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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.

Keywords

The van der Waals attraction nano/micro-electro-mechanical systems spreading residue harmonic balance method

Introduction

Nano/micro-electro-mechanical systems (N/MEMS) refer to the small size of high-tech devices. The N/MEMS are independent intelligent systems and their internal structures are generally micron or nanometer. They are developed on the basis of micro-electronic technology. Consequently, they focus on ultra-precision machining, involving micro-electronics, materials, mechanics, chemistry and other fields. In addition, N/MEMS are used in the production of micro-vibrators, optical sensors, pressure sensors and so on.^1–5 In order to facilitate the research, N/MEMS are modeled by Galerkin method, which can be produced by the nonlinear mathematical models. Therefore, it is also important to study the approximate solutions of these nonlinear mathematical models. In 2009, Zand and Ahmadian⁶ studied the analytic solutions to predict the dynamic pull-in instability of electrostatically actuated microsystems by homotopy analysis method (HAM). They investigated the influences of different parameters. The results obtained by HAM were compared with what obtained by Runge–Kutta method, and they are quite consistent. Fu et al.⁷ put forward the energy balance method (EBM), which has been used to solve the nonlinear oscillator arising in MEMS. HAM was applied to solve the approximate solutions for NEMS under the effect of the Casimir force by Askari and Tahani,⁸ and they used the optimized HAM to accelerate the convergence of the approximate solutions. Simultaneously, Fu and Qian⁹ used the modified HAM to study the single periodic and period-doubling solutions of a two-degree-of-freedom coupled Duffing system. Rezazadeh et al.¹⁰ researched the oscillation of an electrostatically actuated microbeam by variational iteration method (VIM). The oscillation is transformed into a nonlinear differential equation through the Galerkin method. The analytical approximate solutions are obtained by VIM. Additionally, they drew a conclusion that when applying AC voltage and adjusting its frequency, the microbeam can be stabilized. However, when the applied DC voltage is equal or greater than the pull-in value, the microbeam is unstable.

N/MEMS devices are affected by electrostatic actuated in the aforementioned contents. N/MEMS may also be affected by Casimir and van der Waals (vdW) attractions except the electrostatic actuated. Soroush et al.¹¹ studied the effects of vdW attractions and Casimir force on the instability of cantilever nano-actuator. They arrived at an analytical solution by the Adomian decomposition, which indicates a reliable method. Koochi et al.¹² employed the homotopy perturbation method (HPM) to derive analytical approximate solutions for the NEMS under vdW forces. They found that vdW forces reduce the NEMS pull-in deflection and voltage. Abadyan et al.¹³ investigated the elastic boundary condition on the pull-in instability of beam-type NEMS subjected to vdW attraction and introduced the modified Adomian decomposition (MAD) method. Similarly, Soroush et al.¹⁴ considered the effect of vdW attraction on cantilever and doubly supported beams by MAD method. Due to the high elastic stiffness of nano-beam, the influence of vdW attraction on the instability of double-supported nano-beams is weak. Tahani and Askari¹⁵ introduced a formula to predict the value of MEMS under vdW attraction, and compared the obtained results with those of other literature; it proves the validity and accuracy of the pull-in formulas. In 2016, Askari and Tahani¹⁶ converted electrostatically actuated N/MEMS under the effect of vdW force into a nonlinear equation through Galerkin decomposition method. Then they analyzed the stability of the former by HPM. In 2017, Askari et al.¹⁷ studied the doubly clamped beam-type N/MEMS under the vdW attraction. They solved this system through HAM and analyzed the frequency criterion of this system. Tajaddodianfar et al.¹⁸ stressed the approximate solutions for the frequency response by HAM, and they exemplified two different types of MEMS: one is MEMS resonator with straight microbeam, the other is MEMS resonator with initially curved microbeam.

Many methods have been mentioned above for solving the analytical approximate solutions of nonlinear systems, including HAM, EBM, VIM, HPM, the parameter expanding method and so on. Of these, the HPM was mentioned in Koochi et al.¹² and Askari and Tahani.¹⁶ Besides, Xu¹⁹ applied the parameter expansion method to solve the strongly nonlinear oscillators. Özen Zengin et al.²⁰ also used the parameter expansion method to study the different strongly nonlinear oscillators and obtained the accurate solutions. The nonlinear amplitude frequency relationship of the nonlinear oscillators was obtained by using the parametric expansion method in Tao.²¹ Kimiaeifar et al.²² employed the parameter expansion method to obtain the exact solutions of non-linear modified van der Pol oscillator. The comparison between the results of the parameter expansion method and the results of the perturbation method showed that the parameter expansion method is effective and convenient. This paper presents another method—the spreading residue harmonic balance method (SRHBM). In the beginning, Guo et al.²³ proposed the residue harmonic balance method (RHBM), which considers Fourier truncated residues to enhance the accuracy of the approximate solutions. They used this method to solve the fractional van der Pol oscillator to prove the effectiveness of this method. Then, they researched the coupled systems exemplified by the damped Duffing resonator driven by a van der Pol oscillator via RHBM.²⁴ In the same year, they used RHBM to study the nonlinear delayed oscillators.²⁵ They presented two examples: the delay Liénard oscillator and the delay feedback Duffing system, respectively, and they analyzed the bifurcation of the periodic motion of the system. Moreover, they predicted solutions of many autonomous delay differential systems by the former method.²⁶ Simultaneously, Xiao et al.²⁷ studied the fractional order van der Pol oscillator by RHBM. Then, the global RHBM^28,29 was presented by Ju et al. and used to solve the approximate solutions of the strongly nonlinear systems. In 2016, Guo and Zhang³⁰ proposed SRHBM on the basis of RHBM. Lee³¹ addressed the multi-level RHBM, which was used to solve a nonlinear panel coupled with extended cavity. Rahman and Lee³² presented a new modified multi-level RHBM for solving nonlinear single beam and double beam problems. The detail history about the development of parameter expansion technology and the HPM can be seen in He.³³ Likewise, the large amplitude vibration of the doubly clamped beam-type N/MEMS subjected to the vdW attraction is considered in this paper.

The rest of this paper is constructed as follows. In the next section, the mathematical model of the double clamped beam-type N/MEMS subjected to vdW attraction is derived. The subsequent section writes that the nonlinear differential equation is solved by SRHBM.³⁰ The zero-order, first-order, second-order harmonic approximation solutions and the approximate frequencies are obtained, respectively. Then the numerical simulation is carried out. The phase curves and the time history responses are plotted and compared with the results of Runge–Kutta method. Finally, the conclusions are drawn in the last section.

Mathematical formulation

A model of the doubly clamped beam-type N/MEMS subjected to vdW attraction is shown in Figure 1. $L$ , $b$ , and $h$ are the length, width and thickness of the N/MEMS, respectively. $x$ , $y$ , and $z$ are the coordinates of length, width and thickness, $d$ is the distance between the non-actuated beam and the fixed electrode, $W$ is the beam’s deflection. The axial residual force of the system is $F_{r}$ , $F_{e s}$ and $F_{v d W}$ are the electrostatic actuation by polarized DC voltage V and the vdW attraction of the nano/micro-beam. The mathematical formulation¹⁷ obtained from the Galerkin method is

\begin{matrix} (a_{0} + a_{1} w + a_{2} w^{2} + a_{3} w^{3}) \ddot{w} + b_{0} + b_{1} w + b_{2} w^{2} + b_{3} w^{3} + b_{4} w^{4} + b_{5} w^{5} + b_{6} w^{6} = 0 \\ w (0) = A, \dot{w} (0) = 0 \end{matrix}

(1)

where

w

is the normalized midpoint deflection of the beam,

\dot{w}

and

\ddot{w}

represent the first derivative and the second derivative of

t

A

is maximum amplitude. In addition, the expressions of the coefficients are

a_{0} = \int_{0}^{1} ψ^{2} d x

(2.1)

a_{1} = - 3 \int_{0}^{1} ψ^{3} d x

(2.2)

a_{2} = 3 \int_{0}^{1} ψ^{4} d x

(2.3)

a_{3} = - \int_{0}^{1} ψ^{5} d x

(2.4)

b_{0} = - (λ_{3} + β) \int_{0}^{1} ψ d x

(2.5)

b_{1} = \int_{0}^{1} ψ ψ^{''''} d x - N \int_{0}^{1} ψ ψ^{″} d x + β

(2.6)

b_{2} = - 3 \int_{0}^{1} ψ^{2} ψ^{''''} d x + 3 N \int_{0}^{1} ψ^{2} ψ^{″} d x

(2.7)

b_{3} = 3 \int_{0}^{1} ψ^{3} ψ^{''''} d x - α \int_{0}^{1} {ψ^{'}}^{2} d x \int_{0}^{1} ψ ψ^{″} d x - 3 N \int_{0}^{1} ψ^{3} ψ^{″} d x

(2.8)

b_{4} = - \int_{0}^{1} ψ^{4} ψ^{''''} d x + 3 α \int_{0}^{1} {ψ^{'}}^{2} d x \int_{0}^{1} ψ^{2} ψ^{″} d x + N \int_{0}^{1} ψ^{4} ψ^{″} d x

(2.9)

b_{5} = - 3 α \int_{0}^{1} {ψ^{'}}^{2} d x \int_{0}^{1} ψ^{3} ψ^{″} d x

(2.10)

b_{6} = α \int_{0}^{1} {ψ^{'}}^{2} d x \int_{0}^{1} ψ^{4} ψ^{″} d x

(2.11)

where

ψ (x) = \frac{1}{φ_{1} (0.5)} φ_{1} (x)

(2.12)

φ_{i} (x) = \cosh (\sqrt{{\bar{ω}}_{i}} x) - \cos (\sqrt{{\bar{ω}}_{i}} x) - \frac{\sinh (\sqrt{{\bar{ω}}_{i}}) + \sin (\sqrt{{\bar{ω}}_{i}})}{\cosh (\sqrt{{\bar{ω}}_{i}}) - \cos (\sqrt{{\bar{ω}}_{i}})} [\sinh (\sqrt{{\bar{ω}}_{i}} x) - \sin (\sqrt{{\bar{ω}}_{i}} x)]

(2.13)

Figure 1.

Model of an electrically actuated nano/micro-beam under the effect of vdW force.¹⁷

Application of the SRHBM

To apply the SRHBM²⁶ in equation (1), we must bring in a new independent variable $τ = ω t$ . Equation (1) can be rewritten as

\begin{array}{l} ω^{2} (a_{0} + a_{1} w + a_{2} w^{2} + a_{3} w^{3}) w^{″} + b_{0} + b_{1} w + b_{2} w^{2} + b_{3} w^{3} + b_{4} w^{4} + b_{5} w^{5} + b_{6} w^{6} = 0 \\ w (0) = A, w^{'} (0) = 0 \end{array}

(3)

It is worth mentioning that $w^{″}$ indicates the derivative with respect to $τ$ and $ω$ is the angular frequency.

Since equation (3) exists in the periodic solution, the solution can be expressed as a set of base functions

{\cos [(2 k - 1) τ] | k = 1, 2, 3, \dots}

(4)

and the steady state solution and the angular frequency are represented, respectively, in the forms^33,34

\begin{array}{l} w (τ) = w_{0} (τ) + p w_{1} (τ) + p^{2} w_{2} (τ) + \dots \\ ω^{2} = ω_{0}^{2} + p ω_{1} + p^{2} ω_{2} + \dots \end{array}

(5)

Where

p

and

ω_{i} (i = 0, 1, 2, \dots)

are the unknown parameters. The zero-order, first-order, second-order harmonic approximate solutions and the approximate frequencies are analyzed as follows.

The zeroth-order harmonic approximation

The initial guess solution of $w (τ)$ can be given by equation (3), so the initial guess solution is written as

w_{0} (τ) = A \cos (τ), τ = ω_{0} t

(6)

Substituting equation (5) into equation (3), and extracting the coefficients preceding $p^{0}$ . When equation (6) is substituted, the residual term is yielded

\begin{matrix} R_{0} (τ) = ω_{0}^{2} (a_{0} + a_{1} w_{0} (τ) + a_{2} w_{0}^{2} (τ) + a_{3} w_{0}^{3} (τ)) {w^{″}}_{0} (τ) + b_{0} + b_{1} w_{0} (τ) \\ + b_{2} w_{0}^{2} (τ) + b_{3} w_{0}^{3} (τ) + b_{4} w_{0}^{4} (τ) + b_{5} w_{0}^{5} (τ) + b_{6} w_{0}^{6} (τ) \\ = (b_{1} A + \frac{3 b_{3} A^{3}}{4} + \frac{5 b_{5} A^{5}}{8} - a_{0} A ω_{0}^{2} - \frac{3 a_{2} A^{3} ω_{0}^{2}}{4}) \cos (τ) \\ + (\frac{b_{3} A^{3}}{4} + \frac{5 b_{5} A^{5}}{16} - \frac{a_{2} A^{3} ω_{0}^{2}}{4}) \cos (3 τ) + \frac{b_{5} A^{5}}{16} \cos (5 τ) \end{matrix}

(7)

According to Galerkin procedure, equation (7) cannot contain secular term $\cos (τ)$ . In order to eliminate the secular term, the coefficient of $\cos (τ)$ should be zero. Then we obtain

b_{1} A + \frac{3 b_{3} A^{3}}{4} + \frac{5 b_{5} A^{5}}{8} - a_{0} A ω_{0}^{2} - \frac{3 a_{2} A^{3} ω_{0}^{2}}{4} = 0

(8)

where

ω_{0}

can be solved by equation (8), which is

ω_{0} = \sqrt{\frac{8 b_{1} + 6 b_{3} A^{2} + 5 b_{5} A^{4}}{8 a_{0} + 6 a_{2} A^{2}}}

(9)

The zeroth-order approximation solution can be obtained as follows

w_{0} (τ) = A \cos (ω_{0} t)

(10)

The first-order harmonic approximation

Similarly, equation (5) is substituted in equation (3) and we present the coefficient of parameter $p^{1}$ , yields

\begin{matrix} ϕ_{1} (τ) = ω_{1} (a_{0} + a_{1} w_{0} + a_{2} w_{0}^{2} + a_{3} w_{0}^{3}) {w^{″}}_{0} + ω_{0}^{2} (a_{1} w_{1} + 2 a_{2} w_{0} w_{1} + 3 a_{3} w_{0}^{2} w_{1}) {w^{″}}_{0} \\ + ω_{0}^{2} (a_{0} + a_{1} w_{0} + a_{2} w_{0}^{2} + a_{3} w_{0}^{3}) {w^{″}}_{1} + b_{1} w_{1} + 2 b_{2} w_{0} w_{1} + 3 b_{3} w_{0}^{2} w_{1} \\ + 4 b_{4} w_{0}^{3} w_{1} + 5 b_{5} w_{0}^{4} w_{1} + 6 b_{6} w_{0}^{5} w_{1} \end{matrix}

(11)

It is worth mentioning that equation (11) is linear with respect to $ω_{1}$ and $w_{1} (τ)$ . According to equation (4), we suppose the first-order approximation solution

w_{1} (τ) = c_{3, 1} [\cos (τ) - \cos (3 τ)]

(12)

where

c_{3, 1}

is unknown .

Substituting equation (12) in equation (11) and to improve the accuracy, the residual term $R_{0} (τ)$ is added. So we could consider

R_{1} (τ) = ϕ_{1} (τ) + R_{0} (τ)

(13)

Equation (12) is calculated as

\begin{matrix} R_{1} (τ) = (b_{1} c_{3, 1} + \frac{3}{2} b_{3} A^{2} c_{3, 1} + \frac{25}{16} b_{5} A^{4} c_{3, 1} + \frac{1}{2} a_{2} A^{2} ω_{0}^{2} c_{3, 1} - a_{0} ω_{0}^{2} c_{3, 1} \\ - \frac{3}{4} a_{2} A^{3} ω_{1} - a_{0} A ω_{1}) \cos (τ) + (\frac{b_{3} A^{3}}{4} + \frac{5 b_{5} A^{5}}{16} + 9 a_{0} ω_{0}^{2} c_{3, 1} \\ + \frac{19}{4} a_{2} A^{2} ω_{0}^{2} c_{3, 1} - b_{1} c_{3, 1} - \frac{1}{4} a_{2} A^{3} ω_{0}^{2} - \frac{1}{4} a_{2} A^{3} ω_{1} - \frac{3}{4} b_{3} A^{2} c_{3, 1} \\ - \frac{5}{16} b_{5} A^{4} c_{3, 1}) \cos (3 τ) + (\frac{b_{5} A^{5}}{16} + \frac{11}{4} a_{2} A^{2} ω_{0}^{2} c_{3, 1} - \frac{3}{4} b_{3} A^{2} c_{3, 1} \\ - \frac{15}{16} b_{5} A^{4} c_{3, 1}) \cos (5 τ) + (- \frac{5}{16} b_{5} A^{4} c_{3, 1}) \cos (7 τ) \end{matrix}

(14)

Because $\cos (τ)$ and $\cos (3 τ)$ cannot appear on the right side of the equation (14), their coefficients are equal to zeros. We obtain two equations containing two unknowns $ω_{1}$ and $c_{3, 1}$

ω_{1} = \frac{A^{2} (4 b_{3} + 5 b_{5} A^{2} - 4 a_{2} ω_{0}^{2}) (16 b_{1} + 24 b_{3} A^{2} + 25 b_{5} A^{4} + 8 a_{2} A^{2} ω_{0}^{2} - 16 a_{0} ω_{0}^{2})}{16 N}

(15)

c_{3, 1} = \frac{16 a_{0} b_{3} A^{3} + 12 a_{2} b_{3} A^{5} + 20 a_{0} b_{5} A^{5} + 15 a_{2} b_{5} A^{7} - 16 a_{0} a_{2} A^{3} ω_{0}^{2} - 12 a_{2}^{2} A^{5} ω_{0}^{2}}{4 N}

(16)

where

\begin{matrix} N = 16 a_{0} b_{1} + 16 a_{2} b_{1} A^{2} + 12 a_{0} b_{3} A^{2} + 15 a_{2} b_{3} A^{4} + 5 a_{0} b_{5} A^{4} \\ + 10 a_{2} b_{5} A^{6} - 144 a_{0}^{2} ω_{0}^{2} - 188 a_{0} a_{2} A^{2} ω_{0}^{2} - 55 a_{2}^{2} A^{4} ω_{0}^{2} \end{matrix}

(17)

Therefore, the first-order harmonic approximation solution and frequency are written as follows

ω_{(1)} = \sqrt{ω_{0}^{2} + ω_{1}}, w_{(1)} (τ) = w_{(0)} (τ) + w_{1} (τ) = (A + c_{3, 1}) \cos (τ) - c_{3, 1} \cos (3τ), τ = ω_{(1)} t

(18)

The second-order harmonic approximation

According to the above calculation process, we can derive the second-order harmonic approximation solution and frequency. Firstly, substituting equation (5) in equation (3) yields the coefficients of the $p^{2}$

\begin{matrix} ϕ_{2} (τ) = ω_{2} (a_{0} + a_{1} w_{0} + a_{2} w_{0}^{2} + a_{3} w_{0}^{3}) {w^{″}}_{0} + ω_{1} (a_{1} w_{1} + 2 a_{2} w_{0} w_{1} + 3 a_{3} w_{0}^{2} w_{1}) {w^{″}}_{0} \\ + ω_{0}^{2} (a_{2} w_{1}^{2} + 3 a_{3} w_{0} w_{1}^{2} + a_{1} w_{2} + 2 a_{2} w_{0} w_{2} + 3 a_{3} w_{0}^{2} w_{2}) {w^{″}}_{0} + ω_{1} (a_{0} + a_{1} w_{0} \\ + a_{2} w_{0}^{2} + a_{3} w_{0}^{3}) {w^{″}}_{1} + ω_{0}^{2} (a_{1} w_{1} + 2 a_{2} w_{0} w_{1} + 3 a_{3} w_{0}^{2} w_{1}) {w^{″}}_{1} + ω_{0}^{2} (a_{0} + a_{1} w_{0} \\ + a_{2} w_{0}^{2} + a_{3} w_{0}^{3}) {w^{″}}_{2} + b_{2} w_{1}^{2} + 3 b_{3} w_{1}^{2} w_{0} + 6 b_{4} w_{0}^{2} w_{1}^{2} + 10 b_{5} w_{0}^{3} w_{1}^{2} + 15 b_{6} w_{0}^{4} w_{1}^{2} \\ + b_{1} w_{2} + 2 b_{2} w_{0} w_{2} + 3 b_{3} w_{0}^{2} w_{2} + 4 b_{4} w_{0}^{3} w_{2} + 5 b_{5} w_{0}^{4} w_{2} + 6 b_{6} w_{0}^{5} w_{2} \end{matrix}

(19)

Secondly, we choose the second-order harmonic approximation solution

w_{2} (τ) = c_{3, 2} [\cos (τ) - \cos (3τ)] + c_{5, 2} [\cos (τ) - \cos (5τ)]

(20)

where

c_{3, 2}

and

c_{5, 2}

are unknowns. Substituting equation (20) in equation (19), considering the accuracy of the solution, we introduce the following equation

R_{2} (τ) = φ_{2} (τ) + R_{1} (τ)

(21)

and produce

\begin{matrix} R_{2} (τ) = (b_{1} c_{3, 2} + \frac{3}{2} b_{3} A^{2} c_{3, 2} + \frac{25}{16} b_{5} A^{4} c_{3, 2} + \frac{1}{2} a_{2} A^{2} ω_{1} c_{3, 1} + \frac{1}{2} a_{2} A^{2} ω_{0}^{2} c_{3, 2} + \frac{9}{4} b_{3} A c_{3, 1}^{2} \\ + \frac{9}{4} b_{3} A^{2} c_{5, 2} + \frac{15}{4} b_{5} A^{3} c_{3, 1}^{2} + \frac{45}{16} b_{5} A^{4} c_{5, 2} + b_{1} c_{5, 2} - a_{0} A ω_{2} - a_{0} ω_{1} c_{3, 1} - a_{0} ω_{0}^{2} c_{3, 2} \\ - a_{0} ω_{0}^{2} c_{5, 2} - \frac{3}{4} a_{2} A^{3} ω_{2} - \frac{25}{4} a_{2} A ω_{0}^{2} c_{3, 1}^{2} - \frac{9}{4} a_{2} A^{2} ω_{0}^{2} c_{5, 2}) \cos (τ) + (9 a_{0} ω_{1} c_{3, 1} \\ + \frac{19}{4} a_{2} A^{2} ω_{1} c_{3, 1} + \frac{41}{4} a_{2} A ω_{0}^{2} c_{3, 1}^{2} + 9 a_{0} ω_{0}^{2} c_{3, 2} + \frac{19}{4} a_{2} A^{2} ω_{0}^{2} c_{3, 2} + \frac{5}{16} b_{5} A^{4} c_{5, 2} \\ + 6 a_{2} A^{2} ω_{0}^{2} c_{5, 2} - \frac{1}{4} a_{2} A^{3} ω_{2} - \frac{9}{4} b_{3} A c_{3, 1}^{2} - \frac{5}{2} b_{5} A^{3} c_{3, 1}^{2} - b_{1} c_{3, 2} - \frac{3}{4} b_{3} A^{2} c_{3, 2} \\ - \frac{5}{16} b_{5} A^{4} c_{3, 2}) \cos (3 τ) + (\frac{b_{5} A^{5}}{16} + \frac{11}{4} a_{2} A^{2} ω_{0}^{2} c_{3, 1} + \frac{11}{4} a_{2} A^{2} ω_{1} c_{3, 1} + \frac{11}{4} a_{2} A^{2} ω_{0}^{2} c_{3, 2} \\ + \frac{27}{2} a_{2} A^{2} ω_{0}^{2} c_{5, 2} + \frac{3}{4} a_{2} A ω_{0}^{2} c_{3, 1}^{2} + 25 a_{0} ω_{0}^{2} c_{5, 2} - \frac{3}{4} b_{3} A^{2} c_{3, 1} - \frac{3}{4} b_{3} A c_{3, 1}^{2} - \frac{3}{4} b_{3} A^{2} c_{3, 2} \\ - \frac{15}{16} b_{5} A^{4} c_{3, 1} - \frac{5}{2} b_{5} A^{3} c_{3, 1}^{2} - \frac{15}{16} b_{5} A^{4} c_{3, 2} - \frac{3}{2} b_{3} A^{2} c_{5, 2} - \frac{25}{16} b_{5} A^{4} c_{5, 2} - b_{1} c_{5, 2}) \cos (5 τ) \\ + (\frac{3}{4} b_{3} A c_{3, 1}^{2} + \frac{5}{8} b_{5} A^{3} c_{3, 1}^{2} + \frac{27}{4} a_{2} A^{2} ω_{0}^{2} c_{5, 2} - \frac{5}{16} b_{5} A^{4} c_{3, 1} - \frac{19}{4} a_{2} A ω_{0}^{2} c_{3, 1}^{2} - \frac{5}{16} b_{5} A^{4} c_{3, 2} \\ - \frac{3}{4} b_{3} A^{2} c_{5, 2} - \frac{5}{4} b_{5} A^{4} c_{5, 2}) \cos (7 τ) + (\frac{5}{8} b_{5} A^{3} c_{3, 1}^{2} - \frac{5}{16} b_{5} A^{4} c_{5, 2}) \cos (9 τ) \end{matrix}

(22)

Finally, the coefficients of $\cos (τ)$ , $\cos (3 τ)$ , $\cos (5 τ)$ are supposed to be zeros, and then we obtain three equations. According to these equations, three unknown $c_{3, 2}$ , $c_{5, 2}$ and $ω_{2}$ can be solved. Thus, the second-order harmonic approximation solution and frequency are represented by

ω_{(2)} = \sqrt{ω_{0}^{2} + ω_{1} + ω_{2}}, w_{(2)} (τ) = w_{(0)} (τ) + w_{1} (τ) + w_{2} (τ), τ = ω_{(2)} t

(23)

By the zero-order harmonic approximation, the first-order harmonic approximation, and the second-order harmonic approximation, we can sum up that the kth-order harmonic approximation solutions and frequencies are

\begin{array}{l} w_{(k)} (τ) = w_{(k - 1)} (τ) + w_{k} (τ) \\ ω_{(k)} = \sqrt{ω_{(k - 1)}^{2} + ω_{k}}, k = 2, 3, 4, \dots \end{array}

(24)

and

\begin{matrix} w_{(k - 1)} (τ) = w_{(k - 2)} (τ) + w_{k - 1} (τ) \\ ω_{(k - 1)} = \sqrt{ω_{(k - 2)}^{2} + ω_{k - 1}} \\ w_{k} (τ) = \sum_{i = 1}^{k} a_{2 i + 1, k} [\cos (τ) - \cos ((2 i + 1) τ)] \\ w_{(0)} = A \cos (τ), ω_{(0)} = ω 0 \\ k = 2, 3, 4, \dots \end{matrix}

(25)

Results and discussion

To illustrate the effectiveness of SRHBM for solving the nonlinear equation, we compared the approximate solution obtained by the second-order SRHBM with the solution obtained by Runge–Kutta method. According to equations (2.1) to (2.13), we only need to choose different $N$ , $α$ , $β$ , $λ_{3}$ , $A$ . We mainly select four different sets of data, and then we plot their phase curves and time history responses. The phase curves and the time history responses are shown in Figures 2 to 5. The solid lines denote the solutions of the Runge–Kutta method and the circles mean the approximate solutions of the second-order SRHBM.

Figure 2.

Comparison of second-order SRHBM solution and Runge–Kutta method solution for $A = 0.1$ , $N = 10$ , $α = 0.06$ , $β = 0.1$ , $λ_{3} = 0.2$ : phase curve and time history response. SRHBM: spreading residue harmonic balance method.

In Figure 2, we plotted the phase curve and the time history response when parameters are $N = 10$ , $α = 0.06$ , $β = 0.1$ , $λ_{3} = 0.2$ , $A = 0.1$ . The observation shows that the second-order SRHBM is in good agreement with the Runge–Kutta method, whether it is the phase diagram or the time history curve. From the phase curve, it reveals that the nonlinear equation has a stable periodic solution. It also shows that it is effective to use SRHBM to approximate the Runge–Kutta numerical solution. Next, we increase the values of the parameters $N$ and $α$ , and the phase curve and the time history response are drawn for $N = 20$ , $α = 6$ , $β = 0.1$ , $λ_{3} = 0.2$ , $A = 0.1$ in Figure 3. From the image, we know that when $N$ and $α$ are increased, it does not change the consistency between the second-order SRHBM and the Runge–Kutta numerical solution.

Figure 3.

Comparison of second-order SRHBM solution and Runge–Kutta method solution for $A = 0.1$ , $N = 20$ , $α = 6$ , $β = 0.1$ , $λ_{3} = 0.2$ : phase curve and time history response. SRHBM: spreading residue harmonic balance method.

To explain the size of the parameter $N$ which does not affect the accuracy of the solution, we have selected two sets of data, which are $N = 50$ , $α = 0.06$ , $β = 0.1$ , $λ_{3} = 0.2$ , $A = 0.1$ and $N = 100$ , $α = 6$ , $β = 0.1$ , $λ_{3} = 0.2$ , $A = 0.1$ , respectively. The phase diagrams and the time history responses are shown in Figures 4 and 5. As observed in Figures 4 and 5, the solutions of the two methods are quite consistent. This further proves that the SRHBM is effective and the proposed method can help us find the solution of the nonlinear equation precisely.

Figure 4.

Comparison of second-order SRHBM solution and Runge–Kutta method solution. For $A = 0.1$ , $N = 50$ , $α = 0.06$ , $β = 0.1$ , $λ_{3} = 0.2$ : phase curve and time history response. SRHBM: spreading residue harmonic balance method.

Figure 5.

Comparison of second-order SRHBM solution and Runge–Kutta method solution. for $A = 0.1$ , $N = 100$ , $α = 6$ , $β = 0.1$ , $λ_{3} = 0.2$ : phase curve and time history response. SRHBM: spreading residue harmonic balance method.

Conclusions

The paper adopts the SRHBM to solve the doubly clamped beam-type N/MEMS under the vdW attraction. The zero-order, the first-order and the second-order approximate solution of this system was obtained by SRHBM. Then, we drew the second-order harmonic approximation phase curves and time history responses for different parameters $N$ , $α$ , $β$ , $λ_{3}$ , $A$ , which were compared with the Runge–Kutta method. It is found that the second-order harmonic approximation solutions are in good agreement with the solutions of Runge–Kutta method and the SRHBM are convergent quickly. In addition, this explains that the SRHBM is valid for solving the approximate solution of nonlinear differential equations.

Authors' contributions

All the authors contributed equally and significantly in writing this paper. All the authors read and approved the final manuscript.

Footnotes

Acknowledgements

We are grateful to the anonymous reviewers for their constructive suggestions.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/ or publication of this article: The first author gratefully acknowledges the support of the National Natural Science Foundations of China (NNSFC) through grant No. 11572288. The author J. S. Wang acknowledges the support of the Teachers' professional development projects of visiting scholars in Zhejiang Province through grant No. FX2017090.

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

Abstract

Keywords

Introduction

Mathematical formulation

Application of the SRHBM

The zeroth-order harmonic approximation

The first-order harmonic approximation

The second-order harmonic approximation

Results and discussion

Conclusions

Authors' contributions

Footnotes

Acknowledgements

Declaration of conflicting interests

Funding

References