# Effect of Overheat and Direct Flow Loading on the MEMS Bistable Flow Sensor

Ivan Litvinov  Dan Refaeli  Alex Liberzon  and Slava Krylov

###### Abstract

We present the findings from an experimental study of a MEMS flow sensor, in
which an initially curved, double-clamped bistable microbeam serves as the
primary sensing element. Our research focuses on exploring how the overheat
ratio, direct flow loading, and turbulence-induced vibration, affect the
sequential snap-through (ST) buckling and snap-back (SB) release of an
electrostatically actuated beam heated by an electric current. The sensor is
fabricated from highly doped single-crystal silicon using a silicon-on-
insulator (SOI) wafer. Positioned at the chip’s edge, the microbeam is exposed
to airflow, enabling concurrent dynamic response measurements, with a laser
Doppler vibrometer and a video camera.

Our research demonstrates that for this sensing principle, the overheat ratio
can be significantly lower compared to conventional thermal sensing elements,
pointing to the potential for substantial energy savings. We also emphasize
the significant impact of flow angles and vibrations on the critical ST and SB
voltages, which are vital for the flow sensor’s output. Additionally, we
introduce the previously inaccessible experimental observation of the beam
profile’s time history during the snap-through transition.

The potential impact of this research lies in the development of more robust
MEMS flow sensors with enhanced sensitivity and a better understanding of
their response to environmental factors, which could have broader applications
in fields such as aerospace, environmental monitoring and industrial process
control.

###### keywords:

MEMS , bistable microbeam , bifurcation-based sensor , flow sensor

††journal: Sensors and Actuators: Physical

[inst1]organization=School of Mechanical Engineering,addressline=Tel Aviv
University, city=Tel Aviv Jaffo, postcode=6997801, country=Israel

##  1 Introduction

Microscale flow sensors play a key role in advancing scientific and
technological endeavors. Among the various types of flow sensors, those based
on microelectromechanical systems (MEMS) have become increasingly important
for monitoring gas flow in industrial environments, sensing environmental
flows, detecting marine conditions and analyzing biomedical flows [1]. These
sensors have a wide range of applications in technical fields, such as “smart
skin”, the Internet of Things (IoT), and flight control (e.g., [2, 3]).

The utilization of MEMS-based flow sensors presents several advantages,
including compactness, power efficiency, high sensitivity, and compatibility
with mass production requirements. Consequently, a multitude of MEMS flow
sensing methods have emerged over the past three decades [4, 5, 6]. These
sensors operate based on either thermal or mechanical principles, with a
majority of the previously developed flow sensors relying on the thermal
principle [6]. Thermal flow sensors offer robust and straightforward
manufacturing processes, linear sensor characteristics, and lack of moving
parts, distinguishing them from mechanical flow sensing methods. However,
their major drawbacks include high power consumption and undesired heat
dissipation. Recent advancements in microelectronic device design and
fabrication have aimed to mitigate these limitations of thermal flow sensors
[7]. Nonetheless, mechanical flow sensors continue to play a vital role in
water flow, especially for laminar flow. For example, sensors based on
micro/nano-cantilevers have found diverse applications in chemical and
biological sensing [8]. Mechanical flow sensors often rely on drag or lift
force sensing [9, 10, 11], frequency monitoring [12], or artificial hair
sensor designs [13]. Although various MEMS flow sensors are under research,
specific sensing capabilities are necessary for each technical application,
contingent on environmental and flow conditions.

Previous studies [14, 15] introduced an alternative flow sensing
principle,combining thermal and mechanical responses in a bifurcation-type
sensor. These studies showed that the electrostatically and electrothermally
(Joule heated) actuated initially curved double-clamped silicon microbeam
exhibits bistable behavior, in other words, remaining in one of the two
equilibrium position states under the same load. As a result of this
bistability, the curved beam shows high responsiveness to external stimuli
near two equilibrium states and has great potential for sensor implementation.
This mechanical structure is pushed to its stability limits, and is highly
sensitive to even the slightest changes in heat transfer caused by flow. As a
result, reducing the overheat ratio (the temperature difference between the
beam and the surrounding airflow) can reduce power consumption. In addition, a
curved microbeam device at the chip’s edge can be used as a spatial velocity
gradient sensor [16].

The feasibility of employing such devices for airflow sensing has been
confirmed through a combination of theoretical analysis and experimental
testing. However, to effectively utilize the laboratory prototype in turbulent
airflow measurements within practical scenarios, it is imperative to enhance
our understanding of the device’s response to various influencing factors.
These factors include overheat ratios (the absolute temperature to which the
microbeam is heated as a multiple of the ambient temperature), the sensitivity
of the microbeam to direct flow loading, as well as noise and vibrations that
may arise during turbulent flow measurements, such as those over aircraft
wings or wind turbine blades.

This study aims to investigate into how the sensor interacts with these
factors. To achieve this, we have expanded the diagnostic system introduced in
[16] to incorporate a multipoint Doppler laser vibrometer. This addition
allows us to visualize the beam shape during transitions between two
equilibrium states. The outcome of this extension has revealed a previously
only theoretically predicted phenomenon – asymmetric snap-through motion, as
described in [17]. Furthermore, this finding sheds light on the limitations of
the model presented in the paper, emphasizing the need for further
refinements.

##  2 Materials and methods

###  2.1 Sensor and chip design

The bistable flow sensor based on an initially curved microbeam was introduced
in [16, 18]. We briefly review the device design and its operational principle
for clarity and completeness. We use an initially curved bell-shaped
electrostatically actuated double-clamped beam, as shown in Figure 1, as the
sensing element. The beam is fabricated from the device layer of a silicon-on-
insulator (SOI) wafer, using the SOIMUMPsTM process [19]. The beam, anchors
and electrode are all made of highly doped single-crystal silicon with a
resistivity of 0.02 $\Omega\cdot$cm. The beam is designed to perform an in-
plane, parallel to the substrate, motion and to deflect in the $z$-direction.
The nominal, as designed dimensions of the beam are given in Table 1 and
definitions are according to Figure 1.

Figure 1: (a) A scanning electron microscope micrograph of the device (the scale bar is 100 $\mu m$) and device 3D schematics. The double-clamped curved beam is shown in the initial upward curved configuration. The values $V_{\mathrm{h}}$ and $V_{\mathrm{ES}}$ are the heating and electrostatic voltages, respectively (G is a ground). (b) Schematic front view of the beam, its main geometric parameters, and the airflow direction (not to scale).  Table 1: The nominal, as designed, dimensions of the beam used in the experiments. Parameter |  Size ($\mu$m)
---|---
Length, $L$ | $500$
Width, $b$ | $25$
Thickness, $d$ | $2$
Initial elevation, $h$ | $2.5$
Beam-electrode distance, $g_{0}$ | $10$

The operational principle of the device relies on coupled thermal and
mechanical sensing. It begins with an initially curved bistable microbeam,
subjected to controlled heating, using a weak electric current regulated by a
constant heating voltage, denoted as $V_{\mathrm{h}}$. Simultaneously, cooling
is achieved through the presence of an airflow, as illustrated in Figure 1.
The voltage difference $V_{\mathrm{h}}$ is applied across the two beam
anchors, with one of them serving as the ground (G in Fig. 1a).

The time-periodic electrostatic force acting on the beam is controlled by the
actuating voltage $V_{\mathrm{ES}}$, applied to an electrode, as marked in
Figure 1. Increasing $V_{\mathrm{ES}}$ beyond a critical snap-through (ST)
threshold, denoted as $V_{\mathrm{ST}}$, triggers a dynamic shift from the
microbeam’s first to the second stable equilibrium position. Conversely,
reducing the voltage below the lower critical threshold, $V_{\mathrm{SB}}$,
causes the beam to snap back (SB). The interaction between Joule heating and
cooling via the airflow influences the curvature of the beam and the critical
voltage values. Measurements of $V_{\mathrm{ST}}$ and $V_{\mathrm{SB}}$ are
used to determine the air velocity. This device offers enhanced performance
owing to its enhanced sensitivity near the ST and SB points [14, 15].
Additionally, it possesses a unique capability to measure flow at two closely
spaced points at two stable positions of the microbeam, allowing for the
assessment of flow velocity gradients, as discussed in [16].

###  2.2 Setup

The experimental setup is illustrated in Figure 2(a). During the experiments,
an AFG3022C waveform generator from Tektronix is employed. It provides a
voltage through channel 1, which is subsequently amplified through a signal
amplifier (Trek PZD350A). This amplified voltage is then used to apply
$V_{\mathrm{ES}}$ to the electrode. Additionally, to generate the electric
current required for Joule heating within the beam, a constant heating voltage
$V_{\mathrm{h}}$ is applied across the beam’s anchors, with this voltage being
supplied by the same waveform generator through channel 2.

Figure 2: (a) Overview of the experimental setup. (b) Optical image of the
experimental setup and (c) schematics of the chip from the perspective of the
side camera, with the laser from the LDV, and the airflow at an angle
$\alpha^{\circ}$ relative to the beam.

The real-time displacements of the beam’s center are recorded using a single-
point laser Doppler vibrometer (LDV, Polytec, OFV534). The LDV focuses its
laser beam on the side face of the beam, which has a width of $b=25\;\mu$m.
The values of $V_{\mathrm{ST}}$ and $V_{\mathrm{SB}}$ are extracted from the
time series recorded with an oscilloscope (DSOX2004A, Keysight), synchronously
with the LDV signal.

To regulate the flow rate, we employ a pressure gauge (P-30PSIG-D/5P, Alicat)
to monitor the pressure in the air supply system. A rectangular nozzle (1”
High Power, EXAIR) generates a wide, uniform jet airflow directed towards the
microbeam. The air supply is calibrated using a commercial anemometer (CTV
110, Kimo). Our experiments encompass a range of air velocities spanning from
0 to approximately 15 m/s. We estimate the systematic uncertainty of these
measurements to be 2.8 m/s. Additionally, it is worth noting that the airflow
can be applied at various angles, denoted as $\alpha^{\circ}$ in Figure 2.

The beam is strategically positioned at the chip’s edge, as depicted in Figure
2(b,c). This positioning offers the advantage of simultaneously capturing
real-time beam velocity from a top view through LDV, while also enabling
direct imaging of ST and SB events using a custom-built optical arrangement.
The optical setup comprises a video camera (UI-6250SE-M-GL, IDS) and a
horizontally mounted tube microscope (Navitar). In most of our experiments,
the side camera’s use is primarily limited to the initial setup preparations,
as the bulk of quantitative measurements are conducted through the LDV.

###  2.3 Measurement methodology

Figure 3(a) illustrates the time history of a cyclic voltage $V_{\mathrm{ES}}$
applied to the electrode (labeled in blue, left vertical axis) and the
corresponding $z$ value of the beam midpoint ($x=L/2$), which is obtained
using LDV (labeled in red, right vertical axis). The electrostatic voltage
signal $V_{\mathrm{ES}}$ is a linear ramp with a constant rate of about 170 V
per period. The analog output from the LDV controller and the input voltage
signal are looped back into a digital storage oscilloscope, allowing for the
synchronized recording of their time histories. The determination of
$V_{\mathrm{ST}}$ and $V_{\mathrm{SB}}$ values occurs during the post-
processing phase. This is achieved by identifying the displacement threshold
transition events within the recorded data, as illustrated in Figure 3(b).
These threshold events correspond to a 5 $\mu$m jump in displacement, serving
as the triggers for ST and SB events.

Figure 3:  (a) Cyclic time history plot of the beam displacement (red) versus
the voltage $V_{\mathrm{ES}}$ (blue) between the beam and the electrode
($V_{\mathrm{h}}$=0 V). (b) Displacement as a function of the actuating
voltage on the beam. (c) Electrostatic voltage (blue) and beam displacement
(the other colors) as a function of time. In contrast to (a) and (b), a track
filter is activated here to filter out fluctuations. (d) A close-up of the
beam displacement from six different heating levels $V_{\mathrm{h}}$=0, 0.33,
0.66, 1, 1.33, and 1.66 V.

The actuation process is repeated for four cycles, operating the beam at a
frequency of 400 Hz. Averaging is employed to mitigate data variability,
facilitating the calculation of the mean value and its associated standard
error. It is important to note that the mechanical response can be considered
quasi-static due to the significantly lower frequency of the electrical signal
chosen, which is of the order of $\mathcal{O}(10^{5})$ Hz compared to the
fundamental frequency of the beam.

The time history of the beam center displacement is presented in Figure 3(c)
for different $V_{\mathrm{h}}$ voltage levels. Crossing of dashed lines with
the electrostatic voltage linear ramp signal help to indicate the values of
$V_{\mathrm{ST}}$ and $V_{\mathrm{SB}}$ ,at the moments of transitions, for
the case of the heating voltage $V_{\mathrm{h}}$ equal to 0 V. A closer
examination of the zoom in view in Figure 3(d) reveals a noticeable shift in
the time when the threshold is crossed for increasing heating voltage
$V_{\mathrm{h}}$ in the range 0 $\div$ 1.66 V. This shift results in
distinctly different critical electrostatic voltages $V_{\mathrm{ST}}$ and
$V_{\mathrm{SB}}$. We will address this shift quantitatively in the following,
but we first address the bias in these measurements, due to the LDV heating of
the beam.

###  2.4 LDV-Induced Beam Heating Bias

The specific setup explained above enables us to obtain unique results,
including direct measurement of the beam’s response time history during the
transition using LDV. However, the LDV system comes with a drawback known as
the “loading effect,” which inadvertently heats the beam during measurements,
as discussed in [20]. To assess the bias on the measured $V_{\mathrm{ST}}$ and
$V_{\mathrm{SB}}$ values, we employ digital camera imaging of the beam and
compare the results obtained with and without LDV. It is important to note
that the image processing method for detecting ST and SB events is less
precise compared to the LDV approach. Therefore, in these experiments, we
extended the period to 60 seconds to achieve more accurate results. This
extension was necessary because of the video camera’s lower temporal
resolution, which demanded a longer measurement duration for precise
observations. Consequently, the bias due to LDV heating could be larger than
in the case of high-frequency measurements in our experiments, due to a
significantly longer time period of the measurement.

The figures in Figure 4(a, b) show the results with “LDV on” and

Figure 4: (a) Critical snap-through voltage ($V_{\mathrm{ST}}$) and (b)
critical snap-back voltage ($V_{\mathrm{SB}}$) as a function of the flow
velocity $U$. Circle markers indicate measurements with LDV, while square
markers indicate measurements without LDV. (c) Sensitivity of the snap-back
event, a dimensionless measure of the change in voltage with respect to the
reference voltage $V_{\mathrm{SB}}$ at $U=0$ m/s. Dashed lines represent a
quadratic fit.

“LDV off,” represented by circles and squares. Using the LDV system increases
both $V_{\mathrm{ST}}$ and $V_{\mathrm{SB}}$ by around 4 V and 6 V,
respectively. To further quantify the influence of LDV heating, we examine the
relative change in SB voltage, denoted as $\Delta
V_{\mathrm{SB}}=100\times(V_{\mathrm{SB}}-V_{\mathrm{SB}}|{U=0})/V_{\mathrm{SB}}|{U=0}$
in Figure 4(c). Notably, $\Delta V_{\mathrm{SB}}$ increases from 6% to 10% at
an airflow velocity of 12 m/s when the LDV is active. We will discuss this
bias further in Section 3.1, where we explore how Joule heating affects the
behavior of the flow velocity sensor.

###  2.5 Analytical model

####  2.5.1 Beam equilibrium

To understand the sensor’s behavior and interpret our experimental findings,
we explore a reduced-order model of the double-clamped heated microbeam with
initial curvature, as shown in Figure 1. While previous models in [21, 14]
addressed specific aspects of sensor phenomena, such as aerodynamic loading,
overheat ratio, air cooling, or dynamic effects, each model focused on
distinct aspects. In this work, we present a comprehensive model that
describes the combined influence of all these effects and their interactions.

The motion of the beam subjected to an axial force $N$ (positive in
compression) and a distributed actuating force $f_{a}$ can be described by the
following equation

$\begin{split}\rho
A\ddot{z}\\!+\\!c\dot{z}+EI_{yy}\left(z^{{}^{\prime\prime\prime\prime}}-z_{0}^{{}^{\prime\prime\prime\prime}}\right)+\left[N-\frac{EA}{2L}\int_{0}^{L}\left(z^{\prime
2}-z_{0}^{\prime 2}\right)dx\right]z^{\prime\prime}=f_{a}(z),\\\
z(t,0)=z(t,L)=0,\;z(0,x)=z_{0}(x)\end{split}$ (1)

Here, $z(x,t)$ and $z_{0}(x)$ are the deformed and the initial (as-designed)
elevations of the beam above its anchored ends, respectively (see Figure
1(b)); $EI_{yy}$ and $EA$ are the bending and axial stiffness of the beam,
where $I_{yy}={bd^{3}}/{12}$ is the second moment of area of the cross-
section, $A=bd$ is the cross-sectional area of the beam, $c$ is the viscous
damping coefficient, $E$ is the Young’s modulus, $\rho$ is a mass density and
$\dot{(\;)}$, $(\;)^{\prime}$ denote the derivative with respect to the time
and the axial coordinate $x$, respectively.

The axial force

$N=\sigma_{\text{res}}A+\frac{\alpha EA}{L}\int_{0}^{{L}}\theta
d{x}=\sigma_{\text{res}}A+\alpha EA\overline{\theta}$ (2)

is generated by both the beam heating of the beam [22] and by a residual
stress $\sigma_{res}$, estimated to be a few MPa [14, 15]. In Eq. (2),
$\alpha$ is the thermal expansion coefficient of Si, and $\overline{\theta}$
is the length-averaged difference between the beam temperature $T({x})$ and
the ambient temperature $T_{\infty}$. In our case, we use Joule heating by
applying a voltage $V_{\mathrm{h}}$ between the ends of the beam to induce
axial loading.

The beam is actuated by an electrostatic force, which is parameterized by the
voltage $V_{\mathrm{ES}}$, and can be described by the following equation

$f_{a}(z)=\frac{\varepsilon_{0}bV_{\text{ES}}^{2}}{2(g_{0}-z)^{2}}\left(1+0.65\,\frac{\langle
g_{0}-z\rangle}{b}\right),$ (3)

where $\varepsilon_{0}$ is the vacuum permittivity and $\langle
g_{0}-z\rangle=\int_{0}^{L}(g_{0}-z)\,dx/L$ is the averaged distance between
the beam and the electrode. Here, we use a first-order fringing field
correction [23], because the term $\langle g_{0}-z_{0}\rangle/b\approx 0.4$ is
not negligibly small.

We construct the single-degree-of-freedom reduced-order model associated with
Eq. (1) using the Galerkin decomposition based on the following approximations
of $z(x,t)$ and $z_{0}(x)$:

$z(x,t)=q(t)\phi(x),\quad z_{0}(x)=-{h}\phi({x}).\;$ (4)

where $\phi({x})=1/2[1-\cos(2\pi x/L)]$ is the base function, $q$ and $h$ are
the deformed and the initial midpoint elevations of the beam, respectively.
Note that $h$ is positive and the minus sign is due to the direction of the
positive $z$-axis defined in Figure 1b. Substituting the representation Eq.
(4) into Eq. (1), multiplying the resulting expression by $\phi(x)$,
integrating by parts, and using the orthogonality of base functions in
Galerkin’s formalism, we obtain the following equation expressed with non-
dimensional variables

$\displaystyle\ddot{w}_{m}+\frac{\dot{w}_{m}}{Q}+w_{m}\left(1-\\!{n}+\frac{\eta^{2}}{8}\right)-\frac{3w_{m}^{2}\eta}{16}+\frac{w_{m}^{3}}{16}=\\!-{n\eta}+\frac{\beta}{\sqrt{(\gamma+\eta-
w_{m})^{3}}}.\;\;$ (5)

Non-dimensional quantities used in the reduced order model are summarized in
Table 2.

Table 2: Quantities used in the development of the reduced-order model.  $r=\sqrt{I_{yy}/A}$ | Gyration radius
---|---
$\gamma=g_{0}/r$ | Normalized beam-electrode distance
$\eta=h/r$ | Initial midpoint elevation
$w_{m}=\eta+{q}/r$ | Midpoint deflection
$N_{e}=4\pi^{2}EI_{yy}/L^{2}$ | Euler’s buckling force
$n=\frac{N}{N_{e}}=(\sigma_{\text{res}}A+\alpha EA\overline{\theta})/N_{e}$ | Normalized axial force
$\beta=\frac{\varepsilon_{0}bL^{4}V_{\text{ES}}^{2}}{8\pi^{4}r^{3}\sqrt{\gamma}EI_{yy}}(1+0.65\,\frac{\langle g_{0}-z\rangle}{b})$ | Voltage parameter
$Q=4\pi^{2}/(c{L}^{2})\sqrt{EI_{yy}\rho A/3}$ | Quality factor

For the case of quasi-static actuation, we can omit the terms containing time
derivatives, then Eq. (5) could be solved in respect of
$w_{m}(V_{\mathrm{ES}})$.

Equation (5) helps to highlight the flow sensor operational scenario. When the
device is exposed to an airflow, the temperature of the beam changes, changing
the average temperature difference between the beam and the air,
$\overline{\theta}$. This leads to a variation in the axial force ($n$ in
Table 2) and alters the critical ST and SB voltages expressed through $\beta$.
To predict the $V_{\mathrm{ST}}$ and $V_{\mathrm{SB}}$ values and the flow, it
is necessary to quantify the convective heat transfer and relate
$\overline{\theta}$ to flow velocity over the beam.

For simplicity of analytical prediction, we limit our analysis to steady-state
heat transfer, and neglect radiation. In this simplified case, the heat
transfer equation in the beam is governed by the equation

$-(\kappa{\theta}^{\prime})^{\prime}={J}^{2}\rho_{{e}}-\frac{P\tilde{h}}{A}{\theta}\;,\;\theta(0)=\theta(L)=0,$
(6)

where $\kappa$ is the thermal conductivity, $\tilde{h}$ is the average
convective heat transfer coefficient, and $P=b+2d$ is the perimeter of the
beam’s cross-section. $J=I/A=V_{\mathrm{h}}/(\rho_{e}L)$ and $\rho_{e}$
represent the electric current density and the beam’s material resistivity,
respectively. Assuming that $\kappa$ and $\rho_{e}$ are temperature-
independent around the operating point, we can find the averaged solution of
Eq. (6) as follows

$\overline{\theta}=\frac{V_{\text{h}}^{2}}{4\kappa\rho_{e}m^{2}}\left(\\!1-\\!\frac{\tanh(m)}{m}\right),\;\;m=\frac{{L}}{2}\sqrt{\frac{P\tilde{h}}{\kappa
A}}.$ (7)

In Eq. (5), $\overline{\theta}$ is given by Eq. (7), and $\tilde{h}$ can be
estimated using the convective heat transfer theory. Due to the small size of
the beam, the relevant dimensionless parameter $Re=UL/\nu$ is small and we can
use laminar boundary layer analysis. $\tilde{h}$ can be determined using the
non-dimensional Nusselt number, $Nu$ as $\tilde{h}=Nu\,\kappa_{\text{air}}/L$.
In the laminar boundary layer analysis $Nu=0.664\,Pr^{1/3}\,Re^{1/2}$. Here,
the nominal values for airflow can be used: Prandtl number, $Pr=0.7$, the
thermal conductivity of air ${\kappa_{\text{air}}=0.0257}$ W/(mK), and
kinematic viscosity $\nu=1.5\times 10^{-5}$ m2/s [14]. The material properties
of single crystal silicon and the rest of the constants required for the
application of the model are given in Table 3.

Table 3: Properties of single crystal silicon (correspond to $T=300$K).  Parameter | Value
---|---
Young’s modulus, $E$ | 169 GPa
Thermal expansion, $\alpha$ |  $2.6\times 10^{-6}\;1/^{\circ}$C
Thermal conductivity, $\kappa$ |  $150$ W/(mK)
Vacuum permittivity, $\varepsilon_{0}$ |  $8.85\times 10^{-12}$ F/m
Resistivity, $\rho_{e}$ |  $0.02\;\Omega$ cm
Density, $\rho$ |  $2.33$ g/cm3
Residual stress, $\sigma_{res}$ |  -$2.7$ MPa

In this study, we revisit the conclusions of the works [15, 14, 16, 18] and
perform detailed experimental investigations of the influence of several
additional, not addressed in the previous works, effects on the sensor. In the
case of perpendicular (to the beam’s axis) flow, the drag force was found in
[14] to influence the beam behavior. In the present work, the flow direction
is varied, but most of the experimental results are obtained for the flow
parallel to the beam axis ($\alpha=0^{\circ}$), where the contribution of the
drag force is negligible. Therefore, we have not included the corresponding
term into the analytical model.

####  2.5.2 Frequency response

Besides the quasi-static analysis, we also investigate the beam’s frequency
response, which involves exploring the effects of beam oscillations after ST
and SB transitions. This encompasses studying the fundamental frequency of the
free vibrations, $f_{\text{res}}$, corresponding to the linearized version of
Eq. (5):

$f_{\text{res}}=\frac{\omega_{0}}{2\pi}\sqrt{1-n+\frac{\left(3w_{m}^{2}+2\eta^{2}-6\eta
w_{m}\right)}{16}-\frac{3\beta}{2\sqrt{(\gamma+\eta-w_{m})^{5}}}},$ (8)

where $\omega_{0}=4\pi^{2}\sqrt{EI_{yy}/(3\rho AL^{4})}$ is the fundamental
frequency approximation of the initially straight unforced beam and
$w_{m}(\beta,\overline{\theta})$ is the solution of the equilibrium equation
Eq. (5). More information about this derivation can be found in [24, 25, 18].

##  3 Results and discussion

In this section, we outline the results of measurements revealing various
factors on sensor behavior during operation, such as the overheat ratio, flow
loading, noise and vibrations. Finally, for the first time, we demonstrate the
real-time beam shape during ST-SB transitions.

###  3.1 Joule heating effect

We present first the quantitative outcomes from various runs conducted at
different Joule heating voltages ($V_{\mathrm{h}}$=0, 1, 1.33, and 1.66 V). In
Figure 5(a, b), we illustrate the variations in $V_{\mathrm{ST}}$ and
$V_{\mathrm{SB}}$ in response to Joule heating at different $V_{\mathrm{h}}$
values across a flow velocity range of 0 to 15 m/s. Notably, our results show
that Joule heating induces a consistent offset in all curves. Interestingly,
we find that sensitivity remains consistently high across all voltage levels,
reflecting the robustness of the sensor’s performance. Note that for all these
measurements, we used LDV with the inherent bias in $V_{\mathrm{ST}}$ and
$V_{\mathrm{SB}}$ due to laser heating of the beam.

Figure 5: (a) Critical snap-through voltage ($V_{\mathrm{ST}}$) and (b)
critical snap-back voltage ($V_{\mathrm{SB}}$) as a function of the flow
velocity $U$ at different $V_{\mathrm{h}}$ levels. Dashed lines emphasize
linear trend lines.

To provide a more comprehensive understanding of the impact of Joule heating
voltage playing a key role in determining the flow sensor’s overheat ratio, we
thoroughly investigate the beam’s response to a range of electrostatic
voltages ($V_{\mathrm{ES}}$). We employ an analytical model, which is detailed
in Sect. 2.5, to systematically compare the model’s predictions and the
experimental results. This in-depth analysis allows us to assess how varying
electrostatic voltages influence the sensor’s behavior, shedding light on its
performance in different operational scenarios.

Figure 6(a) shows the analytical predictions of changes in the electrostatic
voltage $V_{\mathrm{ES}}$ as a function of the normalized midpoint
displacement $w_{m}$ of the beam.

Figure 6: (a) Modeled beam response to the electrostatic voltage
$V_{\mathrm{ES}}$ at various heating voltages $V_{\mathrm{h}}$, with a few
curves shown for clarity. Insets show the convex (before ST) and concave
(after ST) configurations of the beam. (b) Comparison between experimental
(symbols) and modeled (solid curve) voltage $\tilde{V}$ as a function of
voltage $V_{\mathrm{h}}$. Error bars represent a standard error of the mean.
(c) The critical voltage indicator ($\tilde{V}$) as a function of the flow
velocity $U$ at different $V_{\mathrm{h}}$ levels. The model results are
represented by lines (dotted, dash-dot, dashed, and solid for
$V_{\mathrm{h}}$=0, 1, 1.33, and 1.66 V, respectively). Error bars represent a
standard error.

The other limit point (local minimum, SB) on the curve corresponds to the SB
voltage $V_{\text{SB}}$ at which the beam returns to its original state. It is
important to note that bistability can manifest even without additional beam
heating ($V_{\text{h}}=0$), attributed to the beam’s initial curvature.

As $V_{\mathrm{h}}$ increases, $V_{\text{ST}}$ increases, while the
$V_{\text{SB}}$ decreases. Once a certain level of heating is reached,
$V_{\text{SB}}$ reaches zero. The meaning is that the beam can stay in the
switched configuration at zero actuating voltage. At even higher
$V_{\mathrm{h}}$ , a force in the opposite (negative $z$) direction should be
applied to switch the beam back. This behavior is known as the latching effect
[26].

To quantify the relationship between critical voltages $V_{\text{ST}}$ and
$V_{\text{SB}}$, with respect to both flow velocity and Joule heating, we
introduce a universal critical voltage indicator denoted as $\tilde{V}$,
calculated as
$\tilde{V}=(V_{\text{ST}}-V_{\text{SB}})/(V_{\text{ST}}+V_{\text{SB}})$.

This introduction of a universal indicator serves two key purposes. First, it
consolidates the information into a single value, streamlining the analysis
rather than dealing with separate ST and SB voltage values. Second, it
attempts to disentangle the influence of the overheat ratio effect from the
impact of varying ambient temperatures, a factor that can fluctuate in
different experimental conditions.

Figure 6(b) compares the analytical model with the experimental results,
illustrating the changes in $\tilde{V}$ as a function of $V_{\mathrm{h}}$. The
model predictions are consistent with the experimental observations.
Furthermore, it predicts the value of the heating voltage value
($V_{\text{h}}=2.2$ V) corresponding to latching, which was not reached in the
experiment.

The Joule heating effect in terms of $\tilde{V}$ as a function of the flow
velocity $U$ is shown in Figure 6(c). The figure also shows, represented by
the different types of lines, the theoretical results obtained using Eq. (5).
There is no perfect agreement between the model and the experiment, especially
for the low heating case ($V_{\mathrm{h}}$=0 and 1 V). Note that the reduced-
order model could not take into account the effect of velocity when the Joule
heating is “off” ($V_{\mathrm{h}}$=0 V), but it is possible that there is a
small temperature difference between the airflow and the microbeam.
Nevertheless, the reduced-order model accurately captures the trends in the
critical voltage change, indicating that the primary factor in velocity
measurement is the change in average beam temperature, as suggested by the
model. The graphs also show that all curves have nearly the same slope angle
of 0.015 V/(m/s). This suggests that the beam overheat ratio is not a critical
parameter for the sensor, which raises the possibility of significantly
reducing energy consumption at the required sensitivity level.

###  3.2 Flow angle sensitivity

This section delves into the influence of flow angle on the sensor’s response.
The sensor’s angle sensitivity is evaluated in the plane parallel to the
sensor, as depicted in Figure 2(b, c), with the sensor’s long side aligned
with the flow direction. The results of this impact, illustrated in Figure 7,
reveal that the $\tilde{V}$ values remain consistent at low velocities.
However, as flow velocities increase, deviations appear for different flow
directions defined by the angles $\alpha=0^{\circ},15^{\circ}$, and
$45^{\circ}$.

Figure 7: Influence of flow direction by changing the angle $\alpha^{\circ}$
($V_{\mathrm{h}}$=1.33 V). Dashed lines represent visual trends. Error bars
represent a standard error of the mean.

The effect of flow direction on the $\tilde{V}$ value is noticeable. Our
sensor behaves similarly to miniature hot-wire sensors used for measuring
turbulent flows, as shown in reference [27]. Due to its simpler manufacturing
process and reliable angular response, the bistable microbeam has potential as
a component in future multi-array flow sensors, capable of measuring all three
flow velocity components.

###  3.3 Frequency response and sensitivity to vibrations

In Figure 4, it is evident that the sensors respond to flow velocity even when
no LDV or Joule heating is involved. This suggests the existence of an
additional mechanism associated with airflow in our experiment. Since the flow
runs parallel to the flow sensor, it is unlikely that this effect is
attributed to mean flow drag or lift forces, as these forces are negligible in
this particular configuration.

We considered that the device might be sensitive to turbulent velocity
fluctuations or chip vibrations. To investigate this, we initially determined
the natural frequency of the beam’s vibration around various equilibrium
configurations using LDV, following a similar approach to our previous study
[18].

To achieve this, we introduced time-harmonic (AC) components to the steady
(DC) electrostatic actuating voltage, resulting in the equation
$V_{\text{ES}}(t)=V_{\text{DC}}+V_{\text{AC}}\sin{(2\pi f_{a}t)}$. Here,
$V_{\text{DC}}$ represents the DC bias voltage, and $V_{\text{AC}}$ is the
amplitude of the time-harmonic AC component, set at 0.5 V.

During the frequency up-sweep, the frequency $f_{a}$ increases linearly with
time, ranging from 30 to 120 kHz. We determined the natural frequency of the
beam’s vibration, denoted as $f_{n}$, for different equilibrium deflections
determined by the DC bias voltage. The measured resonant frequency of the
beam, when there is zero electrostatic voltage applied, is approximately 111
kHz.

In Figure 8, we present the normalized resonant frequency of the beam as a
function of the normalized bias voltage.

Figure 8: Frequency of free vibrations around equilibrium as a function of the
increasing/decreasing DC bias voltage. The baseline value $f_{0}$ represents
the structure’s frequency in the initial unloaded configuration. The bias
voltage is normalized by the snap-through voltage $V_{\mathrm{ST}}$.
Experimental results for increasing and decreasing voltages are represented by
circle and square markers, respectively. The solid and dash-dot lines
represent the calculated increasing/decreasing curves.

The experimental data is depicted as circles and squares, while the calculated
results, obtained using Eq. (8), are represented by solid and dash-dot lines.
These dependencies are displayed for increasing and decreasing DC voltage
($V_{\text{DC}}$) values, covering the range encompassing the values at which
ST and SB events (represented by $V_{\text{ST}}$ and $V_{\text{SB}}$) occur.
Notably, the experimental findings align well with the results generated by
the reduced-order model.

As expected, the vibration frequency significantly decreases near the ST and
SB points. However, it is crucial to note that the experimental data does not
reach absolute zero values, which would entail zero frequency and stiffness.
This contrasts with the model’s predictions [24].

The measured frequencies at the ST and SB branches reveal an exceptionally low
beam stiffness in close proximity to the critical $V_{\mathrm{ST}}$ and
$V_{\mathrm{SB}}$ points. Consequently, even a minor vibration and noise level
can induce an ST transition at voltages lower than initially predicted.

We can derive consistent conclusions by examining the beam’s frequency
response in the context of electrostatic actuation with a linear voltage ramp.
In this analysis, we utilized the same voltage ramp signal, denoted as
$V_{\mathrm{ES}}$, as previously detailed in Section 2.3, with a 60-second
duration, allowing for a thorough examination of the beam’s vibrations. The
LDV signal was sampled at a rate of 1 MHz.

Figure 9 illustrates the beam’s frequency response, particularly during the
occurrences of snap-through (ST) and snap-back (SB) transitions. This power
spectral density (PSD) spectrogram was generated through the application of
the overlapped segment averaging spectral estimation (using the ’spectrogram’
Matlab function) with a segment length of $2^{18}$ samples and a 75% overlap.

The spectrogram reveals that the fundamental frequency closely mirrors the
curves in Figure 8. Notably, a sharp change in the fundamental frequency is
evident during the transition between equilibrium states, specifically during
snap-through and snap-back. Moreover, we observe the emergence of a higher
(second) harmonic vibration mode, surpassing the intensity of the first mode
by a factor of 2.75. These results affirm our findings regarding the sensor’s
high sensitivity in proximity to the beam’s equilibrium state. They underscore
the fact that even a minor level of vibration or noise can trigger snap-
through or snap-back transitions. These factors are known to hinder the
experimental voltage-frequency dependency curves from reaching the theoretical
zero level, as documented in references [28, 29, 30].

Figure 9: Frequency response of the beam vibrations in the form of a
spectrogram during the voltage actuation ramp $V_{\mathrm{ES}}$ (60 s). The
level of the PSD is normalized to the maximum value. The overheat ratio is
$V_{\mathrm{h}}=$0 V. The line that corresponds to $1.8f_{0}$ is a regular
noise signal that comes from the voltage amplifier.

To gain a deeper insight into this effect, we performed an experiment designed
to emulate the influence of the noise on the ST and the SB voltages. Namely,
we add an electrical white noise signal generated by the wave generator to the
baseline DC electrostatic voltage $V_{\mathrm{ES}}$ used in all other
experiments.

The noise amplitude, ranging from 0 to 5 V, should be viewed in the context of
the base DC voltage, which extends from 0 to 200 V in the electrostatic
voltage ramp signal. The measurements of $V_{\mathrm{ST}}$ and
$V_{\mathrm{SB}}$ are carried out under conditions without flow ($U=0$ m/s)
and are depicted in Figure 10.

Figure 10: Effect of noise on sensor performance ($V_{\mathrm{h}}$=0 V). The
error bars represent a standard error of the mean.

As the amplitude of noise increases, its influence becomes apparent in the
absolute value of $\tilde{V}$. This provides direct evidence that the beam
exhibits remarkable sensitivity to fluctuations, including those induced by
the flow, near the critical equilibrium point. Consequently, it is well-suited
for sensing turbulent flows. This capability has the potential for
applications beyond the measurement of average flow velocity. It can also be
used to capture turbulent fluctuations, such as in shear stress measurements,
particularly when the sensors are flush with walls. Notably, similar sensors
based on hot-wire technology have been developed [31, 32, 33]. Our findings
open up new possibilities for applying bifurcation-based sensors in this
field.

###  3.4 Dynamics of the ST transition

In contrast to previous studies [16, 18], where a single-point LDV system was
employed to measure only the midpoint deflection, our approach in this study
takes a significant step forward. We utilize the Micro System Analyzer
(Polytec, MSA 600).

The MSA operates in a scanning mode, allowing us to capture the complete shape
of the entire beam during the transient ST and SB events. To achieve this, the
MSA records a 36 milliseconds long velocity signal from 56 points distributed
along the length of the beam. We actuate the beam buckling with a linear ramp
of $V_{\mathrm{ES}}$ ranging from 0 to 200 V at a frequency of 400 Hz. The
displacement of the beam at each point is determined by integrating the
velocity signal over time. Phase averaging is applied to calculate the average
displacement at each measurement point on the beam, effectively eliminating
free vibrations following ST and SB.

In Figure 11, we present snapshots of the beam’s shape $z(x)$ along the beam’s
length $x$ during the snap-through and snap-back transitions.

Figure 11: Snapshots of beam deformation during snap-throught transition (a),
and snap-back transition (b).

The arrows indicate the direction of the transitions, and the color coding
reflects the progression from blue to magenta over time. We have marked
positions at various arbitrary times, evenly distributed for clarity.

This represents a real-time beam profile measurement during the dynamic snap-
through and snap-back transitions. Notably, we observe a slight asymmetry in
the shape of $z(x,t)$ during both transitions concerning the midpoint,
$z(x=0.5L)$. This implies that the transitions occur at lower
$V_{\mathrm{ST}}$ and higher $V_{\mathrm{SB}}$ values than what would be
anticipated from a symmetric first-mode transition. The presentation of these
asymmetric snapshots of beam deformation during ST and SB events is a novel
contribution, supporting the hypothesis presented in [17] regarding the nature
of this transition. Additionally, it provides insight into the shape of free
vibrations following snap-through, in line with findings from [34].

Notably, during the beam transition, it is evident that the center of the beam
lags behind the rest of the beam, with the exception of the edges, which
remain static. Particularly, the sections between the center and the edges,
around $0.2<x/L<0.4$ and $0.6<x/L<0.8$, lead the motion during a significant
portion of the deformation. This shape likely reflects a combination of
various modes and underscores the contribution of the second symmetric mode.
This discovery encourages further exploration in this direction.

##  4 Summary and conclusions

This study delved into critical aspects of the sensor’s behavior. Firstly, we
initiated by quantifying the “loading effect” induced by the laser Doppler
vibrometer, shedding light on the impact of laser heating on voltage
measurements.

Secondly, we investigated the beam’s response to varying levels of Joule
heating. While the heating voltage $V_{\mathrm{h}}$ influenced the absolute
values of $V_{\mathrm{ST}}$ and $V_{\mathrm{SB}}$ voltages, the device’s
sensitivity remained notably consistent. Our analytical model supported these
findings and revealed a latching effect with increasing Joule heating. These
results suggest that the beam overheat ratio may not be as pivotal as assumed
for most thermal MEMS sensors, promising substantial energy consumption
reduction.

Thirdly, we investigated the influence of the flow angle on the performance of
the beam. We concluded that the device performs similarly to previous hot-
wire-based microsensors and can measure flow direction.

We conducted a detailed analysis of the beam’s response to fluctuations at
different frequencies, typically present in various applications. Our results
confirmed that this device, operating near critical transition points, is also
sensitive to turbulent flow velocity fluctuations.

Finally, we obtained the first direct beam shape measurements during
transitions and confirmed the theoretical predictions of asymmetric, higher-
than-first-mode transitions. We can confirm that the single degree-of-freedom
approximation of the single of beam needs further improvement.

Overall, this study lays the groundwork for designing a series of flow sensors
and multi-sensor arrays for velocity and velocity gradient measurements using
initially curved bistable microbeams. However, unanswered questions still
require further investigation, such as the sensor’s response to out-of-plane
flow directions, its performance in realistic environments, and the noise
levels at which it can accurately measure low and high flow velocities for
different applications.

## Acknowledgments

The authors acknowledge the financial support of the Israeli Innovation
Authority, NOFAR program and of RAFAEL LTD, Advanced Defense Systems. S.
Krylov acknowledges the support of the Henry and Dinah Krongold Chair of
Microelectronics. The authors would also like to thank Gal Spaer Milo, Grigori
Gulitski, and Stella Lulinsky for their assistance.

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