Download Foreword Word Problems

Homework # 9 (STRICTLY EXTRA CREDIT)
ENGN 330—Winter 2016
Due: Friday, Apr 13, 2016
Foreword
This homework is strictly extra credit. It’s all about accelerometer design. How many points can
I earn, you ask? You may earn whichever is greater of i) half of the points you missed on the first
exam, or the equivalent of 5 pts on the homework. In essence, you’ll be credited points according
to whatever scenario is more beneficial for your overall course grade.
Word Problems
1. Accelerometer Design: As we have seen, accelerometers have a great many applications.
The fact that they can now be manufacturered and packaged in small dimensions means
they are everywhere—airbag systems, bridge monitoring, football helmets, even video game
controllers. As we discussed in class, a basic accelerometer consists of a proof mass connected
to beams acting as springs. The interdigitated fingers create a capacitor arrangement, wherein
the electrical signal output (voltage) is proportional to acceleration (see Figure 1). Note the
size scale of the device is on the order of tens of microns.
All length dimensions can be reasonably estimated from Fig 1, with exception of the length
of the beam (proof mass): Assume it is 400 µm long.
In terms of damping, recall that the squeeze film effect is primarily responsible for determining the damping constant. Basically, a thin film of air interacts with the moving pieces
to create damping. The damping constant is given as :
c=
µ
LW 3
h3o
where µ is the dynamic viscosity of air (≈ 1.98 × 10−5 kg/m s) and ho is the equilibrium
distance between the fixed and movable plates of the capacitive fingers. With regard to the
squeeze film damping geometrical parameters, see Fig 2.
Now, given your training in vibrations, you are in a great position to verify the design of
engineers at Analog Devices meets various design criteria! Making a detailed/careful model
of the system, please answer the following:
(a) What is the natural frequency ωo of the vibrating element in the accelerometer? Carefully
show estimations/calculations for mass and spring constant.
(b) What is the damping factor ζ for this device? Show all estimates/calculations.
(c) What is the maximum frequency for which the measured acceleration can be measured
to within 0.2% of the true acceleration? This figure is commonly called the bandwidth
of the device.
(d) Given this maximum frequency that can be measured to within 0.2% (from part c), what
is the maximum possible phase distortion, |φ| that occurs? How severe is this phase shift,
will it significantly distort the true acceleration you are trying to measure? To make this
concrete, assume the true acceleration is given by ÿ(t) = 0.5 cos(3000t) + 0.5 cos(10000t).
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Compute the resulting output signal of the accelerometer. On the same plot, overlay
the true and measured acceleration. Comment on how well the accelerometer measured
the true acceleration.
(e) What is the shock resistance of the device? This is basically defined as the maximum
magnitude of acceleration the device can withstand before the capacitive fingers start
touching other (creating an electrical short). State any assumptions and show all relevant
calculations.
(f) Thus far, we have only dealt with a 1-D system. That works for measuring vibratinos
in one direction. Getting the second dimension is fairly easy given the planar structure
of micro-electromechanical systems (MEMS) fabriaction. Just make a second capacitive
comb structure perpendicular to the first. How about the 3rd dimension? That takes a
bit more work, as one needs ot measure out of plane capacitive changes. For example,
see slides 8 and 9 from this very nice presentation on 3-axis accelerometers: http://
ieee-sensors2013.org/sites/ieee-sensors2013.org/files/Serrano_Accels.pdf.
Naturally, this also makes for a beautiful N-DOF problem. Specifically, we have motion
in the plane and out of the plane. Now build a 2-DOF model to capture translations in
the plane in the x-direction and out of the plane in the z-direction. Design a device—such
as the LIS3DH that you have used in class,—that can measure up to 16g accelerations
to within 0.2% accuracy in BOTH x and y directions. Of course, this means that you
need to solve for the natural modes of the device as well. Be sure to derive equations of
motion, solve for natural modes (draw them too) and show that your device can make
accurate measurements as described above.
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Figure 1:
Accelerometer in concept and design.
Upper left:
cartoon of fibrating proof mass with interdigitated capactive fingers.
Lower Left:
zoomed highlighting dimensions of capactive finger arrangement.
Lower right:
SEM of an actual device manufactured by Analog Devices.
Upper right: SEM of a slightly different design. Image sources: http://low-powerdesign.com/donovansbrain/2011/03/27/
mems-motion-sensors-the-technology-behind-the-technology/, http://www.analog.com/
library/analogdialogue/archives/33-01/accel/index.html
Figure 2: Squeeze film damping effect between fixed object and movable mass. In this case, the
interdigited capacitive fingers cause the damping. Image source: A. Berny, Substrate Effects in
Squeeze Film Damping of Lateral Parallel-Plate Sensing MEMS Structures.
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