Stability analysis of a capacitive micro-resonator with embedded pre-strained SMA wires

Abstract

This paper presents a study on the effects of the SMA wires’ characteristics on tuning the stability of a capacitive micro-resonator. In the proposed model, pre-strained SMA wires have been embedded in a double clamped resonant microbeam which is actuated electrostatically. The governing equations of the system have been introduced and then an eigen-value problem has been adopted to investigate stability. Galerkin-based numerical methods have been used to solve the governing equation of motion for obtaining the motion trajectories of the beam. The effects of the number of SMA wires, their diameter, pre-strain and temperature on the pull-in instability and frequency response of the resonator have been shown. Critical values of recovery stress and SMA temperature for avoiding instability, with and without applying DC voltage have been obtained. The results have shown that by changing each of the SMA parameters, one can reach a needed magnitude of recovery stress as well as transmitted longitudinal force to the host beam, and consequently control and tune the stability and resonance frequency of the micro-resonator.

Introduction

MEMS (Microelectromechanical systems) is a technology that in its most general form can be defined as miniaturized electro-mechanical elements (i.e., devices and structures) that are usually made of silicon material using the micro-fabrication techniques. MEMS technology, due to its merits such as small weight and dimensions, low cost and energy consumption, high reliability and etc. is widely used in a broad variety of industries, including the automotive, communication, medical and many other industries. Electrostatically-actuated (capacitive) devices such as capacitive micro-resonators form a broad class of MEMS due to their simplicity, as they require few mechanical components, small actuation voltage and excellent compatibility with the integrated circuits (ICs) (Senturia 2001; Azizi et al. 2013). Microbeams under voltage driving are widely used in many MEMS devices such as capacitive resonant micro-sensors. These devices are fabricated, to some extent, in a more mature stage than some other MEMS devices. Fixed–fixed microbeams due to their high natural frequencies are widely used in resonant sensors and actuators.

From elasticity point of view classical theories of elasticity are not able to present a proper mechanical model for them in most cases (including granular materials) and non-classic theories should be used (Fathalilou et al. 2014a). This is because of the strong size dependence in deformation behavior of them. Size dependent behavior is mainly due to the dominance of atomic structures of the material neglected in classic elasticity. Voigt was the first who tried to interpret the size dependent behavior by taking into account of the assumption that interaction between the two parts through an area element inside the body is transmitted not only by a force vector but also by a moment vector giving rise to a ‘couple stress theory’ (Voigt 1887). The complete theory of asymmetric elasticity was developed by Cosserat and Cosserat (1909). After a gap of about fifty years, Cosserats theory drew attention of researchers and several Cosserat-type theories were developed independently, e.g., Mindlin and Tiersten (1962), Toupin (1962) and Eringen (1966) among several others. In recent years many researchers paid attention to these non-classical theories e.g., Voyiadjis and Abu Al-Rub (2005), Lam et al. (2003) and Fathalilou et al. (2014a, b, c) among several others.

In spite of the importance of the size dependent behavior in most of the micro structures, for some materials such as silicon this behavior has not remarkable effects in micro-scale, because the length scale parameter of the silicon is in order of nanometer (Nix and Gao 1998; Fathalilou et al. 2014a) and classical theory of elasticity can be used to model the mechanical behavior of the structure.

In capacitive structures one of the most important issues that has always been of interest to researchers, is the pull-in instability. The static pull-in instability is a discontinuity related to the interplay of the elastic and electrostatic forces (Azizi et al. 2011). When a microbeam is imposed to a static DC voltage, the beam acts as a plate of a parallel capacitor, and the electrostatic force between the microbeam and the stationary substrate causes the microbeam to bend. This force is nonlinear and increases nonlinearly by increasing the voltage and reducing the distance between the electrodes. On the other hand, elastic restoring force resists bending. But this force is linear, therefore, with a further reduction in distance, the restoring force cannot balance with the electrostatic force and hence the beam falls to the lower plate and the distance between the electrodes becomes zero. The critical voltage is known as the static pull-in voltage. It is worth mentioning that the pull-in can be regarded as a static or divergence instability in which, a transition from stability (in which all eigenvalues of the system are purely imaginary) to this type of instability necessarily occurs by vanishing the imaginary parts as well as acquiring real parts in eigen values of the system. More details about this instability will be given in Sect. 2.2.

On the other hand, application of the smart materials in engineering structures have drawn serious attention recently. Smart materials are referred to materials that can react appropriately by understanding their environment and surrounding conditions (Azizi et al. 2014; Ghazavi et al. 2010; Abbasnejad and Rezazadeh 2012). Shape memory alloys (SMA) are one of the interesting smart materials which can illustrate both shape memory effect (SME) and superelasticity (SE) (Jani et al. 2014; Zainal et al. 2015). The reason for this is the unique feature of this alloy in retrieving its original shape under certain temperature conditions. At the temperatures below the phase change temperature (the martensitic phase) this alloy has a very low yield strength that easily deforms. By heating the alloy to the phase change temperature, its crystalline structure changes from the martensitic phase to the austenite phase (stable phase). This phase change is the reason for the return of the alloy in its original form. Other properties of this alloy include: high recovery force, material property changeability, extensive temperature range and extraordinary damping properties (Asadi et al. 2015). From all alloys with a memory-based behavior, Ti–Ni based alloys have been used frequently. Because of its unique properties, SMA can be a good candidate for control of smart structures (Asadi et al. 2015; Sohn et al. 2009). To this end, generally, pre-strained SMA wires as actuators are embedded in a structure. By passing the electric current through them, heat generation occurs due to the electrical resistance of wires. So, a large recovery force is generated and transmitted to the host structure by returning the SMA wires to their original shape. In addition to the transmitted recovery force, variations in the elasticity modulus and density of the SMA wires by phase transformation, can change the mechanical properties of the host structure.

Since the early 1980s, the use of memory alloys has been considered among researchers and engineers, and this intelligent alloy has been used in a wide range of areas, including modulating the satellite’s aeroelastic behavior, controlling vibration of space structures, controlling vibration levels of airplane, and etc (Lee et al. 2013). Birman (1997) compared the effect of composite and SMA stiffeners on the stability of composite cylindrical shells and rectangular plates which are subjected to compressive loads. He has shown that composite stiffeners are more efficient in cylindrical shells, whereas SMA stiffeners are preferable in plates or long shallow shells. Lau (2002) investigated the vibration characteristic of SMA composite beam considering different boundary conditions using Finite Element Method. Park et al. (2004) investigated the effect of SMA on vibrational behavior of thermally buckled composite plate. The results depicted that SMA fibers have significant influence on increasing the critical buckling temperature. Zhang et al. (2006) explored the vibrational characteristics of a laminated composite plate containing unidirectional fine SMA wires and laminated composite plates with embedded woven SMA layer experimentally and theoretically. They showed the influence of both SMA arrangement and temperature on the vibrational characteristics. Kuo et al. (2009) used the Finite Element Method to study the buckling of laminated composite plate embedded with SMA fibers. According to their findings, the concentration of these fibers on the middle of plate improves buckling load. Dos Reis et al. (2010) studied vibration attenuation in an epoxy smart composite beam with embedded NiTi shape memory wires. Asadi et al. (2013, 2014a, b) proposed an analytical solution for free vibration and thermal stability of SMA hybrid composite beam. They found that increase of temperature can postpone the critical thermal buckling temperature of plate. Malekzadeh et al. (2014) studied the effect of some geometrical and physical parameters on the response of free vibration of rectangular laminated composite embedded with SMA fibers.

In spite of many valuable researches about the application of SMA in a wide variety of structures, there is not enough investigation on the effect of these alloys on the pull-in instability and other key specifications of electrostatically actuated micro-resonators. Tunability of the resonance frequency of the electrostatic resonators by SMA wires as well as applied DC voltage can be more interesting and practical. This work considers the SMA wires embedded in a capacitive resonant microbeam and analyzes their effects on the stability and resonance frequency of the micro-resonator as well as critical conditions to avoid the instability. The results show that beside the DC voltage, changing the SMA parameters can tune the dynamic characteristics of a capacitive micro-resonator.

Mathematical modeling and stability analysis

Model description

Figure 1 shows an electrostatically-actuated resonant microbeam with embedded pre-strained SMA wires that are aligned along the beam’s longitudinal axis. Geometrical notations of the system including the geometry of the movable electrode (top beam), initial gap between top and bottom (stationary) electrodes and embedded SMA wires are shown in the figure. When a voltage is applied between the top and bottom electrodes, the upper deformable beam is pulled down due to the electrostatic pressure.

Fig. 1
figure1

an electrostatically-actuated microbeam with embedded SMA wires

The governing equation for the transverse displacement of the beam under applied voltage, V, can be obtained using both Hamilton and Newton’s methods as (Rezazadeh et al. 2011):

$$EI\frac{{\partial^{4} w}}{{\partial x^{4} }} + \rho A\frac{{\partial^{2} w}}{{\partial t^{2} }} + c\frac{\partial w}{\partial t} - F\frac{{\partial^{2} w}}{{\partial x^{2} }} = \frac{{e_{0} bV^{2} }}{{2\left( {g_{0} - w} \right)^{2} }}$$
(1)

with the following boundary conditions:

$$w(0,t) = \frac{\partial w}{\partial x} + (0,t) = w(L,t) = \frac{\partial w}{\partial x} + (L,t) = 0$$
(2)

where E and \(\rho\) are equivalent elasticity module and density, respectively, which can be evaluated by mixture rule as (Lau 2002):

$$E = E_{B} + (D - E_{B} )\frac{{I_{SMA} }}{I},\quad \rho = \rho_{B} + (\rho_{SMA} - \rho_{B} )\frac{{A_{SMA} }}{A}$$
(3)

B and SMA indices represent the host beam and embedded SMA, respectively. Also \(A_{SMA} = N\pi d^{2} /4\) is the cross section of the wires in which N and d are number of the embedded wires and their diameters, respectively and D is the elasticity module of the SMA and is written as (Ostachowicz et al. 1998):

$$D = \xi D_{m} + (1 - \xi )D_{a}$$
(4)

where \(D_{m}\) and \(D_{a}\) are elasticity module of the SMA in martensite and austenite phases, respectively. \(\zeta\) is one of the most important properties of SMA, martensite fraction, that is the ratio of the amount of alloy in the martensitic phase to the total alloy, which can vary from zero to one. It is defined as following for transformation from martensite to austenite phase (Brinson et al. 1997):

$$\xi = exp\left[ {b_{a} \left( {A_{s} - T} \right) + c_{a} \sigma \left( {\xi ,T} \right)} \right]$$
(5)

An axial force, F, exerts on the beam which comes from recoverable pre-strain of emerged SMA wires. As mentioned before, due to the pre-strain in the wires, with the change in temperature the recoverable stress occurs in the wires, which exerts a large amounted force to the axis of the beam (Lau 2002). Magnitude of this stress can be theoretically obtained using the constitutive models introduced in literature. According to the Brinson model, the one-dimensional constitutive equation for SMA wires is defined as following (Brinson 1993):

$$\sigma - \sigma_{0} = D(\varepsilon - \varepsilon_{0} ) + \varTheta (T - T_{0} ) + \varOmega (\xi - \xi_{0} )$$
(6)

where \(\varTheta\) denotes the thermal coefficient of expansion and \(\varOmega\) is transformation tensor which are defined as (Ostachowicz et al. 1998):

$$\begin{aligned} \varTheta & = \xi \varTheta_{m} + \left( {1 - \xi } \right)\varTheta_{a} \\ \varOmega & = - \varepsilon_{L} D \\ \end{aligned}$$
(7)

in which \(\varepsilon_{L}\) is considered as the strain recoverable limit (Lau 2002).

In Fig. 2 the experimental curves for the Nitinol fibers recovery stresses versus temperature for four different initial strains are presented (Ostachowicz et al. 1999).

Fig. 2
figure2

Recovery stress in SMA as a function of temperature and initial strains (\(\varepsilon_{0}\)) (Ostachowicz et al. 1999)

It should be noted that the force, which are generated by thermal expansion of the host structure, was normally ignored in previous literature reports (Birman 1997; Brinson 1993). Indeed, this can be due to two reasons; first the smaller value of the thermal expansion coefficient of the silicon as a host structure compared to the used SMA material. Second is the fact that the small compressive thermal force generated in the composite beam is much lower than the recovery force of the SMA phase transformation and can be neglected in modelling.

Now, by introducing the axial force of Eq. 1, F, we can analyze this equation. For convenience in analysis, the governing equation should be written in dimensionless form with the nondimensional parameters as following:

$$\bar{w} = \frac{w}{{g_{0} }},\quad \bar{x} = \frac{x}{L}\;{\text{and}}\;\bar{t} = \frac{t}{\tau }\;{\text{with}}\;\tau = \sqrt {\frac{{\rho bhL^{4} }}{EI}}$$
(8)

Substituting these parameters into Eq. 1, the following dimensionless equation is obtained:

$$\frac{{\partial^{4} \bar{w}}}{{\partial \bar{x}^{4} }} + \frac{{\partial^{2} \bar{w}}}{{\partial \bar{t}^{2} }} + \bar{c}\frac{{\partial \bar{w}}}{{\partial \bar{t}}} - \bar{F}\frac{{\partial^{2} \bar{w}}}{{\partial \bar{x}^{2} }} = \frac{{\alpha V^{2} }}{{(1 - \bar{w})^{2} }}$$
(9)

where

$$\bar{c} = \frac{{L^{4} }}{EIT}c,\quad \bar{F} = \frac{{L^{2} }}{EI}F\;{\text{and}}\;\alpha = \frac{{e_{0} bL^{4} }}{{2EIg_{0}^{3} }}$$
(10)

It is worth to mention that by eliminating the time dependent terms from Eq. 9, one can reach the governing static equation as follow:

$$\frac{{\partial^{4} \bar{w}}}{{\partial \bar{x}^{4} }} - \bar{F}\frac{{\partial^{2} \bar{w}}}{{\partial \bar{x}^{2} }} = \frac{{\alpha V^{2} }}{{(1 - \bar{w})^{2} }}$$
(11)

Stability analysis

To analyze the stability of the micro-resonator with embedded SMA wires, we should consider the dynamic behavior of the system because stability is essentially a dynamical concept.

First, we apply a DC voltage on the system and investigate the motion that occurs after some initial perturbations about the equilibrium state of the applied voltage, and from the properties of the motion we can infer or deny local stability of the system. The practicality of this approach depends crucially on the linearization of the equations of motion of the perturbation. By linearizing we can express the perturbation motion as the superposition of complex exponential elementary solutions. The linearized equation of motion of the beam about its equilibrium position (\(\bar{w}_{s}\)) under applied DC voltage can be expressed as:

$$\frac{{\partial^{4} \bar{w}}}{{\partial \bar{x}^{4} }} + \frac{{\partial^{2} \bar{w}}}{{\partial \bar{t}^{2} }} - \bar{F}\frac{{\partial^{2} \bar{w}}}{{\partial \bar{x}^{2} }} - \frac{{2\alpha V^{2} }}{{(1 - \bar{w}_{s} )^{3} }}\bar{w} = 0$$
(12)

By applying the Galerkin method, the reduced order equations of motion can be written as (Rezazadeh et al. 2009):

$$\left[ \varvec{M} \right]\varvec{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\ddot{q}} } + \left[ \varvec{K} \right]\varvec{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{q} } = {\mathbf{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{0} }}$$
(13)

where M is a positive definite mass matrix and K is a symmetric stiffness matrix in which their elements are defined as:

$$\text{M}_{ij} = \left\langle {{\varphi }_{i} ,{\varphi }_{j} } \right\rangle ,\quad {\rm K}_{ij} = \left\langle {{\varphi }_{i} ,{\varphi }_{j}^{iv} } \right\rangle - \left\langle {{\varphi }_{i} ,\bar{F}{\varphi^{\prime\prime}}_{j} } \right\rangle - \left\langle {{\varphi }_{i} ,\frac{{2\alpha V^{2} }}{{(1 - \bar{w}_{s} )^{3} }}{\varphi }_{j} } \right\rangle$$
(14)

where \(\left\langle {f,g} \right\rangle = \int\limits_{0}^{1} {f(\hat{x}).g(\hat{x})d\hat{x}}\) represents the inner product of two function over nondimensional length domain. In this paper, \(\varphi_{j} \left( {\hat{x}} \right)\) is selected as jth undamped linear mode shape of the straight microbeam which is normalized as \(\left\| {\varphi (\hat{x})} \right\| = 1\).The following eigen-modal can be assumed to treat the obtained linear set of equations:

$$\varvec{q}(t) = \sum\limits_{i = 1}^{n} {\varvec{q}_{i} (t)} = \sum\limits_{i = 1}^{n} {A_{i} \varvec{u}_{i} e^{{s_{i} t}} }$$
(15)

in which index i ranges over the number of degrees of freedom. The si are generally complex numbers called the characteristic exponents whereas the corresponding column vectors ui are the characteristic vectors. It is worth mentioning that only the eigenvalues si are of interest in stability assessment and the initial conditions that are used to obtain the unknown Ai, are irrelevant in that regard. The characteristic exponents of these solutions can be determined through a characteristic value problem or eigen problem. The set of characteristic exponents gives complete information on the linearized local stability of the system at the given equilibrium configuration. In practical studies the characteristic exponents are functions of the control parameter λ. In presented model, the applied DC voltage or one of the SMA specifications such as wires’ diameter, number, pre-strain and temperature can be considered as the control parameter.

Substituting Eq. 15 into 13, the following eigen-value problem is obtained that governs linearized local stability.

$$\left( {\varvec{K} + s_{i}^{2} \varvec{M}} \right)\varvec{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{u} }_{i} = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{0}$$
(16)

The square roots of the eigenvalues of (16) yield the characteristic exponents si of the eigen modal expansion.

If all roots s 2 i of (16) are negative real, their square roots are purely imaginary numbers (\(s_{i} = \pm j\omega_{i}\), the nonnegative real numbers \(\omega_{i}\) are the natural frequencies of free vibration) and the system is stable. If so the system will simply vibrate, that is, perform harmonic oscillations about the equilibrium position.

If s 2 i is positive real, then one of its square roots will give rise to an aperiodic, exponentially growing motion. The other root will give rise to an exponentially decaying motion, while the combination of them leads to an exponentially growing motion. As mentioned before the eigen-values can be a function of a control parameter. The transition from stability (in which all roots are negative real) to this type of instability necessarily occurs when a finite eigenvalue s 2 i (λ), moving from left to right region of s2 complex plane as λ varies, passes through the origin s2 = 0. This type of instability is called divergence or static instability. The pull-in instability in capacitive microbeams can be regarded as this type of instability (Rezazadeh et al. 2009).

It is worth mentioning that, to investigate the global stability of such a system, one should consider the phase portrait of the system (Fathalilou et al. 2014a).

Results and discussion

Model validation

Based on the authors’ knowledge, there is not theoretical or experimental results for the pull-in (divergence instability) voltage of electrostatic micro-resonators with embedded SMA wires. So, the presented procedure is validated by Rezazadeh et al. (2011), in which their used residual stress equates to present recovery stress. To this end a silicon microbeam with the specifications of Rezazadeh et al. (2011) is considered and the pull-in voltage of it, is calculated for the recovery stress of the SMA wires, \(\sigma_{rec} = - 238.7\,{\text{MPa}}\) which is equivalent to the compressive residual stress of the beam, \(\sigma_{res} = - \,25\,{\text{MPa}}\). The results of comparison are shown in Table 1.

Table 1 The comparison between the calculated pull-in voltages with previous works for equivalent compressive stresses

As shown, there is a good agreement among the results. It is worth to mention that the reason for the difference of 1.84 V in the results of stressed beams is due to the changes in equivalent E and I of SMA embedded beam compared with purely silicon beam.

Effects of the SMA wires’ specifications on pull-in instability

We begin our results by illustrating the effect of SMA wires’ properties on the pull-in instability of the system. To this end, a capacitive resonant microbeam with the specifications of Table 2 is considered. Also, physical properties of the SMA wires are given in Table 3.

Table 2 Physical and geometrical specifications of the resonant microbeam
Table 3 Specifications of SMA wires (Ostachowicz et al. 1998)

The pre-strain, defined in the wires is tensile. Therefore, the axial force created in the beam by the SMA wires is a compressive force. To show physical meaning of the pull-in instability as a saddle node, bifurcation diagram of the system without embedded SMA wires versus applied DC voltage is plotted in Fig. 3a. This figure is plotted by solving Eq. 11 using direct Galerkin method (Rezazadeh et al. 2011).

Fig. 3
figure3

Pull-in or divergence instability in the capacitive microbeam, a bifurcation diagram, b phase portrait, and Root locus plot on the complex s2 plane for c before pull-in, d pull-in

In the state-control space, the stable and unstable branches of the fixed points, with increasing applied voltage, meet together at a saddle-node bifurcation point. It is a local stationary bifurcation which can be analyzed based upon locally defined eigen-values. The voltage corresponding to the saddle-node bifurcation point is a critical value (\(V = 31\,{\text{V}}\)), which is known as the static pull-in voltage in the MEMS Literature. To show the stability of fixed points, the phase portrait of the system for a given applied voltage with various initial conditions is plotted in Fig. 3b. By paying attention to this figure, it can be found that for a given applied voltage the first equilibrium position is a stable center and the second is an unstable saddle-node. As shown, there are two basins of attraction and repulsion of the stable center and unstable saddle node, respectively. Moreover, it can be seen that we have a bigger basin of attraction for the singular point of the system (position of the substrate), it means, any motion started out of the basin of attraction of the center point, placed in the basin of attraction of the singular point. Depending upon the location of the initial condition, the system globally may be stable or unstable.

On the other hand, in this point we have a divergence instability because all roots of s2 transitioned from left region of s2 complex plane to the origin s2 = 0. This problem is well shown in root locus plots on s2 complex planes of Fig. 3c, d.

Now, it is considered that the resonant microbeam with embedded SMA wires is exposed to an electrostatic actuation as depicted at Fig. 1. To show the effect of the SMA wires’ specifications on the pull-in instability of the electrostatic resonant microbeam, we apply the presented eigen-value problem with emphasis on first vibrational mode. Figure 4 illustrates variations of both positive and negative imaginary part of the system’s eigen-values versus applied DC voltage as a control parameter, for different numbers of the embedded wires. The circles on the plot show position of the pull-in or divergence instability. As shown, increasing the wires’ number decreases the pull-in voltage as well as fundamental frequency of the system. Because the wires are exposed to a pre-strain prior to temperature increase up to 70 °C, increasing the wires’ number increases the resultant force (due to the shape recovery effect of SMA) which is applied on the beam.

Fig. 4
figure4

The effect of the number of wires on the pull-in instability as well as fundamental frequency at T = 70 °C, diameter of the wire d = 1.5 μm and the pre-strain ε0 = 4%

It is expected that with the constant number of SMA wires, the change in the diameter of the wires will also cause variations in the pull-in voltage of the beam. This problem is verified in Fig. 5a. It is worth to mention that beside the recovery stress of the pre-strained SMA wires, the change in effective physical properties of the composite also affects these variations. In this figure, T = 70 °C, N = 20, and ε0 = 4%, are considered. As shown, the pull-in voltage can be reduced about 50% by increasing the diameter of SMA wires from 0.1 to 1.5 µm. Figure 5b illustrates the effect of the pre-strain of SMA wires on the pull-in voltage of the system. As shown, increasing the pre-strain decreases the pull-in voltage. This is due to the increasing the recovery stress of the SMA in higher pre-strains. In this figure diameter of the wire is considered to be d = 1.5 μm. As given in the SMA constitutive relations, the temperature of the SMA wires can affect the martensitic fraction and results in the change in produced recovery stress as well as the physical properties of the composite. This is verified in Fig. 5c, where the effect of the temperature of SMA on the variations of the eigen-values of the system versus applied DC voltage is shown for the pre-strain ε0 = 4%. As given in this figure, increasing the SMA temperature, also decreases the pull-in voltage of the system. This decrease is considerably greater for the temperatures between 40 °C and 70 °C. Because from 40 °C to near 70 °C, the phase transformation occurs from martensite to austenite, and hence strongly increases the recovery stress. But, above 70 °C, the whole SMA is transformed to its stable phase, austenite, and hence the rate of increasing the recovery stress with temperature, decreases considerably. Moreover, in phase transformation interval, due to the change in embedded SMA properties, the equivalent physical properties of the system such as E and ρ are changed considerably.

Fig. 5
figure5

variations of the eigen values of the system versus applied DC voltage for different values of a SMA diameter, b SMA pre-strain and c SMA temperature

Figures 4 and 5 confirm that, emerging the pre-strained SMA into an electrostatic resonant microbeam, can change the pull-in voltage of the system and one can control the pull-in instability by active changing the SMA dependent parameters such as number and diameter of wires, pre-strain of SMA as well as its temperature. On the other hand, from dynamical point of view, applying the pre-strained SMA and applied DC voltage simultaneously, can be more effective to control the fundamental frequency of the micro-resonator.

In following, the effect of SMA properties on the vibrations of the micro-resonator under step DC actuation will be shown. To this end we solve Eq. 9 using Galerkin-based reduced order model (Fathalilou et al. 2014a, b, c; Rezazadeh et al. 2011; Abbasnejad and Rezazadeh 2012). Figure 6a, b illustrate the effects of the pre-strain on vibrational response and phase portrait of the microbeam for V = 13.91 V which is about 90% of the static pull-in voltage of the system with 4% pre-strain for SMA wires. As it is shown, increasing the pre-strain from 1 to 3% decreases the oscillations frequency while increases its amplitude. But, for 4% pre-strain, it is seen that the beam collapses on the stationary electrode. In MEMS literature, this phenomenon is introduced as dynamic pull-in and the critical magnitude of the step voltage corresponding to the instability, is referred to a dynamic pull-in voltage (Rezazadeh et al. 2011). In problems of this type, usually, the loss of stability is characterized by the development of a divergent motion when the structure collapses on the actuating electrode and the maximally achievable displacement is bounded by the actuating electrode itself or by a mechanical constraint provided intentionally in order to prevent an electrical short or break down (Krylov 2007). With attention to Fig. 6b, indeed the periodic orbit in the phase portrait is ended at dynamic pull-in voltage where a homoclinic orbit is formed. In another words when applied voltage approaches dynamic pull-in voltage, the periods of the closed orbits tend to infinity. It can be said that there happens a homoclinic bifurcation (Kuznetsov 1997; Lin and Zhao 2003; Guo and Zhao 2006. It is worth mentioning that the position of the stationary electrode in this structure can be regarded as a singular point which we have introduced it as a strong motion attractor (Fathalilou et al. 2014a, b), because any motion which is started from out of the finite basin of attraction of the center point, approaches this point with infinite velocity. Anyway, a conclusion about the stability of the system with these conditions is made based on the examination of the response. It is clear that applying initial conditions influence the dynamic pull-in instability. This figure is plotted for zero initial conditions.

Fig. 6
figure6

Effect of the pre-strain on dynamic pull-in at T = 70 °C, N = 20 and d = 1.5 μm. a Time history, b phase portrait

This figure can be generalized by considering other SMA parameters such as N, d and T, where it can be concluded that changing each of these parameters in order to the stiffness decreasing can result in dynamic pull-in in the system.

Effects of the SMA parameters on the frequency response of the micro-resonator

As it is known, frequency response curves have an important role in the analysis of micro-resonators. In Fig. 7, effects of the temperature of SMA wires on the frequency response of the resonator is shown. It is assumed that the microbeam is excited by a small amplitude AC voltage with nondimensional excitation frequency of \(\bar{\varOmega }\). As it is shown, increasing the SMA temperature decreases the resonance frequency while increases the vibration amplitude. It should be mentioned that in this figure, N = 20, d = 1.5 µm and ε0 = 4%. Similar to Fig. 7, we can plot the frequency response curves considering the other SMA specifications (N, d and \(\varepsilon_{0}\)) as the varying parameters as well. A comparison of the results confirms the effectiveness of dynamic response of the resonator by changing each of these parameters. Therefore, SMA parameters can be accounted as a useful tool for tuning the frequency response of a micro-resonator. For example, increasing the wires’ number or diameter or pre-strain, decreases total stiffness of the system and shifts the curve to left.

Fig. 7
figure7

Variations of frequency response curves versus SMA temperature

Critical recovery stress

As it is described in previous sections, the resonant beam is under compressive axial force due to the pre-strain of SMA wires and hence, can experience instability even in the absence of voltage appliance. In the following, the critical recovery stress of each wire for avoiding the pull-in instability is obtained with and without presence of DC voltage.

In Figs. 8 and 9 variations of the recovery stress of the SMA versus wires’ number and diameter are shown without applying DC voltage. As the figures show, increasing the number of wires or their diameter, decreases the critical value of the recovery stress which means that by increasing the wires’ number and diameter, smaller force in each wire is needed for the buckling of the beam.

Fig. 8
figure8

Variations of the critical recovery stress versus the SMA wire number at T = 70 °C, diameter of the wire d = 1.5 μm and the pre-strain ε0 = 4%

Fig. 9
figure9

Variations of the critical recovery stress versus the SMA wire diameter at T = 70 °C, N = 20

It is expected that applying the DC voltage reduces the critical recovery stress. In Fig. 10 we show the variation of the critical recovery stress versus applied DC voltage. As shown, by applying the voltage, smaller values of the recovery stress are needed for buckling of the system. This is due to the simultaneous effect of the SMA wires and DC voltage on reducing the stiffness of the system. As shown, the higher the applied voltage, the higher the recovery stress decreasing rate.

Fig. 10
figure10

Variations of the critical recovery stress versus the applied DC voltage at T = 70 °C, d = 1.5 μm and the pre-strain ε0 = 4%

As obtained from the constitutive equations, the recovery stress is produced due to the SMA temperature change. Variation in SMA temperature, changes the physical properties of the system during the phase transformation from martensite to austenite. Figure 11 shows the variations of the critical SMA temperature versus applied DC voltage. This figure illustrates the maximum allowable temperature of the SMA wires in presence of the electrostatic actuation for avoiding the buckling of the system.

Fig. 11
figure11

variations of the critical SMA temperature versus the applied DC voltage for d = 1.5 μm and the pre-strain ε0 = 4%

Conclusion

Considering the presented results and discussions, it was concluded that the use of shape memory alloys as a pull-in controller in MEMS is a good and practical option. The use of SMA wires in an electrostatic micro-resonator, can change the static and dynamic pull-in voltages of the system. It will also be possible to control the resonance frequency of the system by using SMA wires in the resonant microbeams. By changing the number, diameter and temperature of SMA wires arranged in the beam, the amplitude of the vibration can be increased or decreased, depending on the pre-strain defined on the SMA wires. Finally, the results showed that in the proposed model, a critical value for the recovery stress can be introduced in terms of the SMA wires’ number, diameter and temperature, in which the microbeam undergoes instability without applying any DC voltage, meanwhile applying DC voltage will change the magnitude of critical recovery stress as well as critical SMA temperature.

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Correspondence to Moahmmad Fathalilou.

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Fathalilou, M., Rezazadeh, G. & Mohammadian, A. Stability analysis of a capacitive micro-resonator with embedded pre-strained SMA wires. Int J Mech Mater Des 15, 681–693 (2019). https://doi.org/10.1007/s10999-018-09437-7

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Keywords

  • MEMS
  • SMA
  • Electrostatic
  • Resonator
  • Stability