Bifurcation and dynamics of periodic solutions of MEMS model with squeeze film damping

In this paper, we study the oscillations of an idealized mass–spring model of micro-electro-mechanical system (MEMS) with squeeze film damping. The model consists of two parallel electrodes separated by a gap $d$: one of them is fixed, and another one is movable and attached to a linear spring with stiffness coefficient $k>0$. The oscillation, under the influence of AC–DC voltage $V\left(t\right)=v_{dc}+v_{ac}cos\frac{2\pi }{T}t$, is ruled by the following singular differential equation $m{y}^{\prime \prime }+\left[\frac{A}{{(d-y)}^3}+\frac{A}{d-y}\right]y^{\prime }+ky=\frac{{\theta }_{0}A}{2}\frac{V^2\left(t\right)}{{(d-y)}^2}.$Here, $y$ is the vertical displacement of the moving plate ($y$ is always assumed to be less than $d$), $m>0$ is its mass, $A>0$ is the electrode area, and ${\theta }_0>0$ is the absolute dielectric constant of vacuum. Taking $d$ as the parameter, we show the existence of saddle–node bifurcation of $T$-periodic solutions to the equation in the parameter space. This answers, from certain point of view, the open problem proposed by Torres in his monograph, see Torres (2015, Open Problem 2.1, p. 18). Further, we prove that the equation has exactly two classes of $T$-periodic solutions: as $d$ tends to $+\infty$, one of them uniformly tends to $+\infty$ at the rate of $d$, while the minimum values of the second class tend to, or cross, 0.