Closed-form approximation and numerical validation of the influence of van der Waals force on electrostatic cantilevers at nano-scale separations

Abstract

In this paper the two-point boundary value problem (BVP) of the cantilever deflection at nano-scale separations subjected to van der Waals and electrostatic forces is investigated using analytical and numerical methods to obtain the instability point of the beam. In the analytical treatment of the BVP, the nonlinear differential equation of the model is transformed into the integral form by using the Green's function of the cantilever beam. Then, closed-form solutions are obtained by assuming an appropriate shape function for the beam deflection to evaluate the integrals. In the numerical method, the BVP is solved with the MATLAB BVP solver, which implements a collocation method for obtaining the solution of the BVP. The large deformation theory is applied in numerical simulations to study the effect of the finite kinematics on the pull-in parameters of cantilevers. The centerline of the beam under the effect of electrostatic and van der Waals forces at small deflections and at the point of instability is obtained numerically. In computing the centerline of the beam, the axial displacement due to the transverse deformation of the beam is taken into account, using the inextensibility condition. The pull-in parameters of the beam are computed analytically and numerically under the effects of electrostatic and/or van der Waals forces. The detachment length and the minimum initial gap of freestanding cantilevers, which are the basic design parameters, are determined. The results of the analytical study are compared with the numerical solutions of the BVP. The proposed methods are validated by the results published in the literature.

A new approach using hybrid power series – cuckoo search optimization algorithm to solve electrostatic pull-in instability and deflection of nano cantilever switches subject to van der waals attractions

“…The results show that the Adomian series are capable to obtain a solution for pull-in instability of the nano actuators; however, the accuracy of the results is not good. Hence, Noghrehabadi et al [9] tried to increase the accuracy of the Adomian power series solution by using the Pade approximations. The Pade approximants increased the accuracy of the solution.…”

“…However, Pade approximants transformed the simple power series solution to a rational form, and in some cases the denominator could take zero digits which in this case the results are not applicable. Ramezani et al [9,10] utilized the Green method to obtain a closed-form solution of the pull-in instability in nanocantilevers. They [9,10] assumed a second order shape function for the shape of nanobeam.…”

“…Ramezani et al [9,10] utilized the Green method to obtain a closed-form solution of the pull-in instability in nanocantilevers. They [9,10] assumed a second order shape function for the shape of nanobeam. Recently, artificial intelligence techniques have been utilized to solve various types of differential equations [11][12][13][14].…”

Boundary exponential stabilization of non-classical micro/nano beams subjected to nonlinear distributed forces

“…Most of the researchers deal with this undesired effect by deriving a safe region for system parameters (for example, deriving the maximum input voltage, minimum separation length, and maximum beam length). Ramezani et al [15] studied the static pull-in instability by using the continuous model of electrostatic cantilever at nano-scale separations and by taking the effect of van der Waals force into consideration. Considering different nonlinear effects, Mojahedi et al [16] studied the static pull-in instability of electrostatically actuated microbridges and micro-cantilevers by adopting homotopy perturbation method.…”

An improved model for the cantilever NEMS actuator including the surface energy, fringing field and Casimir effects

“…In order to treat the nonlinearity of the governing equations, Noghrehabadi et al [25], Duan et al [5] and Farrokhabadi et al [7] used the Adomian Decomposition Method (ADM), which has been discussed widely by Rach [26] and Duan et al [27]. Ramezani et al [28] also proposed a second-order polynomial shape function for transverse deflection in their analytical approximate solution by using the Green's function. The proposed Homotopy perturbation method (HPM) [29] as well as reduced-order method by Nayfeh et al [30] are the other approaches which applied to calculate the analytical expressions for the pull-in instability parameters.…”