscillators' g-sensitivity.
The proof mass thickness was also increased to 50 Pm.
Nokes' design also featured an oscillator anchor point that was placed between the
two flexures. This placement allowed the two thermal effects discussed in Section
2.4 to cancel each other, and gave the instrument very low thermal sensitivity. The
oscillator resonant frequency of this device was 21 kHz.
This project's main goal was to overcome the manufacturing difficulties caused
by the SOA-2's large size by designing a new device with the same performance as
the current device, but with a proof mass that was at most one quarter its area. To
accomplish this goal, six objectives were established [17]:
25
Anchor Point
Proof Mass
Anchor
Beam
Connecting
Lever Arm
Oscillating
Mass
Flexure
Base Beam
Anchor Point
Figure 3-1: Concept drawing of force multiplying lever arm connecting the proof mass
to the oscillator
" The new device must have a scale factor of about 100 Hz/g.
" The resonant frequency of the proof mass along the input axis must stay above
4 kHz. This limit is set to prevent measurement errors that would be produced
if external shocks and vibrations excited this mode.
" The temperature sensitivity of the device should be as low as possible. This
objective can be met by balancing the change in Young's modulus with stresses
induced by the thermal expansion mismatch between the silicon device and the
glass substrate.
" The stiffness nonlinearity of the flexures must be as small as possible.
* The oscillator resonant frequencies must be between 20 and 25 kHz.
In order to decrease the size of the proof mass, while retaining the sensitivity of
the device, the concept of a connecting the oscillator to the proof mass with a lever
arm was conceived [15]. A schematic of this idea is presented in Figure 3-1.
26
As depicted in this figure, the force multiplier consists of four beams: a lever
arm that provides the force multiplication, two vertical beams that connect the lever
arm to the oscillator and the proof mass, and an anchor beam, which serves as the
pivot point for the lever. By tuning the force multiplier to provide a 4-to-1 force
amplification from the proof mass to the oscillator, the size of the proof mass can
be cut by a factor of four. In this way, the mass would apply the same amount of
force on the oscillator as in the previous design, and the scale factor would remain
unchanged.
3.1
Previous Work
Much of the groundwork for this project was established in my Bachelor's thesis, [13].
During this project, a Matlab script was developed that allowed structures to be
modeled using one-dimensional beam elements. By assigning material properties to
these elements, such as stiffness and area moment of inertia, the structure's behavior
under loading could be studied.
In the Bachelor's thesis, the focus of the design work was tuning the force multiplier to fit the existing oscillator. In this way, many of the characteristics of this
oscillator, such as a resonant frequency of 21 kHz and the placement of other vibrational modes away from the primary drive mode were retained. Since the Matlab
model used linear beam equations, and since it was also not able to calculate thermal
expansion, only scale factor and axial resonant frequency were investigated.
To use the Matlab script, a beam model of the oscillator and force multiplier was
created with a configuration similar to Figure 3-1. Forces modeling the action of the
proof mass were added to one end of the force multiplier, while the anchor points
were held fixed. Resonant frequencies of the oscillator drive mode were calculated by
displacing the oscillating masses to mimic drive motion. By estimating the size of
the masses and calculating the force required to displace them, a measure of resonant
frequency was obtained. Scale factor was calculated by simultaneously applying a
force mimicking the action of the proof mass to the lever arm, and finding the change
27
Reaction Forces vs. Vertical Connecting Beam Width
R.eaction Force at Lever Anchor Point
0.2S
Input Force From Proof Mass
--
0
-0.1
-
- 0.2 -
~0. u.
- 0.3 ---
-- - -
--- ----
-
- -
--- -
-- -- --- -- - Reaction Force at Oscillator Base Beam
---- - - --- ---- --
Desired Force Level
0
0.5
1
Vertical Connecting Beam Width (non-dimensionalized)
1.5
Figure 3-2: Plot of oscillator reaction forces vs. input force
in resonant frequency. Axial frequency of the proof mass was calculated in a similar
way by applying a displacement to the lever arm and calculating the resultant forces.
The results of this analysis showed that because the beams in the force multiplier
were built into each other, rather than connected with pin joints (as would be the case
for an ideal lever) the dimensions of the lever arm must be greater than four to one.
The extra forces created by this lever are taken up in the bending of the connecting
beams and the lever arm itself. For this reason, the connecting beams were made
much thinner than the lever arm.
Figure 3-2 shows the reaction forces created at both the lever anchor point and
the oscillator base beam (this is the force that is transmitted to the flexures.) Beam
widths in this plot have been normalized against the lever arm width so that a value
of 1 on the horizontal axis indicates that the connecting beams have the same width
as the lever arm. This plot clearly shows that the lever arm must be much wider
than the connecting beams to achieve sufficient force multiplication. In the finished
design for this project, the lever arm was 4.5 times wider than the connecting beams.
With this thickness and with a lever ratio of 4.2 to 1, the structure produced a force
28
P
A
Proof Mass
Input Axis
t
Attaches Here
Anchork
4
3
3
4
X4
X2
Lever Ann
X3
2
Oscillator
1
Anchor
Line of Symmetry
Figure 3-3: Spring model of SOA used in thermal analysis
multiplication of 3.8. The axial resonant frequency of the proof mass, which is also
dependent on the size and thickness of the lever arm, remained at 4 kHz at these
dimensions.
3.2
Thermal Sensitivity
In the SOA-2 design, pictured in Figure 2-1, the temperature sensitivity was controlled
through the placement of the oscillator anchor point so that the thermal stresses induced in the flexure beams would counteract the decrease in Young's modulus with
temperature. In the new SOA-3 design, the presence of two anchors, each moving
with the expansion of the Hoya glass underneath, alters the forces that are transmitted to the flexures. In order to study the temperature sensitivity of the SOA-3
oscillator with a force multiplier, a simple model of the SOA was constructed using
springs to represent the compliance of the various components involved. This model
29
was then used to find the forces transmitted to the flexures due to the thermally induced displacements of both the silicon device and the Hoya substrate. The following
analysis follows the work done by Marc Weinberg at Draper Laboratory [17].
Figure 3-3 shows a spring model of the SOA oscillator that includes the force
multiplier. Only one side of the SOA is depicted, and displacements, x, are given
relative to the line of symmetry separating the two oscillators. In this model, the
compliance of the flexures and base beams shown in Figure 3-1 are modeled as one
spring, k2 . Since these beams, which are not pictured in Figure 3-3, can be modeled
as springs connected in parallel, the total stiffnes of the oscillator is then
k
kbbkbbkf
+ k1
(3.1)
31
where
kbb =
stiffness of the base beams
kf = stiffness of the flexures
This spring is attached to an anchor point on one end and a beam representing the
lever arm on the other. The lever arm is assumed to be rigid, compared to the other
elements of the model.
Also included in the model are the beams attaching the lever arm to the the proof
mass and the second anchor point. The load applied externally by the proof mass is
denoted P,, and internal loads of the springs are denoted Fe, where n corresponds
to the spring number. These internal spring forces are defined as positive when the
spring is in tension and negative when the spring is in compression. All elements of
this model are assumed to be connected with pin joints so that there are no moments
transmitted. The validity of this assumption was addressed in Section 3.1.
We are interested in finding the force F2 applied to the flexures due to the thermal
expansion of the Hoya glass and silicon. This force can be found by first identifying
the displacement relationships within the model and solving the resulting equations.
The displacement of the oscillator anchor, x1 due to the thermal expansion of the
30
glass substrate is
1 = 11aoAT
(3.2)
where
11 = the distance from the line of symmetry to the oscillator anchor point
ag = the coefficient of thermal expansion for the Hoya glass substrate
AT = temperature change
For the oscillator, the internal spring forces are
F2
-- XI -
l 2aAT.
k2
(3.3)
From the lever to the anchor point,
F3
k3
= (l1 + 12 + 13) agAT - X3 - 13aAT,
(3.4)
and from the lever to the proof mass,
F4
-
X5
-
-
l4 cxAT
(3.5)
where as is the coefficient of thermal expansion for silicon. Force balance requires
that
F3 + F4 = F2,
(3.6)
nF 4 = F 2
(3.7)
and torque balance gives
where n is the force multiplication (in this case n = 4.) Finally, for the lever arm,
the nodal displacements have the relationship
(X4 - x 3 ) = n(x 2 - X 3 ) .
(3.8)
The stiffness of the structure along the input axis, kax, can be found by setting
31
AT = 0 and applying a proof mass load Pp = F4 . Solving equations 3.3 through 3.8
yields
1
x5
n2
kax
Pp
k2
ka
ax
=
(n
-1)
- -+
k3
2
+
1
(3.9)
k4
k2 k 3 k 4
k 2 k3 + (n - 1) 2 k 2 k 4 + n 2 ksk 4
.(.0
Since the proof mass is much more rigid than either the oscillator or the force
multiplier, we can assume that it moves independently from them under thermal
expansion. With this in mind, the displacement of the proof mass at the point of
connection with the force multiplier is then
X5 = (11 + 12 + 14)
asAT.
(3.11)
The force transmitted to the flexures due to both acceleration and thermal expansion
can be found by solving this equation and equations 3.2 through 3.8 for F 2
Fflexure = F 2 =
nPp + nkax [(12 + 13) (n - 1) - 11] (ag - a,) AT.
(3.12)
In the SOA-2 design shown in Figure 2-1, the lack of a force multiplier (n =
1) causes the above expression to revert to the expected form, where the thermal
component is described by -k 211 (ag - as) AT.
The axial stiffness term, kax, also
reflects the added compliance produced by the lever arm. Since the area of the proof
mass in the SOA-3 is reduced by a factor of four, the axial stiffness must decrease by
the same amount for the axial resonant frequency to remain the same. With this in
mind, we see that the term nkax remains unchanged between the SOA-2 design and
the SOA-3 design (n = 4).
Since the change in Young's modulus with temperature reduces the natural frequency of the oscillator, the thermal component of equation 3.12 must be positive to
counteract this effect. The coefficient of thermal expansion for Hoya glass is less than
that of silicon, so the term [(12 + 13) (n - 1) - 1i] must then be negative. In the previous SOA-2 design, this term was equal to -li, since there was no force multiplier.
32
Proof Mass
Lever Arm
Base Beam
Oscillating
Mass
Flexure
=--Anchor Point
Figure 3-4: Concept drawing of oscillator and force multiplier with anchor beam
repositioned for thermal compensation
In the new design, the force multiplier anchor must be flipped, as in Figure 3-4, so
that this term has the same value as before.
3.3
Beam Models using ANSYS
As a first step in creating a full model of the SOA-3 using the concepts defined in
the previous section, a second beam model, similar to the Matlab model of Section
3.1, but analyzed using ANSYS 5.5 finite element software, was created. This model,
in which the oscillator was again modeled with one-dimensional beam elements, was
more detailed than the previous model in that it included both the proof mass and
the proof mass anchor beams. Including all of the device's components in this model
allowed easy modeling of the full device later by simply laying the actual geometry
over the beam elements. All geometry was created in Pro/Engineer and exported
into ANSYS for analysis. This model is shown in Figure 3-5.
To create the model, a three-dimensional shape representing the proof mass (the
33
Figure 3-5: Beam model of SOA created in ANSYS
34
shaded, meshed portion in figure 3-5) was first drawn in Pro/Engineer. Datum points
were then drawn and connected with datum curves to form the shape of the oscillator
and proof mass anchor points. Using the Pro/Engineer finite element package, onedimensional beam elements were then laid over the datum curves, and the proof mass
geometry was meshed using 8-node two-dimensional shell elements (shell93 elements
in ANSYS.) The beam elements were then assigned material properties and an area
moment of inertia according to the actual dimensions of the beam that the element
represented. The proof mass elements were also assigned material properties and a
thickness of 50 pm. For this analysis, as well as future analyses, the following material
properties were used:
" E (Young's Modulus) = 1.68 x 10
"
11pN/pm 2
v (Poisson's Ratio) = 0.26
* G (Shear Modulus) = 6.67 x 10 1 0pN/pm 2
"
p (Density) = 2.33 x 10-/g/M2
For these material properties, a Young's modulus corresponding to beams in the
(110) orientation was used, and isotropic properties were assumed for the Poisson's
ratio and shear modulus. This simplification has little effect on the model since a
slender beam approximation for bending, which does not include these properties,
can be used for the flexure beams. Further finite element models using orthotropic
properties have been used to validate this assumption.
Once this model had been created, it was then exported into ANSYS. Oscillator
drive frequency was calculated by first applying fixed boundary boundary conditions
to all anchor points and then using ANSYS' Block-Lanczos solver to find the device's
natural modes and frequencies. The proof mass axial frequency was also found in this
manner. Scale factor was calculated by applying a ig acceleration to the model in
the direction of the input axis. A static analysis was then performed to calculate the
resulting stresses and displacements within the device. This prestress data was then
35
included in the modal analysis to find the device's new oscillator frequencies. Scale
factor was then calculated by taking the difference of the shifted frequencies.
This model predicted an oscillator drive frequency of 25.1 kHz and a proof mass
axial frequency of 5.1 kHz. Because of the added stiffness of the proof mass anchor
beams, the scale factor of this model dropped to 86 Hz/g. This number was increased
by lengthening the lever arm by 30 pm so that the lever ratio was 5:1. This change
resulted in a scale factor of 96 Hz/g.
A thermal analysis was also performed on this model by including a representation
of the Hoya glass substrate. This was done in Pro/Engineer by creating a large block
of Hoya glass directly underneath the proof mass. The oscillator and proof mass
anchor points were then connected to the glass using rigid beam elements, and the
glass substrate was then meshed using the same type of elements as the proof mass.
A coefficient of thermal expansion of 2.533 ppm/ 0 C was then assigned to the Hoya
glass, and one of 2.75 ppm/ 0 C was assigned to the silicon. Since the Hoya substrate
is much thicker than the silicon, the elements representing it were given a much
greater stiffness than silicon. In this way, the movement of the anchor points due to
temperature changes would be driven solely by the Hoya glass.
To find the thermal sensitivity of the model, a temperature increase of 1 'C was
applied. A static analysis was then performed to find the stresses induced in the
oscillator by the mismatch in CTE between the Hoya glass and silicon. A modal
analysis was then performed with the prestress data included, and the resulting oscillator frequency was compared to the unloaded bias frequency. With an overlap
between lever anchor and the oscillator anchor in Figure 3-4 of 6 = 43pum the thermal
sensitivity due to CTE mismatch was calculated at +25ppm/ C. Since Section 2.4
shows that the change in the Young's modulus with temperature for silicon creates a
-25 ppm/ 0 C change in oscillator frequency, this design was shown to have zero temperature sensitivity, in theory. With the proven concept of positioning the anchor to
attain the desired thermal sensitivity, a detailed model of the device could then be
constructed and analyzed.
36
Proof Mass
Anchor Points
Lever Arm
Flexures
Figure 3-6: Finite element model of one half of the SOA
3.4
Detailed Analysis
With the knowledge gained from the preliminary analyses a detailed finite element
half model of the SOA with force multiplier was created. This model is pictured
in figure 3-6. A half model of the device was used in the analysis to reduce both
modeling effort and computational time. This simplification is possible because of
the symmetry of the device along the X-axis. This model was used for all design
iterations, and a full model was then created to verify the performance of the final
design.
To create this model, Pro/Engineer was used to lay the actual beam geometries
of the oscillator components directly over the datum curves of the model created in
section 3.3. The thicknesses of each of the beams were determined initially from the
area moments of inertia of the beams in the final one-dimensional model. Attaching
the beam geometries to the datum curves from the previous model allowed for easy
manipulation of dimensions during design iterations.
37
Once this model had been drawn in three dimensions in Pro/Engineer, its geometry was exported to ANSYS and converted into a two-dimensional model. ANSYS
Plane82 elements were used to mesh the parts. These elements each have eight nodes
and are used to model plane stresses. They were used in all in-plane modal and thermal analyses for this project. The same silicon material properties used in previous
models were used except for the density of the proof mass silicon. Since, in production, holes are placed in the proof mass, its density was reduced by 10% in the model
to reflect the reduced weight.
The primary purpose of this analysis was to accurately characterize the device's
performance. This characterization included finding all natural frequencies and mode
shapes of the device, determining the device's thermal sensitivity, and finding the
nonlinear characteristics of the flexures. Modal analysis was performed in the same
fashion as the analysis in section 3.3. Fixed boundary conditions were placed on the
anchor points, and, depending on the analysis being performed, either symmetric or
anti-symmetric boundary conditions were placed on the proof mass symmetry line. An
anti-symmetric boundary condition allowed free movement of the proof mass parallel
to the input axis, but restricted all other movement. This boundary condition was
used in finding scale factor and proof mass axial frequency, as well as several out-ofplane modes found in later analyses. Symmetric boundary conditions allowed only
movement perpendicular to the input axis. This boundary condition was used to find
the "cross-axis" frequency of the proof mass and also to find out-of-plane modes.
3.4.1
Modal Analysis
The first goal in developing a final design was to fine tune the beam dimensions so
that the design objectives outlined at the beginning of this chapter were met.
The
two beam dimensions studied in this exercise were the base beam width and the lever
arm width. Both of these dimensions play a large role in the determination of scale
factor and proof mass axial frequency. The Figures 3-7 and 3-8 show the results
of design iterations performed with these parameters. The final SOA-3 design was
chosen with a base beam width of 45 pm and a lever arm width of 125 pm. With
38
Effect of Base Beam Width on Axial Freq., Drive Freq, and Scale Factor
4500
6-2 3 50 0 - . . .. . .
3 0 00 -
-
-.
.-. .-
.-..-..-..-..-...-..-..-..-..-..-.. .-.-. .. . . -... -.
-.
25
.
-..-.-.-..-
-.-.-
.-. -.
250020
-. .-.-..-.-.
. .-.-.
.
. -.
.-..-.-..-
. .. . .
4 00 0 - .-.
N
30
35
40
Base Beam Width (microns)
45
50
55
35
40
Base Beam Width (microns)
45
50
55
35
40
Base Beam Width (microns)
45
50
55
21
0
U)
19.5 C) 120.5
20
0..
---
--
-
25
30
95
18.5
. .. . . . . .. .. . .. .
0
8 5 -.
20
-.-.-.-.-.-.
25
30
Figure 3-7: The contribution of base beam width to proof mass axial frequency, drive
frequency, and total scale factor (frequency difference of two oscillators.) Lever arm
width for this analysis is 125 pm.
39
Effect of Lever Arm Width on Proof Mass Axial Freq. and Scale Factor
4500
-
4000
-
-
-
-
-
-
- -
-
I
a, 3500
- -
-- -
- ......
- - - ........
- -
........
........
-..
........-
U-
x
-
3000
. .. .. .
2500
6)
80
70
. . . . . . . . . ..
90
..
.
100
110
120
Lever Arm Width (microns)
.. .
130
. . . . .- . . . . . .
140
150
160
95
90
-.-
85
.
-
-
- ..
-
- .........
- .
-
-
-
-....-....
- - - - . - ......-..
.-
-
.
-.
0
a) 80
-
--
.- -
-
-
- .
......
- - -..
.....
- - . - - - .......
- -..
......
........
- -..
.-
0
Ci)
75
70
6)
I
I
I
70
80
90
I
I
I
100
110
120
Lever Arm Width (microns)
I
I
I
130
140
150
Figure 3-8: The contribution of lever arm width to proof mass axial frequency and
total scale factor (frequency difference of two oscillators.) Base beam width for this
analysis is 45 /Lm.
40
160
these dimensions, the device had a predicted drive frequency of 20.3 kHz, an axial
proof mass frequency of 4.1 kHz, and a scale factor of 91 Hz/g.
These figures show that both parameters have equally important effects on the
axial resonant frequency and scale factor of the device. For each parameter, a 50%
decrease from the final dimension results in a 20% decrease in scale factor and a 35%
decrease in axial frequency. It is observed that the scale factor nearly reaches a steady
state with the chosen dimensions, so any further increase must be accomplished by
increasing the lever ratio. Axial frequency would also reach a steady state if the
thickness of the beams were increased further, but this amount of rigidity was not
needed in the current design. The effect of base beam width on drive frequency is not
as dramatic as the effects on scale factor and axial frequency, but it is still important
in retaining the 20 kHz drive frequency of the previous design.
With these key parameters successfully defined, further modal analysis was conducted to check the placement of other vibrational modes of the device. In-plane
modes were found using the same model as in the above analysis, and out-of-plane
modes were found by replacing the elements with ANSYS Shell93 elements. These
two-dimensional elements allow the modeling of thin structures in three-dimensions,
as long as no torsional forces are present. For the SOA, these elements were an ideal
choice to reduce model complexity, while ensuring accuracy.
Two full models of the SOA-3 were also created to verify the placement of the
in-plane modes. In these models, two oscillator configurations were used. The first
configuration, pictured in a completed unit in Figure 1-1, arranged the oscillators in
a mirrored orientation. The other configuration, shown in Figure 4-1, positioned the
oscillators to be rotationally symmetric about the center of the device. Out-of-plane
modes of the full models were not calculated because the use of shell93 elements
produced very computationally intensive models.
Table 3.1 gives a list of all the relevant modes of the device calculated with the half
and full models. All in-plane modes are given using the results from the full models,
and the out-of-plane modes are given using the half-model results. In this table,
"Hula" refers to the mode in which the flexures oscillate in phase with one another.
41
Fundamental Modes (Hz)
Drive
Axial
20,281.5 20,281.8
4,141
Other In-Plane Modes (Hz)
"Hula" 16,702 16,721
Rotational 18,251 (89% of drive)
Perp. to Flexures . 24,482
Out-of-Plane Modes (Hz)
Low Freq.
5,223
5,900
6,161
11,107 11,510 15,544
Close to Drive
High Freq.
17,933 (88% of drive)
26,436 28,915
Table 3.1: Modal frequencies of SOA with force multiplier. All in-plane modes were
calculated using the full FEA model, and out-of-plane modes were calculated using
the half FEA model.
The "Rotational" mode, which at 18.3 kHz is the closest to the drive mode, refers
to the rotation of the proof mass about its center. Appendix B contains half-model
diagrams of all of these modes.
In each of the full models, two identical oscillators showed a drive frequency separation of 0.3 Hz, which is due to the coupling existing between them through the
proof mass. This coupling prevents the two oscillators from operating independently
from one another, and it creates the possibility that the two frequencies will "lock"
together, making it impossible to take differential frequency measurements. Since
this phenomenon, which is being studied further at Draper Laboratory [5], occurs
when the two drive frequencies are close to one another, the final design submitted
for fabrication contained oscillators with drive frequencies separated by 1000 Hz. This
separation, which was created by thinning the flexures of one oscillator by 0.2 pm,
avoids mode locking by reducing the possibility that the oscillator frequencies will
cross one another during operation.
3.4.2
Thermal Modeling
A thermal analysis similar to the one described in section 3.3 was performed on the
finite element half model, and an optimal length for the force multiplier anchor beam
42
ANSYS 5.5.1
MAY 10 2000
15:12:41
DISPLACEMENT
STEP=1
SUB =4
FREQ=20282
PowerGraphics
EFACET=1
AVRES=Mat
DMX =1
*DSCA=-55. 085
ZV
=1
*DIST=2750
*XF =1250
Z-BUFFER
Drive Mode
Figure 3-9: Drive mode for SOA
43
ANSYS 5.5.1
My 12 2000
13:18:29
NODAL SOLUTION
STEP-1
SUB -2
FREQ-6105
UZ
(AVG)
RSYS-0
PowerGraphics
EFACET-1
. ---
AVRES-Mat
DM
-1.652
SM -- 1.529
smx -1.652
-- 1.352
A
--. 69399
c
_
D
E
a- 291917.
-. 061475
'F-
-4 1 1 V2
I
=1.475
Out of Plane Mode (Symmetric)
Figure 3-10: Out-of-plane mode for SOA
44
was chosen. Rather than create a computationally intensive shell model containing
both the silicon part and the glass substrate, the plane stress model created for modal
analysis was used.
To model the effects of both the glass and silicon expansion,
the anchor points were given fixed boundary conditions, and the silicon part was
given a CTE equal to the difference of silicon and Hoya glass. By doing so, the
silicon expanded the same amount relative to the fixed boundary conditions as it
would relative to the expansion of the actual glass substrate. This simplified analysis
was verified by comparing its results with another analysis performed by displacing
each node along the anchor point by an amount equal to the expansion of the glass
substrate and then assigning the CTE of silicon to its true value. In each of these tests,
the same temperature sensitivity was derived. Final analysis predicted a thermal
sensitivity of +25.87 ppm/OC with an anchor point overlap of 6 = 13pm. Combined
with the change in Young's modulus with temperature, the sensitivity was predicted
to be 0.87 ppm/ 0 C. A change in thermal sensitivity of ±0.14 (ppm/ C)/pm was
predicted for movement of the anchor point around this position.
3.4.3
Nonlinear Analysis
A nonlinear analysis was performed on the oscillator to predict the cubic stiffness
terms introduced in section 2.3. To perform this analysis, a piece of the model from
the above analyses was taken and modified slightly. Figure 3-11 shows the model
with its various components labeled.
In this model plane82 elements were again used, and fixed boundary conditions
were placed on both anchor points. Since the proof mass is much stiffer than the oscillator, the end of the force multiplier connecting to the proof mass was also modeled
as fixed. To find the force vs. displacement curve for the flexures, opposing loads were
applied to the two small proof masses. A nonlinear analysis including stress-stiffening
effects was then performed in ANSYS. During this calculation, the software breaks
the force applied into a number of load steps. Within each step, the displacement of
the flexure resulting from a portion of the total load is calculated, and the internal
stresses resulting from this displacement are also found. The next load step then uses
45
FNF7
Anchor
points
11AMAN
F
Base
Beam
Flexure
Proof mass
Small Proof
Masses
Force
connects
Multiplier
surface. igs
Figure 3-11: Finite element model of oscillator and force multiplier used in nonlinear
analysis
46
Nonlinear Stiffness Curve for SOA Oscillator
SX 108
3
2
..............
- -.
.... ........... .....
.. ..
- .... --.
-
.. ...-
.
0
0
-2
-3
-4
-15
-10
-5
0
Flexure Displacement (um)
5
10
15
Figure 3-12: Predicted nonlinear stiffness curve for flexure displacement
these stresses in its calculation of force and displacement. This process continues
until the entire load has been applied. The result of this analysis is shown in figure
3-12.
Equation 2.6 gives an expression for force vs. displacement for a cubic spring
F = kix + k 2 x 2 + k3 X3 .
(3.13)
The calculated coefficients for this expression are given below. Note that the presence
of a quadratic term means that the flexures experience spring softening for inward
displacements and spring stiffening for outward displacements.
k,
=
20.93 x 106 pN/p-m
k2= 75.65 x 103 pN/pum
2
47
k3 = 61.38 x 103 pN/pm 3
These coefficients have a ratio g-k-that is 4 times greater than those calculated for
the previous SOA design, which had a similar oscillator, but no force multiplier. This
difference can be explained by the change in axial stiffness that occurred between
designs. Since the proof mass was reduced by a factor of four in the new design, the
axial stiffness of the oscillator was also reduced by the same amount so that the proof
mass axial frequency would remain constant. Appendix A gives a brief explanation of
how this decrease in stiffness acts to increase the stiffness nonlinearity of the flexures.
48
Chapter 4
Fabrication and Testing
The final design for the new version of the SOA was fabricated at Draper Laboratory
using a silicon-on-glass bulk dissolved wafer process similar to one described in [7].
Figures 4-1 and 4-2 show typical devices mounted on the glass substrate. Two of the
bond pads, which provide electrical connections for the device, can be seen on the
middle right and left-hand side of the image.
Two configurations of oscillators were fabricated so that both symmetric versions
could be tested.
Figure 4-1 shows one configuration in which the oscillators are
rotationally symmetric with one another, and Figure 4-2 shows the other configuration
in which the oscillators mirror each other. Despite finite element models of each of
these configurations showing agreement with the half model predictions from chapter
3, it was decided that both configurations should be produced to ensure that any
significant, unmodeled properties of the device had not been overlooked.
In addition to the two oscillator configurations, three anchor beam lengths were
also fabricated to test the thermal sensitivity predictions from section 3.4.1. These
three configurations had anchor point overlaps of 6 = -7,
13 (original dimension),
and 33 ytm. Overlaps of this size were predicted to give the device a temperature
sensitivity of 0.9 ± 2.8 ppm/0 C (Young's modulus thermal effect included.)
In order to prepare the devices for testing, all manufactured units were first inspected visually. Units that were free from debris and had all beams, flexures, and
comb teeth intact were tested using a probe station to ensure that all vibrational
49
Figure 4-1: SOA with rotationally symmetric configuration
LOW MAG PHOTO OF THE DEVICE.
Figure 4-2: SOA with mirror configuration
50
modes were functioning properly. The probe station allows temporary electrical connections to be made to the device so that initial characterizations can be performed
without having to package it.
After devices with satisfactory performance had been chosen, they were then packaged inside a leadless ceramic chip carrier (LCCC). This package allows the device
to to be handled more easily and enables it to be connected to the test electronics.
The SOA is mounted inside the LCCC using a small braze pre-form which is placed
between the SOA chip and the LCCC near the center of the SOA. A permanent bond
is then formed by heating these components. Once the SOA has been bonded to the
LCCC, wire bonds are run from the SOA bond pads to the corresponding pins of the
package.
Units are tested by placing the LCCC into a "fuzz button" package, and then connecting this package to the control electronics. The package, which allows LCCC's to
be connected and removed from the electronics with relative ease, contains a preamplifier for the output signal of each oscillator, as well as a temperature sensor. The
fuzz button package is attached to the electronics with a "guillotine" connector, in
which long electrical pins protruding from the fuzz button package are clamped to
the test fixture.
Since, for devices such as the SOA, the air surrounding the device is a major
source of damping, the SOA's are operated in a vacuum environment of less than
5 mTorr. After each SOA is packaged, it is first tested in an evacuated bell jar for
initial characterization of scale factor, drive frequency, and quality factor. If a device
is judged to have performed well on each of these tests, it is sealed in a vacuum inside
its LCCC by brazing a lid on top of it.
4.1
Fabrication Results
In order to determine the beam dimensions of the fabricated devices, destructive
tests were performed on defective units using a scanning electron microscope (SEM)
to measure all critical dimensions. Figure 4-3 shows the profile of a flexure from one
51
Figure 4-3: Profile of flexure beam
of these units.
In this figure, the beam has been inverted so that what would be the top of the
beam is at the bottom of the image. Considering that the designed thickness of the
flexure is 7 pm, evidence of beam thinning due to over-etching of the silicon is clearly
visible in this figure. Since beam thickness is important in determining the stiffness
of the flexure, this thinning is expected to lower the oscillator's natural frequency and
increase the scale factor. The following section discusses the expected effects of beam
thinning on device performance, and it presents experimental data supporting these
predictions. Figure 4-4 shows an SEM image of several comb finger profiles. These
comb fingers were designed to have a thickness of 4 pm. The figure shows that in
an actual device, this thickness has been reduced to as little as 2 pm at the center
of the finger. This thinning of the comb fingers contributes to a significant drop
52
Figure 4-4: Profile of comb finger
53
in the mass of the oscillator masses, and acts to raise the natural frequency of the
oscillator. Other non-destructive measurements have been made on other units using
an optical microscope. These measurements agree with those made by the SEM, and
even suggest that other units may have thinner beams that those shown in the figure.
4.2
Scale Factor and Drive Frequency Testing
The scale factor and oscillator drive frequency of the fabricated devices were tested
by using an HP Dynamic Signal Analyzer 35665A to perform an open loop frequency
response of the oscillator. This was done by driving the inner combs of each oscillating
mass with a varying sinusoidal signal plus a bias voltage. The motion of the masses
was derived from the output of the outer combs, according to the equations in Section
2.5.
Drive frequency was measured while the SOA was positioned in a horizontal orientation so that gravity would not affect the oscillator frequency. Scale factor was
measured by recording the natural frequency in this orientation and then repeating
the measurement with the SOA's input axis oriented vertically. The scale factor results presented below are given in terms of a single oscillator, rather than the device's
total sensitivity. This was done to demonstrate the that each oscillator in a device
was performing equally well.
Figure 4-5 shows a typical frequency response plot from one of these tests showing the resonance at drive frequency. Using the equations given in Section 2.5, an
amplitude of 0.1 pm is calculated at resonance. This agrees well with the expected
amplitude of 0.3 pm, calculated using the quality factor measured in Section 4.3 and
the beam stiffness measured in Section 4.6. The measured natural frequency is significantly lower than the one predicted in table 3.1, but considering the beam thinning
presented in the previous section, this low frequency is expected.
Figure 4-6 shows the drive frequencies and scale factors of twelve units. Each
point on the graph represents a single oscillator. For all units, the frequency separation between oscillators was measured at approximately 1000 Hz, which agrees with
54
X-15.
6222721H
Yc-12. 259
ciR
EF
PREQ
C
Y8. 0
.........
15. E11823k
Yb-~- .258. 37 D~a
FRFQ
RESP
--.-----.
15.
H
31823ri
-440
is.
6 1 18E23k
E31B23k
H=15-
Figure 4-5: Frequency response of an SOA
the intended design mentioned in Section 3.4.1. This section also predicts a single
oscillator scale factor of 46 Hz/g. In Figure 4-6 the low frequency and high scale factor due to beam thinning during production is observed, as is the trend of increasing
scale factor with decreasing frequency. The following analysis shows how this trend
can be predicted using the equations of beam theory.
Equation 2.5 states that the natural frequency of an oscillator is given by
24EI
W
=
12P
m
(4.1)
51
where the beam properties correspond to the flexures and m is the modal mass of
the tuning fork. The two factors that affect the natural frequency of the oscillators
are flexure width and oscillator mass. Given that the area moment of inertia, I, is
Lhw3 , where h is the depth of the flexure and w is the width, the unloaded natural
frequency of the beam (P = 0) is observed to vary by
W.
The load P on each flexure is equal to one quarter of the force produced by the
55
SOA-3 Scale Factor vs. Drive Frequency
160
Measured Values
0 Calculated FEA values
*
140 -
-- *
120
*
0
100a
**
8 0 - - -.--
40
13
- -.-
14
15
.--.
.--.-.
-.
-.
16
18
17
Oscillator Drive Frequency
19
20
(kHz)
Figure 4-6: Scale factor vs. drive frequency for twelve SOA units. Values calculated
using FEA are also shown.
56
large proof mass under acceleration.
P
Ma
(4.2)
4
where M is the mass of the proof mass and a is the applied acceleration. Substituting
this expression into equation 4.1, expanding, and differentiating with respect to a gives
the approximate scale factor for the oscillator:
S.F. ~ M
40
mEI
M
(4.3)
11
Substituting for I reveals that scale factor is expected to vary by
mw
3
.
As a sample calculation a case can be considered where the flexure width has
been reduced from 7 to 5.8 pm and the tuning fork modal mass, which includes both
the oscillating masses and a portion of the flexure mass, has been reduced to 70% of
its original value. For these conditions, the resonant frequency is expected to drop
to 18.3 kHz, and the scale factor is expected to increase to 78 Hz/g. In another
case in which the flexure width is 5 pm and the modal mass is 60% of its original
value, a natural frequency of 15.7 kHz and a scale factor of 107 Hz/g is expected.
Both of these predictions match the measurements displayed in Figure 4-6. While
it is difficult to predict the precise relationship between flexure thickness and mass
loss due to over-etching of the comb fingers, these calculations clearly show that the
mechanism described in the above equations is at work in the tested SOA's.
In addition to these calculations, three-dimensional finite element models have
been created at Draper Laboratory [2] to verify these results. These models started
with beam dimensions similar those measured on actual devices and then uniformly
varied comb finger and flexure beam widths to predict the relationship between scale
factor and bias frequency. The results of these analyses, shown in Figure 4-6, show
an average 20% difference between these calculations and the actual test data. Since
these FEA models were created using the assumption that all beams and combs were
uniformly etched and had rectangular cross-sections, it is clear that models including
57
Typical Ringdown Test
.i5n
1
2
0
C
0
a-
a
ci
0
-1
I
-0.05
I
0
0.05
0.1
0.15
0.2
Time (s)
0.25
0.3
0.35
0.4
0.45
Figure 4-7: Oscillator position signal during ringdown
beam tapering and non-uniform etching would produce results agreeing more closely
with measured data.
4.3
Q
Testing
The quality factor is defined by
1
(4.4)
where ( is the damping ratio for the system. For the control electronics to operate
properly by locking on to the oscillator drive frequency, the
must be high. All units were therefore tested for
Q before
Q factor
of the device
undergoing further tests.
Low Q's in a device can be caused by material damping, physical defects, or an
improper vacuum seal. Some variations in
Q were
noticed between units that had
been fabricated differently, and most of the devices experienced a slight increase in
Q after having
been sealed inside the LCCC.
58
Q
measurements were performed by driving the oscillator open-loop near reso-
nance and at a specified amplitude. The input signal was then quickly shut off, and
the resulting "ring down" of the oscillator signal was recorded using a Tektronix
TDS754C digital oscilloscope. An example of this data is seen in figure 4-7.
The envelope of this ringdown is
X (t)
Xoe~-"Q
(4.5)
where
X(t) =oscillator position
XO =Initial amplitude of oscillator
f, = natural frequency (in Hertz) of oscillator
The position data from a ringdown test can be exported to a program such as MS
Excel or Matlab, where an exponential curve fit can be performed.
Q
The average
factor for the tested sealed units was 88,900, with a maximum of 138,000 and a
minimum of 53,000.
4.4
Temperature Sensitivity
Temperature sensitivity tests were performed by placing sealed units in a temperature
controlled foam container. Heating devices located on the outside of the fuzz button
package were used to control the temperature of the device to with one tenth of
a degree Celsius. Temperature data was taken from sensors placed inside the fuzz
button package. To test a device, the oscillators were driven closed loop, and the
temperature was varied slowly to allow any transients to die away.
Each oscillator
was monitored separately so that any differences between the two could be identified.
Figure 4-8 shows the temperature sensitivity of one oscillator measured from this
test. From the curve fit displayed in the figure, a temperature sensitivity of approximately 4 Hz/ 0 C, or 255 ppm/0 C was measured, which is far from the designed
59
1.572r
Temperature Sensitivity of Single SOA Oscillator
X 104
I
I
I
1 .5 7 1
I
- -
-
--
-
I
- -- -
- -
-
- -
S1.57
4.0017x + 15518
y
0*
1 .56 9
-- -- -
-
.
-.-- .--..-- -.
-.
-.
.. .
. . .. . . . .
0
~1 .5 68 .
1 .567
1.566
36
. .. .
- -
.. . ...
-
-
38
II
40
- -
-- -
-
42
44
Temperature (deg C)
--
. .. . .
-
--
46
- - - -
-
48
50
Figure 4-8: Thermal sensitivity data for one SOA oscillator
sensitivity of 0.87 ppm/0 C. When the differential frequency of two oscillators was
taken, however, this sensitivity dropped by a factor of fifty to -0.069 Hz/G, or 4.4
ppm/'C. This data is shown in figure 4-9. Here we see the advantage of taking the frequency difference of the two oscillators; though both oscillators shift their frequency
significantly, the overall acceleration measurement of the device changes very little.
The average temperature sensitivity for ten tested oscillators (five units) was 196
ppm/'C, and the average common mode temperature sensitivity was 6.6 ppm/0C.
Despite this promising data showing excellent common mode rejection, it is still
desirable reduce the temperature sensitivity. Preliminary finite element models have
shown that the unexpected thermal sensitivity originates from the material that the
Hoya glass substrate is attached to [2].
These models have shown that the braze
material and the LCCC surface, which were both neglected in the thermal modeling
during design, have a significant effect on the overall thermal sensitivity of the device.
Models including these effects predict a temperature sensitivity of 214 ppm/C, which
60
Total Temperature Sensitivity of SOA
-. -.......
654.9
-.-.-..--.--.----.--.--.
654.8
y -0.069
+
657.424
I654.7
654.6
00
654.5
a654.4
**
5 654.3
0
654.2
654.1
*
b!)4 -
36
- -
i
38
40
44
42
46
48
50
Temperature (degC)
Figure 4-9: Thermal sensitivity data of the SOA. Here, the frequency difference has
been taken to show the overall effect of temperature on the device.
61
X- 14. 587kH2z AX-3. 4kHz
Yon-4. 9113
AYc-26. 31
20Avr3
POWER SPECI
%~ipHrr
D2 iv
11.187KHz
--L 10
17,987KHz
L..i..r.
9. 625k
H,1
9.
625k
Figure 4-10: PSD of amplitude modulated drive signal showing the 3.4 kHz axial
frequency of the proof mass.
agrees well with the measured data.
4.5
Axial Frequency
The proof mass axial frequency was measured by externally exciting the axial mode
while operating the oscillators closed loop. This was done by positioning the input
axis in a neutral orientation, and lightly tapping the device along this axis. The
resulting vibration of the proof mass causes the tension in the flexures to change,
thus modulating the frequency output of the oscillators. This motion also causes the
position of the flexures to change, thus modulating the amplitude of the oscillator
position signal. Figure 4-10 shows a PSD of the amplitude modulated position signal.
In this figure the sidebands around the 14.6 kHz drive frequency reveal a 3.4 kHz axial
frequency of the proof mass. This measurement is within 17% of the predicted value
of 4.1 kHz, and the difference can be explained by the reduction of axial stiffness
62
Frequency vs. Amplitude
X 10,
1.644 r
~/
1.642-
1.64
Measured Curve
25.31x + 5.09x + 16327
cr 1.638
A
.
1.636-
Calculated Curve
25.12x 2 + 16327
1.632
0
0.2
0.4
0.6
0.8
1
1.2
Amplitude (um)
1.4
1.6
1.8
2
Figure 4-11: Plot of measured and calculated nonlinear stiffness curves
caused by beam thinning during production.
4.6
Stiffness Nonlinearity
The amplitude stiffness nonlinearity of the flexures was tested by varying the amplitude of oscillation while driving the oscillators closed loop. In order to derive the
amplitude of oscillation from the position output voltage signal, the value of 1 from
Section 2.5 was first found by creating a two-dimensional model of the comb capacitors using the PDEase software package. This software allowed the capacitance of
the actual manufactured parts, which were thinner than designed, to be calculated.
This provided a means to obtain the true change in frequency with drive amplitude.
Figure 4-11 shows the measured nonlinear curve along with the curve predicted
in Section 3.4.3. In this plot, the measured quadratic term differs from the expected
curve by less than 1%. Curves measured on other units did not agree as well as the
one pictured, but overall agreement was satisfactory. In general, the units all had
quadratic terms, which are determined by a, that were higher than those predicted
63
earlier. The equations presented in Appendix A show that since this term is inversely
proportional to the area moment of inertia of the beam, the measured value in tested
devices this should increase due to the thinning of the beams during production. The
tested devices had an average measured quadratic term of 34.26 Hz/Mm 2 , which is
36% higher than the predicted value.
64
Chapter 5
Conclusions and Recommendations
Tests performed on the new SOA-3 design show good agreement with predicted performance characteristics.
Difficulties in fabrication have prevented the units from
being built to specification, and this situation makes it more challenging to truly
characterize the instrument. Further analysis has shown, however, that most differences between the predicted and actual performance originate from these build
imperfections.
From these tests and analyses, it is clear that the following performance objectives
have been met:
* Devices built to design specifications will have a total scale factor of approximately 100Hz/g.
o Oscillators will have 20 kHz drive frequency for devices built to design specifications.
o The proof mass axial resonant frequency is approximately 4 kHz.
o The stiffness nonlinearity of these devices is four times larger than the previous
SOA-2 design, which had no force multiplier.
As fabrication capabilities improve, test results will continue to move towards mirroring the results of finite element analysis.
65
Of the design objectives presented earlier, only the temperature sensitivity of
the device failed to match predictions. This discrepancy developed because of the
incorrect modeling assumption that the Hoya glass is not affected by the thermal
expansion of braze material that it is attached to the surface of the LCCC. Subsequent
thermal models that included these components have confirmed the behavior seen in
testing [2]. This development raises confidence that future designs, which take into
account this effect, will meet the design specification of zero temperature sensitivity.
This will be accomplished by taking advantage of the ability to change the device's
temperature sensitivity by simply changing the placement of the force multiplier
anchor beam (Section 3.2.)
A number of areas of future study have become apparent through the development
and testing of this device. The first involves the level of common moding that is seen
between the oscillators. Thermal sensitivity data presented in Chapter 4 show that
this effect can dramatically reduce errors common to both oscillators. Much of the
device's ability to reject these common sources comes from the exactness to which
the two oscillators are fabricated alike, so future tests of devices with oscillators built
closer to specifications will shed light on the limits to which this effect can be relied
upon for rejecting errors. In addition to the common moding of temperature induced
errors, this effect should also be studied in the context of errors produced by drive
amplitude fluctuations.
Another concern is the sensitivity to cross-axis accelerations (the X-axis of figure 21), which results in a misalignment error of the input axis. Along this axis, two effects
have been shown to cause this error. The first effect involves the "bowing" of the
flexures under acceleration, which has the effect of increasing their natural frequency.
The second, larger effect, involves the movement of the large lever arm. Preliminary
finite element analysis has shown that depending on the direction of the acceleration,
this movement can cause either an increase or a decrease in oscillator frequency [4].
This effect causes the frequencies of oscillators arranged in the 1800 configuration, to
oppose one another under acceleration, and those of oscillators arranged in the mirror
configuration to track one another. It is apparent, then, that the mirror configuration
66
should be used in future devices so that this effect will be rejected through common
moding.
A third area of study involves the presence of "prestress" in the oscillators caused
by heating of the device both while bonding the silicon to the Hoya substrate and
while brazing the substrate to the LCCC. Any stresses created during cooling due to
the CTE mismatch of the silicon and substrate material have the potential to change
the resonant frequency and scale factor of the device. Models produced to predict
the effect of brazing the substrate to the LCCC [2] have shown that these combined
stresses are not prohibitively high. However, it may still be desirable to determine
the exact level of these stresses and perhaps reduce them in the future.
A final area of future investigation involves the possible mode locking of the two
oscillators, which prevents the two drive frequencies from moving independently and
eliminates the ability of the device to measure acceleration. This phenomenon is a
result of the coupling that exists between the oscillators through the proof mass, and it
occurs when the two frequencies approach one another. In the fabricated devices, the
separation of the drive frequencies by 1000 Hz prevents them from coming close to one
another within the 1-g test environment, but higher g-levels may present difficulties
as the frequency separation approaches zero. Further research is needed to develop
methods to prevent locking at these higher g-levels.
67
Appendix A
Predicting Stiffness Nonlinearity
The relationship between the nonlinear characteristics of the previous and current
SOA designs can be derived by investigating the effect that adding a force multiplier
to the end of the oscillator has on axial stiffness of the oscillator. The cubic stiffness
term in beam bending is highly dependent on the axial tension in the beam. Roark
and Young give the solution to a beam with similar boundary conditions as the SOA
flexures. For a beam with fixed ends, loaded transversely with a center load, W, the
maximum lateral displacement, Ymax is
W13
A
Ymax +
1 61
max
27r4 EI
(A.1)
a
(A.2)
and the axial tension in the beam, P is
P
=
7F2 EA
2
412
where
I =area moment of inertia of the beam
A =cross-sectional area of the beam
1 =length of the beam
E =Young's modulus
68
2
Ymax
Proof Mass
X5
kP
4
X2Lever
Arm
(D-I4
2 Oscillator
Anchor
P
1
Figure A-1: Modified spring model of SOA oscillator and force multiplier
If the axial tension in the beam is reduced, the cubic term should also decrease
linearly.
In the case of the new SOA design, some compliance exists at the end of the
flexures in the form of the base beams and the force multiplier.
As the beam is
displaced laterally, some of the resulting axial displacement is taken up by these
elements.
This causes a reduction in axial tension, which then reduces the cubic
stiffness term.
Section 2.3 introduces a simple model of the oscillator that can be modified to
predict the axial strain in the flexures.
Figure A-1 shows the oscillator and force
multiplier, with k2 again representing the compliance of both the base beams and
flexures. External and internal loads P and F, are defined as before. In the previous
model, the goal was to find the axial tension in the flexures caused by the thermal
expansion of the proof mass and substrate. In this exercise, we wish to find the the
axial elongation in the flexures caused by lateral drive motion. This can be approxi-
69
mated by applying an axial displacement, xi to the flexures. In this formulation, the
proof mass is assumed to much more rigid than the beam structure, so it is modeled
as fixed.
The resulting force displacement relationships are similar to before:
k2
-= x1 -
(A.3)
X2
F3
(A.4)
F4
(A.5)
x4
The displacements on either end of the lever arm are defined by:
(A.6)
(x 4 - X3 ) = n (x 2 - x 3 )
Where n is the lever ratio of the force multiplier. Force and torque balance gives:
F3 + F4 = PI
(A.7)
Pi = nF 4
(A.8)
Solving equations A.3 through A.8 gives the overall axial stiffness of the elements
from the point of view of the oscillator, Ka,:
1
x1
1
kaxo
P1
k2
(n-1)2
n 2k3
1
n2 k 2 k 3 k 4
axo
(A.9)
n2k4
n 2 k 3 k4 + (n - 1) 2 k2 k4 + k 2 k 3
(A.10)
Comparing this equation to the stiffness calculated in 3.10, we see that
kax =
70
kax0
(A.11)
where, again, ka, is the axial stiffness of the device from the point of view of the proof
mass (this is the stiffness that determines the axial resonant frequency of the proof
mass.) In other words, the force multiplier is n 2 times as stiff from the point of view
of the oscillator side as it is from the proof mass side.
The new SOA with force multiplier was designed with the same proof mass axial
frequency as the previous design. Since the size of the proof mass was reduced by
n = 4, the axial stiffness, ka, was reduced by the same amount:
kax
where
kprev
(A.12)
= kprey
n
is the axial stiffness of the previous SOA design. Substituting from A.11
gives
kaxo
(A.13)
=rnkprev
This equation reveals that the axial stiffness from the point of view of the oscillator
is n times as large as the axial stiffness of the previous SOA design.
We are interested in the lengthening of the SOA flexures as a result of lateral displacement. Recall from section 2.3 that the oscillator stiffness,
k2
is the combination
of the stiffnesses of the base beams and flexures:
k
(A.14)
kbbkf
kbb+
kf
where
kbb
= stiffness of the base beams
kf = stiffness of the flexures
The lengthening of the flexures, xf, due to the axial displacement x1 is then found
through equation A.13
kaxo
S= x
kprev
(A.15)
k(Akf
" = x1
Since the stiffness of the flexures, kf was not changed between designs, the lengthening
of the flexures, and thus the tension inside them, increases by n. The net result is an
71
increase in the cubic stiffness nonlinearity of the flexures by the same amount.
72
Appendix B
SOA Vibrational Modes
This appendix contains views of all vibrational modes calculated for the SOA. The
in-plane modes are presented first, followed by the out-of-plane modes. Displacements
in the in-plane views are shown greatly exaggerated. The scale on the right hand side
of the out-of-plane views shows relative displacement amplitudes. Notes are added
to some plots for clarity.
73
ANSYS 5.4
SEP 28 1998
Proof Mass
Motion
16:19:45
DISPLACEMENT
STEP=1
SUB =1
FREQ=4122
PowerGraphics
EFACET=1
AVRES=Mat
DNX =1
*DSCA=-191
ZV =1
*DIST=2750
*XF =1250
Z-BUFFER
Axial Mode
Figure B-1: Proof mass axial mode
74
459
ANSYS 5.4
SEP 28 1998
16:12:37
DISPLACEMENT
STEP=1
SUB =3
FREQ=24627
PowerGraphics
EFACET=1
AVRES=Mat
DNX =1.003
*DSCA=-124. 671
zv
=1
*DIST=2750
*XF =1250
Z-BUFFER
Hula Mode
Figure B-2: "Hula" mode in which flexures resonate in phase with one another
75
ANSY9 5.4
SEP 28 1998
16:20:03
DISPLACEMENT
STEP=1
SUB =3
FREQ=18701
PowerGraphics
EFACET=1
AVRES=Mat
DNX =1.004
*DSCA=-249.034
ZV =1
*DIST=2750
*XF
=1250
Z-BUFFER
Proof Mass
Motion
Pivot Point
Rotational Mode
Figure B-3: Rotational mode in which the proof mass rotates about its center point
76
ANSYS 5.5.1
MAY 10 2000
15:12:41
DISPLACEMENT
STEP=1
SUB =4
FREQ=20282
PowerGraphics
EFACET=1
AVRES=Mat
DMX =1
*DSCA=-55.085
zV =1
*DIST=2750
....
~~.....
~
*XF
=1250
Z-BUFFER
Drive Mode
Figure B-4: Drive mode for SOA
77
ANSYS 5.4
SEP 28 1998
i6:I5:32
DISPLACEMENT
STEP=1
SUB =3
FREQ=24627
PowerGraphics
EFACET=1
AVRESmMat
DMX =1
*DSCA=124. 999
ZV =1
*DIST=2750
*XF =1250
Proof Mass
Motion
Z-BUFFER
Cross-Ax is
Figure B-5: Cross-axis mode, similar to axial mode in Figure B-1, but along the axis
perpendicular to the input axis
78
ANSYS 5.4
SEP 28 1998
16:21:30
DISPLACEMENT
STEP=1
SUB =5
FREQ=29742
PowerGraphics
EFACET=1
AVRES=Mat
DMX =1.001
*DSCA=191 .266
ZV =1
*DIST=2750
*XF =1250
Base Beam
Motion
Z-BUFFER
Other Modes
Figure B-6: Another in-plane mode that resembles a "tail wagging" motion of the
oscillator
79
ANSYS 5.5.1
MY 12 2000
13:17:42
NODAL SOLUTION
STEP-1
SUB -1
G
FREQ-5182
(AVG)
UZ
---
RSYS-0
PowerGraphics
-
EFACET-1
AVRES-Mat
DC -1.215
SM --. 078705
SMI -1.215
A
--. 006853
B
-. 136853
C
-. 280558
D
=.424264
567969
9
-.
I
-1.143
Out of Plane Mode (Synnetric)
Figure B-7: First out-of-plane mode. This mode is symmetric to the other half of the
device.
80
ANSYS 5.5.1
MY 12 2000
13:31:01
NODAL SOLUTION
STEP-1
SUB -2
FREQ-5853
UZ
(AVG)
RSYS-0
PowerGraphics
EFACET-1
AVRES-Nat
DIMD -1.747
SMN --. 093015
SM -1.747
A
B
C
rs
-.009212
13666
-.418121
=.622575
-.
-. 027029
F
-1.031
H
I
-1.4-4
=1.645
Out of Plane Mode (Antisymmetric)
Figure B-8: Second out-of-plane mode. This mode is antisymmetric to the other half
of the device.
81
ANSYS 5.5.1
MY 12 2000
13:18:29
NODAL SOLUTION
STEP-1
SUB -2
----
-----
~~FREO-6105
UZ
RSYS-0
(AVG)
PowerGraphics
-
EFACET-1
-
AVRES-Mat
DM
Out of Plane
Mode
-1.652
SM -- 1.529
SM -1.652
-
A
B
-- 1.352
--. 998699
E
F
D --. 291917
-.061475
-. 414
I
I
=1.475
(Symnetric)
Figure B-9: Third out-of-plane mode. This mode is symmetric to the other half of
the device.
82
ANSYS 5.5.1
1MY 12 2000
13:19:05
NODAL SOLUTION
STEP-1
SUB -3
-------------------
FREQ-11076
(AVG)
UZ
RSYS-0
PowerGraphics
EFACET-1
AVRES-Mat
DMx -1.998
SM -- 1.692
SM -1.998
-- 1.487
A
B
C
D
E
--1.077
--. 63671341
-. Z5710~4
=.152927
H
-1.383
=1.793
I
ouoP-- (
Out
of Plane
Mode
(Symmnetric)
Figure B-10: Fourth out-of-plane mode. This mode is symmetric to the other half of
the device.
83
ANSYS 5.5.1
MAY 12 2000
13:36:36
NOD&L SOLUTION
STEP-1
SUB
-3
FREQ-11303
UZ
RSYS-0
-
G aI
(AVG)
PowerGraphics
EFACET-1
AVRES-Mat
DM1 -2.227
SMB --2.068
SM -2.227
A
--1.829
B -- 1.352
C
---
674722
D
--. 397542
E
-.
H
-1.511
079639
-i.9e8
Out of Plane Yode (Antisymmetric)
Figure B-11: Fifth out-of-plane mode. This mode is antisymmetric to the other half
of the device.
84
ANSYS 5.5.1
M4Y 12 2000
13:20:25
NODAL SOLUTION
STEP-1
SUB -4
(AVG)
UZ
RSYS-0
PowerGraphics
EFACET-1
AVRES-Mat
DM -10.923
SM -- 1.476
SM -10.923
--. 78701
A
-. 59044
B
I' =3..546
E
F
-4.724
I
-10.234
-44.I01
Out of Plane Mode (Symmetric)
Figure B-12: Sixth out-of-plane mode. This mode is symmetric to the other half of
the device.
85
ANSYS 5.5.1
MY 12 2000
13:21:27
NODAL SOLUTION
STEP-1
SUB -6
FREQ-17792
(AVG)
UZ
RSYS-0
PowerGraphics
EFACET-1
AVRES-Mat
DMC -3.81
SMW --2.056
SM -3.81
C
--1.73
=-1.079
--. 42692 3
r-
-. 224837
14
I
=3.484
A
B
E
-,876598
-1520
2.832
Out of Plane Mode (Symmetric)
Figure B-13: Seventh out-of-plane mode. This mode is symmetric to the other half
of the device.
86
ANSYS 5.5.1
MRY 12 2000
12:45:57
NODAL SOLUTION
STEP-1
SUB -1
FREQ-26092
-
UZ
(AVG)
RSYS-0
PowerGraphics
\
EFACET-1
AVRES-Mat
DM -2.757
SlM3 --1.922
SPac -2.757
A
--1.662
R
-1.142
-B--C
=-.622014
-
-.41.7582
-
-.
I
-2.497
933
Out of Plane Mode (Symetric)
Figure B-14: First out of plane mode. This mode is symmetric to the other half of
the device.
87
ANSYS 5.5.1
MY 12 2000
14:17:21
NOD&L SOLUTION
STEP-1
SUB -1
FREQ-28287
(AVG)
UZ
-
-RSYS-0
PowerGraphics
EFACET-1
AVRES-Mat
DMX -2.572
SMN --2.415
SM -2.572
--2.138
A
=-1.584
R
[1
E
P
P
I
Out of Plane
Mode
47 57.36
=.078336
=.b324E8
=1.741
-2.295
(Antisymmetric)
Figure B-15: First out of plane mode. This mode is antisymmetric to the other half
of the device.
88
Bibliography
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(VBA). Prototype test results. In Proceedings of the Annual IEEE International
Frequency Control Symposium 1996., 1996.
[2] FEA model created for this project by Bernie Antkowiak at Draper Laboratory.
[3] Neil Barbour and George Schmidt. Inertial sensor technology trends. In Proceedings of the IEEE Symposium on Autonomous Underwater Vehicle Technology,
1998.
[4] FEA work performed for this project by James Bickford at Draper Laboratory.
[5] Mode locking phenomenon is currently being investigated for this project by
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[6] K. Lyon et al. Linear thermal expansion measurements on silicon from 6 to 340
k. Journal of Applied Physics, 48(3), March 1977.
[7] Kevin A. Gibbons. "A Micromechanical Silicon Oscillating Accelerometer". Master's thesis, Massachusetts Institute of Technology, February 1997.
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[9] Values for the thermal expansion coefficient of Hoya SD-2 glass provided by Hoya
Corporation.
89
[10] Rand Hulsing. MEMS inertial rate and acceleration sensor. IEEE Aerospace and
Electronic Systems Magazine, 13(11):17-23, November 1998.
[11] J.Bernstein, S. Cho, A. T. King, A. Kourepenis, P. Maciel, and M. Weinberg. A
micromachined comb-drive tuning fork rate gyroscope. In IEEE Micro Electro
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[12] H. McSkimmin. Measurements of elastic constants at low temperatures by means
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[13] Nathan A. St. Michel. "Force Multiplier Design in a Vibrating Structure". Master's thesis, Massachusetts Institute of Technology, May 1998.
[14] Ali Hasan Nayfeh and Dean T. Mook. Nonlinear Oscillations, chapter 2.3. John
Wiley & Sons, Inc., 1979.
[15] The lever concept is similar to work done by Susan (Xiao-Ping) Su at the University of California, Berkeley. At the time this thesis was written, no submitted
papers had been published.
[16] William C. Tang, Tu-Cuong Nguyen, Michael W. Judy, and Roger T. Howe.
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A-Physical, 21(1):328-331, 1990.
[17] Simple thermal model developed by Marc Weinberg at Draper Laboratory.
[18] Robert D. White. "The Effects of Mechanical Vibration and Impact on the
Performance of a Micromachined Tuning Fork Gyroscope".
Master's thesis,
Massachusetts Institute of Technology, May 1999.
[19] Warren C. Young. Roark's Formulasfor Stress and Strain. McGraw-Hill, Inc.,
1989.
90