On the application of Sturm's theorem to analysis of dynamic pull-in for a graphene-based MEMS model

Abstract

A novel procedure based on the Sturm's theorem for real-valued polynomials is developed to predict and identify periodic and non-periodic solutions for a graphene-based MEMS lumped parameter model with general initial conditions. It is demonstrated that under specific conditions on the lumped parameters and the initial conditions, the model has certain periodic solutions and otherwise there are no such solutions. This theoretical procedure is made practical by numerical implementations with Python scripts to verify the predicted behaviour of the solutions. Numerical simulations are performed with sample data to justify by this procedure the analytically predicted existence of periodic solutions.

Periodic oscillations for a graphene-based electrostatic micro actuator

“…) is an upper solution of periodic problem (18) if for all t ∈ [0, T ] the inequalities in 1 and 2 hold with reversed order. When the inequalities in 1 and 2 are strict the corresponding lower (upper resp.)…”

“…We assume that the lower and upper solutions of the periodic problem (18) have range in ]x 1 , x 2 [. Next we present some needed versions of existence results.…”

“…Then there exists at least one solution x(t) ∈ C 2 ([0, T ]) of the periodic problem (18) such that for all t ∈ [0, T ]…”

“…A particular case of Theorem 3.2 in chapter Theorem 4 Let L and U be constant lower and upper solutions of the periodic problem (18) such that U < L, definē…”

Dynamic pull-in for micro–electromechanical device with a current-carrying conductor

“…4 Mathematical analysis of the one-degree-offreedom lumped parameter model for an electrostatically actuated beam has been provided in our previous works, see Skrzypacz; 7 Omarov. 8 It is well known in physics and engineering that comparing with electrostatic force the magnetic force stated in Ampe´re's law is much larger and therefore in many applications using it as the actuating force has advantages as stated and discussed in Lobato-Dauzier, 9 Imai and Tsukioka, 10 and Xingdong. 11 In particular, in certain devices and applications such as in vibration-based energy harvesting systems, cf.…”