The spreading residue harmonic balance method for studying the doubly clamped beam-type N/MEMS subjected to the van der Waals attraction

Abstract

Since the doubly clamped beam-type N/MEMS subjected to the van der Waals attraction is transformed into a mathematical model through Galerkin method, which is a nonlinear differential equation, the spreading residue harmonic balance method is introduced to solve the approximate solution of the nonlinear problem in this paper. The spreading residue harmonic balance method is developed on the basis of the residue harmonic balance method. The disadvantage of the approach is small parameter assumption is needed. In order to improve the accuracy of the approximate solution, the residual of the former order harmonic approximation is introduced. Besides, we compare the second-order spreading residue harmonic balance method solution with the numerical one by the Runge-Kutta method. This proves the availability and validity of spreading residue harmonic balance method.

Residue Regulating Homotopy Method for Strongly Nonlinear Oscillator

“…In above examples, we use the plain form of the vector and successfully regulate the convergence rate through the uniform residue-regulating parameter c v , which is, in effect, the convergence-control vector of the RRHM since it is somewhat similar in function to h of the traditional HAM [14]. We notice some aforementioned methods [29][30][31][32][33][34][35] exhibit some similarities with the RRHM proposed in this paper in the way of constructing the linear sub-equations. Nevertheless, these works do not include any residue-regulating parameters that can control the convergence of the iteration or homotopy.…”

“…Different from the HAM and HPM, there is a class of methods that only use the concept of homotopy to expand the solution into a power series with respect to q , instead of using auxiliary operators to construct a homotopy of equations, such as the iterative homotopy harmonic balancing approach (IHHBA) [29,30], global residue harmonic balance method (GRHBM) [31,32], spreading residue harmonic balance method (SRHBM) [33,34], and forward residue harmonic balance method (FRHBM) [35]. Although these methods are easy to implement and do not involve any auxiliary operators, they all lack overall theoretical guidance and convergence-control techniques, thus not suitable for solving strongly nonlinear problems.…”

Li-He’s Modified Homotopy Perturbation Method for Doubly-Clamped Electrically Actuated Microbeams-Based Microelectromechanical System

“…The last two decades have witnessed rapid advancement in nonlinear sciences arising in oscillation theory and other fields of physics [1][2][3][4][5]. Several methods were developed to find periodic solutions of nonlinear oscillatory systems, for example, variation iteration method (VIM) [6][7], homotopy perturbation method [8][9], Hamiltonian approach (HA) [10], energy balance method (EBM) [11], spreading residue harmonic balance method (SRHBM) [12], iteration perturbation method (IPM) [13], and other methods [14][15]. HPM was proposed in the later 1990s [16][17] and now has been established into a mature phase for ordinary differential equations [18][19], partial differential equations [20][21], and differential equations of fractional order [22][23][24].…”

“…Microelectronic technology is the origin of these tiny devices used in vibrators, sensors, switches, and so on [41][42]. Spring-base structures [43][44][45], nanotubes [15,46], and microbeams [11][12]14] can be considered as some of the potential and very applicable nano/microstructures in various sensing and actuating devices. These structures are modeled by generally using Galerikin's method and represented by nonlinear mathematical models.…”

“…These structures are modeled by generally using Galerikin's method and represented by nonlinear mathematical models. Different types of forcing nonlinearities such as electrostatic force [11,43], electromagnetic force [41,44], and van der Waals force [12] make the solution process extremely difficult. Therefore, approximate solutions of these nonlinear models are important for predicting their dynamic behavior.…”