Abstract
The influence of the Casimir excitation on dynamic pull-in instability of a nanoelectromechanical beam under ramp-input voltage is studied. The ramp-input actuation has applications in frequency sweeping of RF-N/MEMS. The presented model is nonlinear due to the inherent nonlinearity of electrostatics and the Casimir excitations as well as the geometric nonlinearity of midplane stretching. A Galerkin based reduced order modeling is utilized. It is found that the calculated dynamic pull-in ramp input voltage leads to dynamic pull-in step input voltage by increasing the slope of voltage-time diagram. This fact is utilized to verify the results of present study.
1. Introduction
Nano/microelectromechanical systems (N/MEMS) are mostly used as sensors and actuators. Because of their small size, low power consumption, and the reliability of batch fabrications, there are lots of potential applications in engineering. Clamped-clamped microbeams represent major structural components and play crucial roles in these systems. One of the most important phenomena associated with electrostatically actuated N/MEMS is pull-in instability which occurs when input voltage exceeds its critical value. In this manner, the movable part is suddenly collapsed toward the substrate. This phenomenon was observed experimentally by many researchers. Nathanson et al. [1] and Taylor [2] have investigated this phenomenon experimentally. This instability can occur in both static and dynamic circumstances. If the rate of applied voltage is negligible, the static pull-in instability may be observed; otherwise, one can observe DC dynamic pull-in.
At the nanoscale, the intermolecular forces significantly influence dynamics of nanobeams. The Casimir effect is the most important force at the scale of N/MEMS. It represents attractive force between two flat parallel plates of solids that arises from quantum fluctuations in the ground state of the electromagnetic field [3]. The Casimir interaction becomes operative at separations less than several micrometers and above 20 nm [4]. The influence of Casimir force on the pull-in instability of nano- and microsystems has been investigated by many researchers. Lin and Zhao [5] studied the influence of the Casimir force on static pull-in behaviour of nanoelectromechanical systems using lumped model. Ramezani et al. [6] proposed a distributed parameter model to study the static pull-in instability of nanocantilevers subjected to intermolecular and electrostatic forces. They transferred nonlinear differential equation of the model into the integral form by using Green’s function of the cantilever beam. Koochi et al. [7] investigated the effect of the Casimir attraction on the nonlinear pull-in behaviour of cantilever and double cantilever NEMS using modified Adomian decomposition method (MAD). They neglected the effect of inertia and the von Kármán nonlinearity; in other words, they have investigated the static pull-in case. Dynamic pull-in instability of electrically actuated microbeams in presence of the Casimir force has been investigated by Moghimi Zand and Ahmadian [8]. They consider the von Kármán nonlinearity of midplane stretching, applied axial loading, fringing field effect, and the Casimir attraction and solved the governing equation using nonlinear finite element method (NFEM).
Most of the studies on microstructures have been performed using step-input and harmonic actuations. However, other actuation shapes also have application in microstructures. Ramp-input actuation has applications in frequency sweeping and contact time study of RF-MEMS. Contact time is defined as the time taken by a microstructure to move from the initial position to the position where the deformable part contacts the substrate. It is noted that the interaction of Casimir force and ramp voltage excitation has not been investigated to date.
In present study, the governing equation of motion of nano/microbeams under the combined effect of electrostatic excitation due to ramp-input voltage and the Casimir force has been derived. This model is nonlinear due to the inherent nonlinearity of electrostatic excitation, Casimir attraction, and the geometric nonlinearity of the von Kármán midplane stretching. A numerical analytical method based on Galerkin reduced order modelling has been used to convert the partial differential equation of motion to a set of ordinary differential equations in order to investigate the nonlinear response of double clamped nano/microbeams. The results are in good agreement with those presented in the literature for dynamic pull-in case due to the step input voltage.
2. Modelling and Formulation
Consider a fully clamped nano/microbeam of length , width , thickness , and density under the combined action of the electrostatic excitation due to ramp-input voltage and the Casimir force (Figure 1). The distance between the beam and the stationary electrode is . Also, , , and are, respectively, the coordinate along the length, width, and thickness. is deflection, is time, is the moment of inertia of the cross-section about the axis,is Poisson’s ratio, and is the effective Young modulus of the nano/microbeam which is replaced bywhen .
The electrostatic excitation by ramp-input DC voltage per unit length of the beam can be expressed as where where is the maximum value of input voltage andis the duration in which this voltage is applied (see Figure 2) and is the unit step function.
The Casimir force per unit length of the beam takes the following form [9]: where is dielectric constant of medium, is Planck’s constant divided by , and is the speed of light in vacuum.
Due to the elongation of fixed-fixed nano/microbeam which is called the midplane stretching effect and the mismatch of both thermal expansion coefficient and crystal lattice period between substrate and microbeam film which is unavoidable in surface micromachining techniques, the resultant axial force is applied to the nano/microbeam [10]: The axial force due to the midplane stretching effect takes the following form [11]: and the one due to the residual stress can be defined as So the equation of motion of the clamped-clamped nano/microbeam subjected to the combined effect of ramp-input voltage and the Casimir force is as follows: where dot and prime signs denote derivatives with respect to and , respectively, is equivalent viscose damping coefficient per unit length of the beam due to squeeze film damping [12]. The nano/microbeam deflection is subjected to the following kinematic boundary conditions: The initial conditions are assumed as follows: For convenience, the following dimensionless variables are introduced: where Upon substitution of the dimensionless quantities given in (10) into (7) and dropping the hats, one would get where where for rectangular cross-section one can obtain
3. Solution Procedure
Herein the Galerkin based reduced order modelling is used in order to solve the nonlinear partial differential equation (12) [13]. To this aim, (12) is discretized into a finite degree of freedom system consisting of ordinary differential equations in time. The undamped modeshape of straight clamped-clamped nano/microbeam is used as a basis function in Galerkin procedure. To this end, the deflection is expressed as where is the th generalized coordinate and is the th linear undamped modeshape of the straight clamped-clamped nano/microbeam, normalized such that and expressed as [14] where is defined as and is the th nondimensional natural frequency of the nano/microbeam and governed by Next we multiply (12) by , substitute (15) into the resulting equation, integrate the outcome from to 1, use integration by parts, and obtain where the functionis given by
Equation (19) represents an implicit system of nonlinear second order ordinary differential equations. Using an implicit scheme such as Adams-Moulton implicit methods [15] or transforming them into an explicit system in by multiplying (19) with the inverse of the coefficients of and using fourth order Runge-Kutta method may lead to the solution of (19). The latter approach was used in the present calculations.
4. Results and Discussion
Assuming that the only dominant mode in the response of the nano/microbeam is its first mode (i.e., ), so using the single mode approximation may lead to accurate results. In order to validate the model, the results are compared with those presented in the literature for DC dynamic pull-in case in which the Casimir effect has been taken into account [8]. For this aim we set , , , and , then plot the nondimensional max voltage parameterversus nondimensional Casimir force parameter, and compare the outcome with the results of Moghimi Zand and Ahmadian [8] (see Figure 3).
Consider a polysilicon microbeam with length , thickness , width and gap width . The material properties of this microbeam are , which is replaced by because , and . The DC dynamic pull-in step voltage for this case without considering damping coefficient and Casimir effect is 3.11 V. The response of this microbeam under ramp-input voltage with and without considering the effect of Casimir force before and after bechancing pull-in instability is plotted in Figure 4. From this figure, one can observe the major effect of Casimir force on nano/microstructures.
(a)
(b)
Figure 4 illustrated that neglecting the effect of Casimir force on nano/microstructures may lead to very inaccurate results, so it is very important to consider this effect when the nondimensional Casimir parameter takes noticeable values.
5. Conclusion
In the present paper, reduced order modelling based on the Galerkin procedure was utilized to study the effect of Casimir attraction on the response of a fully clamped nano/microbeam under ramp-input voltage. The model accounted for geometric nonlinearity of von Kármán midplane stretching, applied axial loading, equivalent viscose damping, and inherent nonlinearity of distributed electrostatic and Casimir forces. It was found that considering the Casimir force may lead to early instability in nano/microelectromechanical devices through dynamic pull-in.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.