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Dynamic behavior of smart microelectromechanical systems in industrial applications

Marius Pustan¹, Corina Birleanu¹, Cristian Dudescu¹, and Jean-Claude Golinval² ¹Technical University of Cluj-Napoca, Cluj-Napoca, Romania ²University of Liège, Liège, Belgium

Abstract

Many of the microelectromechanical systems (MEMS) industrial applications require vibrating components that operate to a high-quality factor and small energy dissipation during oscillations. To improve the reliability design of MEMS resonators, the effect of operating conditions on the dynamical response of vibrating components has to be accurately determined. As a function of the operating conditions, the dynamical response and the loss of energy in vibrating MEMS components are influenced by the damping of the surrounding medium and depend on the intrinsic effects of mechanical structure. To differentiate between the extrinsic damping and the intrinsic effects, experiments have to be performed both in ambient conditions and in a vacuum. In this chapter, some analytical models accompanied by experimental tests are presented to estimate the dynamical response and the loss of energy on samples fabricated from polysilicon with different geometrical configurations.

Keywords

Dynamic response; Electromechanical coupling; Energy dissipation; MEMS resonators; Quality factor

14.1. Introduction

Modeling and experimental tests of resonators from microelectromechanical systems (MEMS) are essential for design optimization and device reliability. One of the most important applications of MEMS resonators is mass detection for chemical and biological applications, radio frequency applications (RF MEMS), in the automobile industry and for aircraft conditions monitoring or satellite communications. The reliability and the lifetime are crucial parameters in different MEMS applications and these are strongly dependent on the material properties. Moreover, MEMS resonators need to be designed to perform their expected function with complete accuracy. The accuracy of a response is influenced by a resistor's material properties. As a function of applications, MEMS resonators operate under various conditions including different temperatures, humidity, or pressure of the surrounding medium. The mechanical behavior of resonators strongly depends on the operating conditions. To design a reliable MEMS resonator, experimental investigations are needed to evaluate the accuracy of their mechanical response under different loadings and operating conditions.

Two of the major causes of failure in MEMS resonators, which operate under high-cycle loading, are fatigue and the loss of energy during vibration, based on thermal effects. The high-cycle fatigue life is greatly influenced by microstructural variables such as the grain size, the volume fraction of the secondary phase, and the amount of solute atoms or precipitates. The information related to fatigue failure comes from stress–numbers of cycles curves, which is the traditional way of representing fatigue data. For cyclic motions of a structural material, significant heat generation occurs and energy dissipation is produced because of an energy loss mechanism internal to the material. The temperature gradient generates heat currents that cause an increase in entropy in the resonator and lead to energy dissipation. It would be highly desirable to design a MEMS resonator with as little loss of energy as possible. Experimentally, the loss of energy in MEMS resonators is evaluated considering the frequency response curves and measuring the bandwidth of oscillations under an excitation signal. In this chapter, experimental investigations are performed to estimate the loss of energy in MEMS resonators. Most of the MEMS vibration sensors that were used have a polysilicon microcantilever or microbridge as the sensing element. Although these components are simple geometrical structures, their dynamical behavior needs to be more accurately investigated.

14.2. Resonant frequency response of smart microelectromechanical systems vibrating structures

Mechanical resonators such as microcantilevers and microbridges are very often used as flexible mechanical components in MEMS. There are many applications that require ambient operating conditions and others that require working in a vacuum. As a consequence, during experimental investigations, the samples are successively tested in air and in a vacuum, and the effect of the surrounding medium on the amplitude and velocity of oscillations is estimated accordingly. The dynamic response of samples is changed as a function of the operating conditions.

In this chapter, vibrating MEMS resonators such as microcantilevers (Fig. 14.1(a)) and microbridges (Fig. 14.1(b)) are dynamically investigated and their frequency responses under a harmonic loading are determined under different testing conditions.

Mechanical stiffness is a fundamental criterion of elastically deformable mechanical flexible microcomponents. The bending stiffness of a microcantilever and a microbridge can be computed using the following well-known relations (Lobontiu and Garcia, 2004; Pustan and Rymuza, 2007):

• for a microcantilever,
$k = 0.25 β E$ $k = 0.25 β E$
(14.1)
• for a microbridge,
$k = 16 β E$ $k = 16 β E$
(14.2)
where E is the Young's modulus of material and β = wt³/l³ is the resonator geometrical parameter given by w = width, t = thickness, and l = length of samples.

Figure 14.1 Schematic of a microcantilever and a microbridge under electrostatic actuation. (a) microcantilever and (b) microbridge.

When a DC voltage (V_DC) is applied between the lower electrode and the vibrating MEMS structure, an electrostatic force is set up and the cantilever bends downward and comes to rest in a new position. To drive the resonator at resonance, an AC harmonic load of amplitude V_AC vibrates the cantilever around the new deflected position.

A single degree-of-freedom model, as presented in Fig. 14.2, can be used to simulate the dynamic response of a resonator due to the V_DC and V_AC electric signals. In this model, the proof mass of the resonators is modeled as a lumped mass m_e, and its stiffness is considered as a spring constant k. This part forms the vibrating side of a variable capacitor. The bottom electrode is fixed and considered as the second part of the MEMS structure. If a voltage composed of DC and AC terms as:

V = V_{DC} + V_{AC} \cos (ω t)

$V = V_{DC} + V_{AC} \cos (ω t)$

(14.3)

is applied between the resonator electrodes, the electrostatic force applied on the structure has a DC component and a harmonic component with the frequency ω such that:

F_{e} (t) = \frac{ε A V^{2}}{2 {[g_{0} - u_{z} (t)]}^{2}}

$F_{e} (t) = \frac{ε A V^{2}}{2 {[g_{0} - u_{z} (t)]}^{2}}$

(14.4)

where ε is the permittivity of the free space, A = w_e × w is the effective area of the capacitor, g₀ is the initial gap between the flexible plate and the substrate, and u_z(t) is the displacement of the mobile plate under the electrostatic force F_e(t).

The expression (14.4) evidences two aspects: the electromechanical coupling between the instantaneous value of beam gap and the applied voltage and then the nonlinear dependence between the mechanical displacement and the voltage.

Pull-in voltage, at which the elastic stiffness does not balance the electric actuation and the beam tends to collapse, can be evaluated by establishing the maximum gap allowing the static equilibrium. The spring force and the electrostatic actuation have opposing directions. The instability threshold is found by imposing the two conditions of null total force and the null first derivative with respect to the displacement:

Figure 14.2 Forced vibration model with fixed support used in dynamic investigations.

k u_{z} - \frac{ε A V^{2}}{2 {(g_{0} - u_{z})}^{2}} = 0

$k u_{z} - \frac{ε A V^{2}}{2 {(g_{0} - u_{z})}^{2}} = 0$

(14.5)

k - \frac{ε A V^{2}}{2 {(g_{0} - u_{z})}^{3}} = 0

$k - \frac{ε A V^{2}}{2 {(g_{0} - u_{z})}^{3}} = 0$

(14.6)

The unknown displacement and voltage are:

u_{pull - in} = \frac{g_{0}}{3}

$u_{pull - in} = \frac{g_{0}}{3}$

(14.7)

V_{pull - in} = \sqrt{\frac{8}{27} \frac{g_{0}^{3} k}{ε A}}

$V_{pull - in} = \sqrt{\frac{8}{27} \frac{g_{0}^{3} k}{ε A}}$

(14.8)

where u_pull-in and V_pull-in are the maximum displacement and voltage at which it is possible to have a stable equilibrium configuration and k is the beam stiffness described by Eq. (14.1) for a microcantilever and Eq. (14.2) for a microbridge.

Dynamic analysis of electrostatically actuated microcomponents is performed by linearizing the electrostatic actuation around an equilibrium position. The equivalent stiffness of a MEMS resonator including the mechanical stiffness provided by Eqs. (14.1) and (14.2) can be computed as:

• for a microcantilever,
$k_{eff} = 0.25 β E - \frac{ε A V^{2}}{{(g_{0} - u_{z})}^{3}}$ $k_{eff} = 0.25 β E - \frac{ε A V^{2}}{{(g_{0} - u_{z})}^{3}}$
(14.9)
• for a microbridge,
$k_{eff} = 16 β E - \frac{ε A V^{2}}{{(g_{0} - u_{z})}^{3}}$ $k_{eff} = 16 β E - \frac{ε A V^{2}}{{(g_{0} - u_{z})}^{3}}$
(14.10)

Based on these equations, the resonant frequency of an electrostatically actuated microcantilever and microbridge can be computed as:

ω_{0} = \frac{1}{2 π} \sqrt{\frac{k_{eff}}{m_{e}}}

$ω_{0} = \frac{1}{2 π} \sqrt{\frac{k_{eff}}{m_{e}}}$

(14.11)

where m_e is the equivalent mass of system.

Using the assumption that the kinetic energy of the distributed-parameter system is equal to the kinetic energy of the equivalent lumped-parameter mass, the equivalent mass can be determined (Lobontiu, 2007). The equivalent mass of a microcantilever is 0.235 m and of a microbridge is 0.406 m (m is the effective mass of beams).

The dynamic response of MEMS resonators presented in Fig. 14.1 subjected to a harmonic electrostatic force F_e(t) with the driving frequency ω given by an AC voltage is governed by the equation of motion:

m \cdot {\ddot{u}}_{z} (t) + c \cdot {\dot{u}}_{z} (t) + k \cdot u_{z} (t) = F_{e} (t)

$m \cdot {\ddot{u}}_{z} (t) + c \cdot {\dot{u}}_{z} (t) + k \cdot u_{z} (t) = F_{e} (t)$

(14.12)

where c is the damping factor.

The response of the system under DC and AC voltages is given by the equation:

u_{z} (t) = \frac{u_{z}}{{\sqrt{{(1 - {(\frac{ω}{ω_{0}})}^{2})}^{2} + (2 ξ \frac{ω}{ω_{0}})}}^{2}}

$u_{z} (t) = \frac{u_{z}}{{\sqrt{{(1 - {(\frac{ω}{ω_{0}})}^{2})}^{2} + (2 ξ \frac{ω}{ω_{0}})}}^{2}}$

(14.13)

where ξ is the damping ratio and ω₀ is the resonant frequency of beams given by Eq. (14.11).

Usually, the response is plotted as a normalized quantity u_z(t)/u_z. When the driving frequency equals the resonant frequency ω = ω₀, the amplitude ratio reaches a maximum value. At resonance, the amplitude ratio becomes:

\frac{u_{z} (t)}{u_{z}} = \frac{1}{2 ξ}

$\frac{u_{z} (t)}{u_{z}} = \frac{1}{2 ξ}$

(14.14)

The experimental investigations of the vibrating MEMS structures are performed using a vibrometer analyzer and a white noise signal. The aim of experimental investigations is to determine the frequency response of a microcantilever and a microbridge and the effect of the operating conditions on the velocity and amplitude of oscillations.

The geometrical dimensions of the microresonators from Fig. 14.3 are the following: total length l of beams is 150 μm; width w is 30 μm; and thickness t is 1.9 μm; the gap between the flexible plates and the substrate g₀ is 2 μm; the holes have a diameter of 3 μm; and the width w_e of the lower electrode of microbridge is 50 μm. The microcantilever is fabricated with the full lower electrode under the flexible plate. During tests, a DC offset signal of 5 V and peak amplitude of 5 V of the driving signal are applied to bend and oscillate the samples. The frequency response and the amplitude and velocity of oscillations are measured.

Figure 14.3 Microresonators used in experimental investigations. (a) Microcantilever and (b) microbridge.

Figure 14.4 The bending modes of oscillations of an electrostatically actuated microelectromechanical systems cantilever. (a) Bending mode 1, (b) bending mode 2, and (c) bending mode 3.

The frequency responses of the samples investigated can be monitored for different oscillation modes using a vibrometer analyzer. As presented in Figs. 14.4 and 14.5, three bending modes of oscillation were monitored under an excitation signal.

The tests are performed under ambient conditions and in a vacuum to estimate the damping effect on the velocity and amplitude of oscillations.

To analyze the dynamic response of the MEMS resonators under investigation, only the first bending mode is monitored and analyzed. The frequency response curves of a microcantilever tested in air are presented in Fig. 14.6 and those of microbridge in Fig. 14.7. Figs. 14.8 and 14.9 present the frequency response of the same resonators tested in a vacuum.

Figure 14.5 The bending modes of oscillations of an electrostatically actuated microelectromechanical systems microbridge. (a) Bending mode 1, (b) bending mode 2, and (c) bending mode 3.

Figure 14.6 Frequency response of an electrostatically actuated microelectromechanical systems microcantilever tested in ambient conditions.

Figure 14.7 Frequency response of an electrostatically actuated microelectromechanical systems microbridge tested in ambient conditions.

Figure 14.8 Frequency response of an electrostatically actuated microelectromechanical systems microcantilever tested in a vacuum. Vacuum at 8 × 10⁻⁶ mbar.

Figure 14.9 Frequency response of oscillations of an electrostatically actuated microelectromechanical systems microbridge tested in a vacuum. Vacuum at 8 × 10⁻⁶ mbar.

The dynamic experimental characteristics of the investigated microbridge and microcantilever are presented in Table 14.1.

The experimental results of the MEMS resonators agree with the analytical models presented above. As can be observed in Table 14.1, there are small differences between frequency responses of beams tested in different operating conditions. These differences depend on the damping of the surrounding medium that causes a shift in the frequency response of the beam. Significant differences were observed in the velocity and amplitude of oscillations. The amplitude and velocity of oscillations have small values if the microresonators are tested in ambient conditions based on the damping of the surrounding medium. The air damping changes not only the dynamical characteristics such as resonant frequency, amplitude, and velocity of oscillations but also the quality factor and the loss coefficient of energy, as presented in next section.

Table 14.1

Dynamic experimental characteristics function of testing condition
Resonator type	Resonant frequency (kHz)		Velocity (mm/s)		Amplitude (nm)
Resonator type	Air	Vacuum	Air	Vacuum	Air	Vacuum
Cantilever	100	99.37	0.27	88	0.47	140
Bridge	1003.37	992.81	0.19	31	0.03	4.96

14.3. Quality factor and the loss coefficient of smart microelectromechanical systems vibrating structures

The energy dissipated during one cycle of oscillation can be evaluated based on the quality factor Q. The quality factor is an important qualifier of mechanical microresonators and allows estimation of the loss coefficient of oscillations Q⁻¹ = 1/Q. In terms of energy, it is expressed as the total energy stored in the system divided by the energy dissipated per cycle. At resonance, the quality factor is expressed as (Lobontiu, 2007):

Q_{r} = \frac{m \cdot ω}{c} \approx \frac{1}{2 ξ}

$Q_{r} = \frac{m \cdot ω}{c} \approx \frac{1}{2 ξ}$

(14.15)

and the normalized response given by Eq. (14.14) is equal to Q_r.

The quality factor is also known as “sharpness at resonance,” which is defined as the ratio:

Q_{r} = \frac{ω}{ω_{2} - ω_{1}}

$Q_{r} = \frac{ω}{ω_{2} - ω_{1}}$

(14.16)

where ω₂ − ω₁ is the frequency bandwidth corresponding to 0.707 u_z(t)_max on the amplitude versus resonant frequency curves (as shown in Fig. 14.6).

The total loss coefficient occurring in a microresonator can be separated into two components as:

Q_{total}^{- 1} = Q_{e}^{- 1} + Q_{i}^{- 1}

$Q_{total}^{- 1} = Q_{e}^{- 1} + Q_{i}^{- 1}$

(14.17)

where e denotes the extrinsic losses and i the intrinsic losses.

Some of the extrinsic mechanisms are affected by changes of environment. The air damping can be minimized under ultrahigh vacuum conditions. Intrinsic losses in the resonator material are an important mechanism in accounting for energy dissipation.

A small internal loss is produced by the energy dissipation anchors that attach the resonator to substrate. The clamping losses can be determined by analyzing the vibration energy, which is transmitted from resonator to substrate, and for one anchor it can be computed as (Lobontiu, 2007):

Q_{anchor}^{- 1} = 2.17 {(\frac{0.5 l}{t})}^{3}

$Q_{anchor}^{- 1} = 2.17 {(\frac{0.5 l}{t})}^{3}$

(14.18)

where l is the length and t is the thickness of resonators.

Analytically, the Q-factor of investigated microresonators fabricated with holes as presented in Fig. 14.3 can be expressed using Eq. (14.15) where the damping coefficient c due to squeeze film c_sq of surrounding medium and due to the loss through holes c_holes can be computed as

c = c_{sq} + c_{holes}

$c = c_{sq} + c_{holes}$

(14.19)

The damping coefficient due to the squeeze film is (Pandey and Pratap, 2008; Pustan et al., 2014a,b):

c_{sq} = \frac{16 σ}{π^{6}} \frac{p_{a} χ l^{2}}{ω g_{0}} \sum_{m, n = odd} \frac{\frac{Γ^{2}}{π^{2}} + m^{2} χ^{2} + n^{2}}{{(m n)}^{2} [{(\frac{Γ^{2}}{π^{2}} + m^{2} χ^{2} + n^{2})}^{2} + \frac{σ^{2}}{π^{4}}]}

$c_{sq} = \frac{16 σ}{π^{6}} \frac{p_{a} χ l^{2}}{ω g_{0}} \sum_{m, n = odd} \frac{\frac{Γ^{2}}{π^{2}} + m^{2} χ^{2} + n^{2}}{{(m n)}^{2} [{(\frac{Γ^{2}}{π^{2}} + m^{2} χ^{2} + n^{2})}^{2} + \frac{σ^{2}}{π^{4}}]}$

(14.20)

where σ is the squeeze number that captures the compressibility effect, p_a is the air pressure, χ is the beam aspect ratio (χ = width/length), l is the beam length, g₀ is the gap between resonator and substrate, and Γ is a constant that captures the perforation effect.

The damping coefficient due to the loss through holes can be determined as (Pandey and Pratap, 2008):

c_{holes} = 8 π μ (\frac{h}{Q_{th}} + Δ E \cdot b) \cdot n

$c_{holes} = 8 π μ (\frac{h}{Q_{th}} + Δ E \cdot b) \cdot n$

(14.21)

where μ is the dynamic viscosity of the environment, h is the beam thickness, Q_th is the flow rate factor, which accounts for rarefaction effect in the flow through the parallel plates and through the holes, respectively (for slip flow regime), ΔE is the relative elongation of the hole length due to end effects, b is the holes radius, and n is the number of holes.

For cyclic motions of a structural material, significant heat generation becomes apparent and energy dissipation occurs because of an energy loss mechanism internal to the material (Lobontiu, 2007). The variation of strain in a microresonator is accompanied by a variation of temperature, which causes an irreversible flow of heat. The temperature gradient generates heat currents, which cause an increase in the entropy of the beam and lead to energy dissipation. This process of energy dissipation is known as “thermoelastic damping.” Thermoelastic damping depends on material properties such as the specific heat, coefficient of thermal expansion, thermal conductivity, mass density, elastic modulus, and the temperature and geometry. Thermoelastic damping is recognized as an important loss mechanism at room temperature in microscale beam resonators. The mechanism of thermoelastic damping was first studied by Zener (1937) and later developed by Lifshtz and Roukes (2000) and Yi (2008). He indicates that the phenomenon is induced by irreversible heat dissipation during the coupling of heat transfer and the strain rate in an oscillating system. The Zener model used the classical thermoelastic theory assuming an infinite speed of heat transmission. In a more complex model (Sun et al., 2006) based on generalized thermoelastic theory with one relaxation time, the bending moment on the beam during oscillations is separated into two parts: the first is the well-known moment that arises from the bending of the beam when the temperature gradient across the beam is zero; the second moment is the bending moment that arises from the variation of temperature across the upper and lower surfaces of the beam known as the “thermal moment.” Analytical results (Sun et al., 2006; Kazemnia et al., 2016) showed that thermoelastic coupling influences the amplitude, velocity, and resonant frequency of a beam based on its thermal moment. Over time, the deflection and thermal moment attenuate. The energy dissipation in a microresonator is given by means of the thermal moment variation followed by the attenuation of the amplitude (Sun et al., 2006). The theoretical results were validated by experiments (Pustan et al., 2012).

The total loss coefficient is experimentally determined when the sample oscillates in ambient conditions. The sample response in a vacuum determines the intrinsic losses. For the microresonators with the geometrical dimensions presented above, the experimental tests are performed both in a vacuum and in ambient conditions. Using the frequency bandwidth (ω₂ − ω₁) corresponding to 0.707 u_z(t)_max on the frequency response experimental curves presented in Figs. 14.6–14.9, the quality factor Q and the loss coefficient Q⁻¹ are determined and presented in Table 14.2.

Using Eq. (14.15), the damping ratio of tested samples in ambient conditions can be estimated. A damping ratio of 0.375 is determined for the microcantilever and 0.018 for the microbridge. The damping ratio ξ is any positive real number. For the value of the damping ratio 0 ≤ ξ < 1 as in the experiments, the system has an oscillatory response.

Experimental tests are conducted to estimate the thermomechanical coupling effect on the vibrating structures as function of operating time. The following presents the case of a microbridge resonator.

Figs. 14.10 and 14.11 present the frequency response experimental curves of the microbridge resonator tested in ambient conditions under the same applied signal at beginning of test and after 4 h of continuous excitation.

The velocity of oscillations in ambient conditions decreases from 187 to 65 μm/s after 4 h (Fig. 14.12); in a vacuum, the velocity is attenuated from 31 to 8.4 mm/s (Fig. 14.13). The same decreases in the microbridge resonator displacements as a function of the operating time were also observed.

Table 14.2

Quality factors Q and loss coefficient Q⁻¹ of investigated microresonators
Resonator type	Quality factor Q		Loss coefficient Q⁻¹
Resonator type	Air	Vacuum	Air	Vacuum
Cantilever	1.33	310.5	75 × 10⁻²	32.2 × 10⁻⁴
Bridge	26.78	2239.69	3.73 × 10⁻²	4.46 × 10⁻⁴

Figure 14.10 Initial frequency responses of a microbridge resonator in ambient conditions under a white noise signal.

Figure 14.11 Frequency responses of a microbridge resonator in ambient conditions under a white noise signal after 4 h of continuous excitation.

Fig. 14.14 shows the attenuation of velocity and displacement as a function of the oscillating time. The microresonator oscillated continuously for 4 h and the changes in its dynamic response were observed at hourly intervals. After 4 h the excitation of the sample was stopped. The next test was commenced after 30 min, when the increase in the velocity and displacement of oscillations was observed. After 1 h with no actuation of the beam, the thermal effect decreases and the beam response is improved. The velocity of oscillations increases from 8.4 to 19.6 mm/s and displacement from 1.5 to 3.14 nm (Fig. 14.14).

The tests were repeated three times (in different days) and the same attenuation of velocity and amplitude was observed (Fig. 14.15). The average attenuation of velocity and displacement is about 65%. The attenuation in velocity and amplitude of oscillations are based on the thermoelastic coupling and change of the thermal moment as reported by Sun et al. (2006). The same analytical study revealed that the computed thermal moment is attenuated significantly after a longer time and the deflection amplitude (peak value) decreases by about 50% after an operating time range because the effect of thermoelastic damping is enhanced. Also, over time, the prestressed position given by DC current is changed based on the thermal relaxation of the material; it has an influence on the forces balance equation and on the peak amplitude of oscillation described by Eq. (14.13).

Figure 14.12 Frequency responses of a microbridge resonator in ambient conditions. Depicted in a frequency domain from 500 to 1500 kHz: (a) the initial response and (b) the beam response after 4 h.

Figure 14.13 Frequency responses of a microbridge resonator in vacuum. Depicted in a frequency range from 985 to 999 kHz: (a) the initial response and (b) the beam response after 4 h; vacuum at 8 × 10⁻⁶ mbar.

The thermoelastic effect changes the resonant frequency as presented in Figs. 14.12 and 14.13. The air damping effect can increase the frequency response because of the change of the medium compressibility factor. The air escapes from the gap formed between the movable and fixed components, its compressibility generating the spring behavior (Yi, 2008). The compressibility factor changes with temperature. During testing, the heat propagation from a vibrating sample changes the temperature of the surrounding medium, decreasing the compressibility factor of the medium. As a consequence, extrinsic damping decreases and changes the resonant frequency of the beam.

Figure 14.14 Experimental variation of velocity V (mm/s) and displacement D (nm) in vacuum as a function of operating time.

Figure 14.15 Experimental variation of velocity V (mm/s) in vacuum as a function of operating time in three different days.

The total loss coefficient is experimentally determined when the sample oscillates in ambient conditions. The sample response in vacuum determines the intrinsic losses. Table 14.3 shows the quality factors of a microbridge resonator at the beginning of its operating time (Q_0h) and after 4 h (Q_4h). The changes in the quality factor as a function of operating time can be observed.

Table 14.3

Dependence of quality factors on operating time
Quality factor	Testing conditions
Quality factor	Air	Vacuum
Q_0h	26.78	2239.69
Q_4h	19.86	1943.66

Table 14.4

Dependence of loss coefficients on operating time
Loss coefficient	Initial	After 4 h
$Q_{total}^{- 1}$ $Q_{total}^{- 1}$	3.7 × 10⁻²	5 × 10⁻²
$Q_{i}^{- 1}$ $Q_{i}^{- 1}$	4.46 × 10⁻⁴	5.144 × 10⁻⁴
$Q_{e}^{- 1}$ $Q_{e}^{- 1}$	3.6 × 10⁻²	4.9 × 10⁻²

Table 14.4 shows the changes of the loss coefficient of energy as a function of operating time. The experiments were repeated three times (on different days) and the same change (13% increasing) of thermoelastic losses was obtained. The increases in the total loss coefficient

Q_{tot}^{- 1}

$Q_{tot}^{- 1}$

were different for each day (26%, 19%, 22%) because of the changes in environmental conditions (the ambient conditions were not controlled during testing). The environmental conditions have a considerable influence on the extrinsic loss coefficient when the sample is tested in ambient conditions.

The strain energy method is used in ANSYS Multiphysics to compute the loss coefficient. For the microbridge resonator, a loss coefficient of 5.1 × 10⁻⁴ was determined (Pustan et al., 2012), a value that was close to the experimental measurement.

14.4. Industrial applications

14.4.1. Resonant accelerometer

Resonant accelerometers have many applications in automobiles, inertial navigation systems, avionics, and satellites. They detect external acceleration by measuring the frequency variation of a resonating part. The principle of MEMS resonator is based on the resonance of a mechanical part of the system. The basic mechanical structures for resonant vibration are microcantilevers and microbridges, as shown in Fig. 14.3. Each structure has several different resonant modes, where each mode has its own displacement pattern, resonant frequency, and Q-factor. The advantage of using frequency shift as the sensing parameter is its high-level signal and lower sensitivity to parasitic influences. The resonant accelerometer is based on a differential capacitive sensing structure, as previously described in this section. The resonant sensor is an element vibrating at resonance. The shift of frequency is a function of the parameter to be measured, acting on the resonator by changing properties such as the shape of the sensor, inducing stress or adding mass (Kempe, 2011). These are directly related to the frequency of the sensor. The advantage of using frequency as the sensing parameter is its high-level signal and lower sensitivity to parasitic influences.

The mechanical resonator sensors must be set in one of the vibration modes (Fig 14.4 or 14.5) and the vibration detected. One of the main types of excitation technique is electrostatic actuation and the capacitive detection method as presented in this chapter. This is a vibration excitation technique for a resonator oscillated in both in a vacuum and air. For the resonators operated in air, it is critical to design the movable electrode to ensure free movement of the air. Air damping is reduced if the electrodes have holes through which the air is free to move.

The dynamical response and the loss of energy in vibrating MEMS components are influenced by the damping of the surrounding medium and depend on the intrinsic effects of mechanical structure. The air damping changes not only the dynamical response as resonant frequency, amplitude, and velocity of oscillations but also decreases the quality factor and increases the loss coefficient of energy. For cyclic motions of a structural material, significant heat generation becomes apparent and energy dissipation occurs because of an energy loss mechanism within the material. The attenuation in velocity and amplitude of oscillations under continuous actuation are based on the thermoelastic coupling and change of the thermal moment as presented in this chapter.

To obtain a high Q-factor for good performance, the resonator is normally encapsulated in vacuum, which reduce the damping effects given by air. These resonators achieve good performance levels, with high Q-factor on the order of 10⁴ (Kinnell and Craddock, 2009; Greenwood and Wray, 1993; Ikeda et al., 1990). The same difference was also obtained for the investigated microcantilever and microbridge presented in this chapter. As shown in Table 14.2, the Q-factor increases with 2.3 × 10⁴ for a microcantilever oscillating in vacuum versus ambient pressure and with 0.8 × 10⁴ in the case of the microbridge resonant structure in the same operating conditions. But the vacuum packaging of MEMS resonators is a complex process and fairly costly. To reduce the complexity and cost, the resonant accelerometers can be packaged at atmospheric pressure. At atmospheric pressure, a low Q-factor of the resonator can be achieved, but a low Q-factor does not mean low accuracy. An example of MEMS resonators that are successfully packaged at atmospheric pressure and already implemented on market is represented by Analog Devices' iMEMS ADXRS series gyroscopes with a Q-factors of 45 (Geen et al., 2002).

14.4.2. Mass detection sensor

The other application of vibrating MEMS structure is in mass detection sensor (Pustan et al., 2014a,b). Microcantilevers mass detection sensors are used in chemical, biological, and environmental conditions monitoring. These vibrating structures are strongly influenced by the environmental conditions. The building of a mass sensing sensor is based on oscillating cantilevers, where additional mass loading onto the cantilever results in a change of its resonance frequency. The polymer films absorb moisture and can be used for humidity sensing. Humidity sensors are mainly used for climate control in building and process systems. A 10 μm-thick polyimide film used as sensing layer was exposed at 100% relative humidity (RH). The mass change because of the water absorption decreases the fundamental resonant frequency of cantilever with a sensibility of 2.7 Hz/%RH (Boltshauser et al., 1992).

Figure 14.16 Paddle microelectromechanical systems cantilevers used in mass detection applications.

Paddle MEMS cantilevers used in mass detection sensor are presented in Fig. 14.16. The area of the sensing plate is 40 μm × 40 μm and the thickness is 1.9 μm. The plate is supported by a microcantilever with a width of 18 μm, a thickness of 1.9 μm, and two different lengths (145 and 125 μm). The frequency response of these cantilevers and the quality factor are influenced by the operating conditions. For the longer cantilever, a Q-factor of 0.82 is experimentally determined under electrostatic actuation if the sample oscillates in air and it increases to 459.26 in the case of vacuum condition (8 × 10⁻⁴ mbar) (Pustan et al., 2014a,b). The Q-factor for the short cantilever is changed from 1.06 in air to 497 for vacuum. The operating conditions have a small influence on the frequency response of cantilevers. The resonant frequency for the longer cantilever is close to 56 kHz and around 74 kHz for smaller cantilever (Pustan et al., 2014a,b). It can be noticed that the lower Q-factors are due to the air damping and it is in the order of 10⁴ smaller than the Q-factor of the same sample operating in vacuum.

Acknowledgments

This work was supported by a grant of the Romanian National Authority for Scientific Research and Innovation, CNCS-UEFISCDI, project number PN-II-RU-TE-2014-4-1271.

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Table of Contents for 14. Dynamic behavior of smart microelectromechanical systems in industrial applications

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