Experimental analysis of spring hardening and softening nonlinearities in
microelectromechanical oscillators.
Sarah Johnson
Department of Physics, University of Florida
Mentored by Dr. Yoonseok Lee
Abstract
Micro-electro-mechanical systems or MEMS are used in a variety of today’s technology
and can be modeled using equations for nonlinear damped harmonic oscillators. Mathematical
expressions have been formulated to determine resonance frequency shifts as a result of
hardening and softening effects in MEMS devices. In this work we experimentally test the
previous theoretical analysis of MEMS resonance frequency shifts in the nonlinear regime.
Devices are put under low pressure and swept through a range of frequencies with varying AC
and DC excitation voltages to detect shifts in the resonant frequency. Mathematical models have
predicted greater nonlinearities with higher AC amplitudes and spring softening with high DC
voltages. Further research is being conducted to compare MEMS experimental data to earlier
theoretical models. Our research will assist in confirming a more accurate model for MEMS
devices, which could result in improved micro-technology in products such as gyroscopes,
accelerometers, and displays.
I. Introduction
Micro-electro-mechanical systems or MEMS have a growing impact on a wide variety of
today’s technology. Ranging from 1 to 100 microns in size, about the thickness of a human hair,
their capabilities for miniaturization and multiplicity make them extremely useful in
microelectronics. Cooperative work from many micro-machines is the most efficient way to
perform a task. It is cost-effective to manufacture and arrange many devices to work together.
Not only are MEMS useful when implemented in micro-objects because of their ability to move
freely in small spaces, they are also relatively inexpensive to produce using microelectronic
fabrication techniques. Thin material layers are deposited as a base, patterned, and etched away,
resulting in a microscopic 3-D structure. There are 2 categories of MEMS: sensors and actuators.
Sensors retrieve information from surroundings, and actuators execute commands through
controlled movements. Their small size and practical capabilities help to improve an extensive
variety of technological products including hearing aids, accelerometers, insulin pumps, and
inkjet printers [1].
The ability to quickly and accurately characterize the electro-mechanical parameters of a
MEMS resonator is crucial to its effective application. Nonlinearities in device operation can
result from various MEMS specifications such as excitation voltage, resonator structure, and
device dimensions. Theoretical and numerical analysis on MEMS devices helps to accurately
predict nonlinearities such as spring softening and hardening in the device. Spring softening is a
reduction in the effective spring constant; it manifests as a downward shift in resonance
frequency. Spring hardening is the opposite: an increase in the effective spring constant. The
phenomena could occur simultaneously, but one usually dominates [2]. Analyzing the effects of
nonlinearities will result in improved predictions of the actual behavior of resonators and similar
devices. Previous theoretical analysis of MEMS devices predicts spring softening and hardening
effects. However, the theory and mathematical models have not yet been tested. Here, we present
an experimental analysis of the nonlinearities in MEMS systems in accordance with the
numerical methods previously presented by Elshurafa in [2]. Our experimental analysis will
assist in confirming the accurate characterization of the electro-mechanical parameters of a
MEMS resonator in the nonlinear regime.
II. Device and Detection Scheme
Figure 1: A diagram of a MEMS device used in this work. The plate and moveable electrodes are depicted by the
lighter shaded regions, which are moveable. The darker shaded regions are fixed to the substrate and receive voltage
excitations. The interleaving electrode teeth actuate the device, causing the center plate to oscillate through
capacitive interactions [2].
Our MEMS device is a comb-drive micro-actuator that converts electrical excitation into
a mechanical output using electrostatic forces. Knowledge of the geometry of the MEMS is
crucial to understanding its operation. The device consists of a movable central plate suspended
0.75 µm above the substrate by four flexible serpentine springs [3]. Connected to the plate are six
sets of comb electrodes, three on each side, which interleave with six sets of electrodes fixed to
the substrate to form three sets of integrated parallel capacitors on each side with the plate as the
common electrode [Fig. 1].
Figure 2: A close-up diagram of the plate and fixed electrodes on the MEMS device. The plate electrode is allowed
to oscillate in the x-direction, while the fixed electrode is attached to the substrate and is stationary. Later, we will
refer to the width of the combs in the y-direction as y. Referring to the z-axis depth of the electrodes as z, we treat
𝜀𝜀 𝐴
the comb teeth as pairs of parallel capacitors, each with capacitances of 𝐶 = 0 , where 𝜀 and 𝜀0 are the relative
𝑑
permittivity and the permittivity of free space, respectively, d is the distance between teeth, and 𝐴 = 𝑧𝛥𝑥 [3].
The pairs of interdigitated teeth can be treated as parallel plate capacitors with a
capacitance of 𝐶 =
𝜀𝜀0 𝐴
𝑑
(see Fig. 2). Because the capacitors formed by the teeth are in parallel,
we add their capacitances to obtain the total capacitance. N+1 teeth creates N pairings, so 𝐶 =
𝑁
𝜀𝜀0 𝑧
𝑑
(𝑥0 + ∆𝑥) = 𝛽(𝑥0 + ∆𝑥), where 𝛽 = 𝑁
𝜀𝜀0 𝑧
𝑑
. β is the transduction factor, which relates
the electrical parameters, such as voltage and current, to the mechanical parameters, such as
force and displacement. ∆x changes as the plate displaces from equilibrium, therefore, the
capacitance will vary as a function of x [4].
Figure 3: The circuit used to actuate the MEMS device. The AC and DC voltage sources are necessary for MEMS
oscillation. The lock-in amplifier demodulates the signal from the charge-sensitive amplifier.
The circuit in Figure 3 will give us a better picture of how the device is actuated and
detected. When an AC voltage is applied to the MEMS, differential capacitive forces are created,
causing the comb electrodes connected to the plate to oscillate parallel to the substrate [5]. The
electrodes on the left side of the MEMS will receive an AC voltage of 𝑉𝐿 = 𝑉𝑎𝑐 , while the
electrodes attached to the right side of the MEMS have DC voltage of 𝑉𝑅 = 𝑉𝑑𝑐 . The voltage
1
𝑑𝑈
between pairs of electrodes creates an electric field with energy 𝑈 = 2 𝐶𝑉 2 . Since 𝐹 = − 𝑑𝑥 , an
1
𝑑𝐶
2
𝑑𝑥
attractive force will develop between the electrodes of 𝐹 = 𝑉 2
1
= 𝛽𝑉 2 [3]. We apply the
2
difference between VL and VR this equation to get:
1
𝐹 = 2 𝛽(𝑉𝑎𝑐 2 − 𝑉𝑑𝑐 2 )
(1)
To oscillate the MEMS, we are only interested in the sinusoidal force on the MEMS device. The
magnitude of the oscillatory drive force becomes:
𝐹=
1
𝛽𝑉𝑎𝑐 2
2
Therefore, when we drive at half of the detected frequency, 𝑉𝑎𝑐 = 𝑉0 𝑒
(2)
𝑖𝜔𝑡
2
,
1
𝐹 = 2 𝛽𝑉0 2 𝑒 𝑖𝜔𝑡 .
(3)
III. Damped Linear and Nonlinear Oscillators
The amplitude of the device displacement can be found by solving the equation of motion
for a damped harmonic oscillator [3]. In the linear regime, MEMS forces can be modeled by
comparing the device to a damped driven harmonic oscillator, where the forces applied counter
to its motion are the spring force, -kx, and the linear damping force, -𝛾v. Newton’s second law of
motion tells us that:
𝑚𝑎 = 𝐹 − 𝛾𝑣 − 𝑘𝑥
(4)
where F is the oscillatory driving force on the plate, 𝛾 is the damping coefficient, and k is the
spring constant of the device. We make the substitution 𝑘 = 𝜔0 2 𝑚, where 𝜔0 is the resonance
frequency and m is the mass of the device [4]. Replacing a and v for separable differential
equations of x and rearranging gives us:
𝐹 𝑑 2 𝑥 𝛾 𝑑𝑥
=
+
+ 𝜔0 2 𝑥
𝑚 𝑑𝑡 2 𝑚 𝑑𝑡
(5)
The solution of this equation takes the form:
𝑥 = 𝐴0 𝑒 𝑖𝜔𝑡
(6)
where 𝐴0 is the amplitude of oscillation and 𝜔 is the driving force frequency from (3).
Substituting (6) into (5) results in:
𝐹0 𝑖𝜔𝑡
𝛾
𝑒
= 𝐴0 (−𝜔2 − 𝑖 𝜔 + 𝜔0 2 )
𝑚
𝑚
(7)
Solving for 𝐴0 :
𝐴0 =
𝐹0
1
𝑚 (𝜔0 2 − 𝜔 2 ) − 𝑖 𝛾 𝜔
𝑚
(8)
To obtain the amplitude of the in-phase and out of phase term, we separate the real and complex
components of 𝐴0 [4]:
𝛾
2
2
𝐹0 (𝜔0 − 𝜔 ) + 𝑖 𝑚 𝜔
𝐴0 =
𝑚 (𝜔0 2 − 𝜔 2 )2 + ( 𝛾 𝜔)2
𝑚
(9)
We can plug 𝐴0 back into the equation for x to get an expression for the displacement of the
device.
The lock-in amplifier detects the right-side MEMS voltage amplitude after current flows
through a charge sensitive amplifier, as shown in Figure 3. The charge amplifier’s output voltage
as a function of input charge can be written as:
𝑉 = 𝛼𝑞
(10)
Where 𝛼 is the amplification factor. Since 𝑞𝑅 = 𝐶𝑅 𝑉𝑅 , and 𝐶𝑅 = 𝛽𝛥𝑥, the detected voltage will
be:
(11)
𝑉 = 𝛼𝛽(𝑥0 + 𝛥𝑥)𝑉𝑑𝑐
As previously shown, x varies as a function of time. Substituting 𝛥𝑥 for x and incorporating
equations (2), (6) and (9) to find the detected voltage results in [3]:
𝛾
2
2
𝛼𝛽 2 𝑉𝑑𝑐 𝑉𝑎𝑐 2 (𝜔0 − 𝜔 ) + 𝑖 𝑚 𝜔 𝑖𝜔𝑡
𝑉=
𝑒
𝛾
2𝑚
(𝜔0 2 − 𝜔 2 )2 + (𝑚 𝜔)2
(12)
Since we graph detected voltage as a function of frequency and not time, we neglect the time
dependent term at the end. Separating the real (X) and complex (Y) terms of the detected
voltage:
𝛼𝛽 2 𝑉𝑑𝑐 𝑉𝑎𝑐 2
(𝜔0 2 − 𝜔2 )
𝛾
𝑚
(𝜔0 2 − 𝜔 2 )2 + (𝑚 𝜔)2
(13)
𝛾𝜔
𝛼𝛽 2 𝑉𝑑𝑐 𝑉𝑎𝑐 2
𝑚
𝑉𝑌 =
𝛾
m
(𝜔0 2 − 𝜔 2 )2 + (𝑚 𝜔)2
(14)
𝑉𝑋 =
𝑉𝑋 and 𝑉𝑌 are the in-phase and out-of-phase expressions for detected voltage of the charge
sensitive amplifier. The theoretical outputs are plotted in Fig. 4.
VX
Detected Voltage (V)
4.00E-06
3.00E-06
2.00E-06
1.00E-06
0.00E+00
-1.00E-06
-2.00E-06
-3.00E-06
-4.00E-06
24500
24700
24900
25100
Frequency (Hz)
25300
25500
VY
Detected Voltage (V)
3.50E-06
3.00E-06
2.50E-06
2.00E-06
1.50E-06
1.00E-06
5.00E-07
0.00E+00
24500
24700
24900
25100
25300
25500
Frequency (Hz)
Figure 4: Theoretical plots of the in-phase (𝑉𝑋 ) and out-of-phase (𝑉𝑌 ) components of the detected voltage with 10V
DC and .5V AC when the resonance frequency is 25 kHz, 𝛾 = 10−8 Ns/m, α= 3x108V/C, β=10-9 F/m, and m=10-10
kg.
Spring softening and hardening arise from nonlinearities in the restoring force, meaning
that another term must be added to equation (4) to determine resonance frequency shifts due to
hardening and softening effects on the MEMS resonator. In order to derive expressions for
MEMS oscillator motion in the nonlinear regime, the well-known duffing equation for nonlinear
damped harmonic motion is used:
𝑑2 𝑥
𝑑𝑥
𝑚 2 +𝛾
+ 𝑚𝜔0 2 𝑥 + 𝑘𝑛 𝑥 3 = 𝐹
𝑑𝑡
𝑑𝑡
(15)
where 𝑘𝑛 models nonlinearities in the restoring force. We note that the duffing equation matches
the damped equation for harmonic motion shown previously when 𝑘𝑛 = 0. This equation has
previously been used to calculate an expression for the resonance frequency shift in a MEMS
device:
3
𝛿 = 𝐾𝐴𝑚𝑎𝑥 2 − 2𝛼1 𝑉𝑑𝑐 2 (1.5𝐴𝑚𝑎𝑥 2 + 1)
8
𝑘 𝑥 2
𝑁𝜀 𝐴
where 𝐾 = 𝜔𝑛 20𝑚, 𝛼1 = 2𝑚𝜔 02 𝑥 3, and 𝛿 =
0
0
0
𝜔−𝜔0
𝜔0
(16)
. 𝜔0 is the original resonance frequency, m is the
mass of the device, 𝑥0 is the equilibrium distance between the plate and moveable electrodes, N is
the number of comb teeth on the device, A is the transverse area per comb tooth, and 𝐴𝑚𝑎𝑥 is the
maximum amplitude of the device oscillation, which has also previously been derived to be:
𝐴𝑚𝑎𝑥 = 4𝑉𝑎𝑐 𝑉𝑑𝑐 𝛼2 𝑄
1 𝑑𝐶
where 𝛼2 = − 2 𝑑𝑥 = −𝜀0
𝑁−1 𝑧
2
1
𝑦
(17)
2𝑑
𝑦
(𝑑 + 𝜋 ln([(𝑑 + 1)2 − 1][ 𝑦 + 1]1+𝑑 ))
(18)
y is the y-axis width of the combs, and Q is the quality factor, a dimensionless parameter
describing how under-damped an oscillator is [2]. A high quality factor corresponds to a lower
rate of damping. A depiction of equation for the frequency shift is shown in Figure 5. Increased
AC excitation voltages were mathematically analyzed in [2] to show an increase in spring
hardening and softening effects in MEMS oscillators [Fig. 5(c)]. Softening of the MEMS device
will occur when 𝛿 < 0 and the peak shifts to the left. At low AC excitations, the frequency shift
expression has been rearranged to show that 𝛿 < 0 when
softening. The peak will shift to the right when 𝛿 > 0,
occurs [Fig. 5(b)].
a)
𝐾
8
𝐾
8
<< 𝛼1 𝑉𝑑𝑐 2, resulting in spring
>> 𝛼1 𝑉𝑑𝑐 2 and spring hardening
(c)
b)
Figure 5: Graphs of excitation frequency vs. amplitude of oscillation using models for typical responses of a MEMS
resonator at various parameters. The grey sections of the graphs will follow one of the dotted lines during softening
or hardening, depending on whether the frequency is sweeping up or down, before returning to blue line behavior.
(a) The resonance of a MEMS device in the linear regime. (b) A positive resonance frequency shift due to spring
hardening behavior at low AC excitations with a low DC bias. (c) Response of the device with a high AC excitation
voltage. Spring softening and hardening effects become more apparent [2].
At low enough AC excitations, spring softening and hardening effects are dependent on
only 𝑉𝑑𝑐 . High 𝑉𝑑𝑐 values were mathematically modeled to show spring softening, while low
𝑉𝑑𝑐 values should result in spring hardening [Fig 5b]. Higher AC excitations have been modeled
to show increased nonlinearities in MEMS resonators, as shown in Figure 4c.
IV. Experimental Methods
To confirm the spring softening and hardening expressions previously derived, the circuit
in Figure 3 was created to first measure device resonance before shifting parameters to model
nonlinearities. An EG&G Instruments 7265 DSP Lock-in Amplifier was used as both an AC
voltage source and a lock-in amplifier to detect the signal coming from our pre-amplifier, which
has a 3x108 amplification factor. A 16 Bit Bipolar DAC connected to a 10MΩ resistor was used
for the DC bias. To ensure minimal damping, our MEMS device was enclosed in a cell attached
to a BOC Edwards RV3 vacuum pump, which brought the cell pressure down to 10 mtorr. The
MEMS cell was also connected to a cryo pump, which is an aluminum tube filled with activated
charcoal and cooled to 77K in operation. Activated charcoal has an extremely large surface area
to mass ratio. When it cools to 77K, gas molecules will be physically adsorbed on the large
surface, and the pressure in the cell will drop further to 5 mtorr, which ensures minimal damping
from viscosity [2]. By replacing the vacuum pump during device operation, the cryo pump also
eliminates potential electric noise. We run frequency sweeps using Labview 2013, which
communicates with the lock-in internal oscillator and DC voltage source to set the frequency, AC
amplitude, phase shift, and DC bias while detecting the voltage to the lock-in amplifier. Files
were created containing the parameters and data for frequency versus detected X and Y voltage
amplitude. These files are currently under analysis for spring softening and hardening effects.
Mathematical models predict that higher AC excitations increase nonlinearities in spring
hardening and softening. Theoretical analysis also calculates spring hardening to occur at low
DC bias voltages, and spring softening to occur at high DC bias voltages. Testing of these
models is currently underway. Further investigation is required to more accurately model
frequency shifts in the MEMS device and compare the fits of our experimental data with MEMS
numerical analysis.
IV. Conclusions
MEMS devices are modeled to follow force equations for nonlinear damped harmonic
oscillators. Experimental measurement of MEMS resonance at varying frequencies is expected to
show the device hardening and softening with changing and AC and DC voltage parameters.
Previous numerical analysis on MEMS resonators have modeled the frequency shift caused by
nonlinearities to follow:
3
𝛿 = 𝐾𝐴𝑚𝑎𝑥 2 − 2𝛼1 𝑉𝑑𝑐 2 (1.5𝐴𝑚𝑎𝑥 2 + 1)
8
If our experimental data agrees with the theoretical predictions in [2], at low AC
(19)
excitations
𝐾
8
<< 𝛼1 𝑉𝑏𝑖𝑎𝑠 2 during spring hardening, and
𝐾
8
>> 𝛼1 𝑉𝑏𝑖𝑎𝑠 2 during spring softening.
Experiments will be conducted with changing AC and DC parameters of MEMS devices to test
whether the detected resonance peak shift follows previous theoretical and numerical analysis.
Experimental confirmation of the models for nonlinearities in MEMS devices will help to better
analyze and model these micro-machines, improving the quality of the cutting-edge technology
in which they are implemented.
Acknowledgement
This research is supported in part by NSF DMR-1461019. Special thanks to Dr.
Yoonseok Lee for his constant support and expertise in MEMS, as well as his positive attitude
throughout the experiment. Much appreciation to Terrence Edmonds, who taught me valuable
experimental techniques with MEMS while allowing me to assist him in summer research.
Thank you to Colin Barquist for his consistent help and guidance, especially during experimental
difficulties. I would like to thank Dr. Selman Hershfield and the University of Florida as a whole
for providing me with a valuable research experience this summer that will greatly prepare me
with the skills to excel in my future physics endeavors.
References
[1] MEMS Industry Group. 2009, Feb 17. Introduction to MEMS "Micro-Electro-Mechanical
System." Retrieved from https://www.youtube.com/watch?v=CNmk-SeM0ZI
[2] A M Elshurafa et al., J. Micromechanic. Sys. 4, 20 (2011).
[3] C S Barquist et al., Signal analysis and characterization of a micro-electro-mechanical
oscillator for the study of quantum fluids.
[4] J Bauer, Undergraduate Honors Thesis, University of Florida (2014).
[5] M. González, P. Zheng, E. Garcell, Y. Lee, and H. B. Chan, Rev. of Sci. Instrum.
84, 025003 (2013).