On the application of Sturm ’ s theorem to analysis of dynamic pullin for a graphene-based MEMS model

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. c © 2018 University of West Bohemia. All rights reserved.


Introduction
An important phenomenon in Micro-Electro-Mechanical Systems (MEMS) is the so called pullin instability.For certain values of the voltage the system is in a stable operation regime in which a moving part, e.g., a charged plate, approaches a stable steady state, the so called static pull-in, and remains separate from the fixed part, e.g., a substrate.When the voltage is increased beyond a critical value (static pull-in voltage), the device is in the touch-down regime, i.e., the moving part collapses onto the fixed part.Therefore, the analysis of pull-in voltage of such devices is very crucial for the right calibration and use of the MEMS devices.Numerous research results have been already obtained about pull-in conditions.The first mass-spring model for an electrostatically actuated device has been introduced by Nathanson et al. [7].The analysis of pull-in voltage of linear materials for MEMS have been throughly discussed in [8,14].For the mass-spring system, Zhang et al. [15] specify the dynamic pull-in and describe it as the collapse of the moving structure caused by the combination of kinetic and potential energies.In general, the dynamic pull-in requires a lower voltage to be triggered compared to the static pull-in threshold, see [2,15].The preliminary results for the static pull-in voltage have been stated in [12] by considering the quadratic stress-strain equation which is validated to be important for graphene with applications in MEMS.Exact conditions for the dynamic pull-in were discussed in [11].However, the shortcoming of aforementioned paper is that it provides condition for mass lumped parameters only in the case of zero initial conditions.
The purpose of this work is to derive the conditions for the dynamic pull-in in the case of general initial conditions in the one degree of freedom (DOF) spring-mass system which represents the graphene-based MEMS.Our mass-spring model can be also considered as one DOF approximations to solutions of MEMS problems which usually require applications of advanced finite element solvers, cf.[3,4].
In Section 2, brief description of general model will be given, in Section 3 Sturm's theorem with its proof will be presented, in Section 4, application of Sturm's theorem to our model will be demonstrated, and in Section 5 conclusions will be drawn.

The model problem
We consider the graphene-based MEMS model for a parallel plate capacitor.The equation for the motion of the capacitor plate proposed by [12] reads as follows where m is mass of the flat plate, x(t) the axial displacement, E the Young's modulus, A c the cross-sectional area of graphene sheet, L the length of graphene sheet, D the third-order elastic stiffness constant, ε 0 is electric emissivity, A is the area of plate, V DC is the applied voltage, and d is the gap between the plate and the substrate, see Fig. 1.The derivation of (1) follows from the standard framework of lumped-parameter modelling for longitudinal motion of elastic rod, see, e.g., [13,Ch. 9].The third order elastic constant and the quadratic stress-strain equation which our model ( 1) is based on were validated by the experimental work of Zhou and Huang [16] which considered in plane deformation and then followed by the theoretical validation by Cadelano et al. [1].Experimental validations were also done for the out-of-plane case by Lee et al. [6].
In order to transform the model equation ( 1) into the dimensionless form, let us introduce the following non-dimensional variables x, t and the lumped parameters K > 0, α > 0 are defined as Hence, the model equation ( 1) can be written in the dimensionless form as follows ( For the rest of the paper we will write x and t instead of x and t, respectively.After multiplication of (2) by x (t) and integration with respect to time, we obtain the conservation of energy which implies for all t ≥ 0, where .
Here, x 0 and x 0 are the initial conditions: From this expression the pull-in time can be found in terms of elliptic integral .
In order to find the conditions for the existence of periodic solutions to problem (2) with initial conditions (4), we need to discuss the function where Hence there is no periodic solution and pull-in happens.
Case 2. f (s) is negative for some s ∈ (0, 1).It means that there are some roots of f (s) in interval (0, 1).As a result, there is periodic solution and pull-in does not happen.
Therefore, it is crucial to know whether there is a root or not for different values of α > 0 and K ≥ 0. Sturm's theorem discussed in the next section can be a very useful tool for solving this task.

Sturm's theorem
Let p 0 (x) and p 1 (x) denote the polynomial p(x) and its derivative p (x), respectively.Then, using the Euclidean algorithm we define the Sturm sequence Proof.We will follow the proof from [9].Near any root c of p(x), p(x) is negative on one side of c and positive on another side of c. Hence σ(x) can change only if it pass through a root of one of the p i (x) and σ(x) looses one sign change.We have to study the following two cases: if one of the interior polynomials p i has a root at a, then p i−1 and p i+1 are both nonzero and opposite signs.In addition, in a sufficiently small neighborhood p i−1 and p i+1 have constant signs.
Case 2. Let p 0 (x) = 0, then p 1 (x) has constant sign in some interval sufficiently small interval [c, d] such that: Hence σ decreases by one.
Thus, σ looses one sign change if and only if x passes through a root of p 0 (x), which is initial p(x).Hence it is derived that the number of sign changes, i.e. losses in the interval [a, b] counts the number of real roots of the polynomial p(x).

Zero initial conditions
In the first case initial conditions are set to zero, i.e. x 0 = x 0 = 0.This results in C = −K and (5) becomes: Notice that f (0) = 0.In order to find periodic solution we need to analyze the condition when f (s) = 0 for s ∈ [0, 1], that is the same as h(s) = 0 from (6), namely Using the Euclidean algorithm we find the following Sturm sequence of polynomials The number of roots of h(s) in the interval [0, 1] is determined using Table 1.The difference between the number of sign changes in the second column and the number of sign changes in third column of Table 1 states the number of roots.Algorithm 1 in Appendix provides Python script [10] to accomplish this task computationally for different α and K values.In addition, algorithm plots numerical solution solved using Runge-Kutta method (see [5]) for particular α > 0 and K > 0 values in order to verify derived conclusion.Figs.2-4 illustrate examples of periodic and non-periodic solutions.

Non-zero initial conditions
In the case of non-zero initial conditions the Sturm polynomials for see Algorithm 2, are given as follows The difference between the number of sign changes in the second column and the number of sign changes in third column of Table 2 states the number of roots of p(s) = g pos (s) in the interval [0, 1].In the case of non-zero initial conditions, checking existence of root in the (0, 1) interval is not enough.We have to perform two more steps.
First, we will check the existence of a root in the interval (0, 1) using Sturm table as it was done for the case of zero initial conditions.If there is no root, then pull-in occurs.If there is a root, then we go to the first additional step.In this step we check the sign of the function p 0 (s) at s = 0, see Algorithm 2 from Appendix.If the sign of 6(C + K) is negative, then there is pull-in.However, if the sign is positive, then we go to the second additional step.In this step we check the existence of a root in the interval (−∞, 0) using Sturm algorithm for p(s) = g neg (s) = 2α s 4 + (3 − 2α)s 3 − 3s 2 − 6Cs + 6(K + C) , see Algorithm 2. If there is no root, then pull-in occurs.Otherwise, we conclude that there is periodic solution.Algorithm 2 from Appendix provides reader with Python code [10] which performs the same analysis in the case of nonzero initial conditions for the given α, K, x 0 , and x 0 values.For example, Figs. 5, 6 and 7 demonstrate the periodic and pull-in solutions for various parameters α, K and initial conditions.

Conclusions
We developed a simple and effective procedure to identify a priori the periodic and pull-in solutions to the MEMS problem of parallel plate capacitor.It was shown that the periodicity of solution depends on the lumped parameters α, K, and the initial conditions.The procedure is supplemented numerically by using Python codes.In general, this technique is useful for determination of the periodicity of solution to higher order differential equations and it can be applicable for analogous singular equations of dynamical systems.Further research can be done to establish other analytical or numerical methods in order to determine solutions of the problem.Another research area is to find the methods for computing the periods of solutions when no pull-in occurs.

Theorem 1 (
Sturm's Theorem).The number of distinct real zeros of a polynomial p(x) with real coefficients in [a, b] is equal to the excess of the number of changes of sign in the sequence p 0 (a), . . ., p k−1 (a), p k (a) over the number of changes of the sign in the sequence p 0 (b), . . ., p k−1 (b), p k (b) .
Let σ(a) is the number of sign changes in the sequence p 0 (a), . . ., p k−1 (a), p k (a) and σ(b) is the number of sign changes in the sequence p 0 (b), . . ., p k−1 (b), p k (b) .

Table 1 .
Sturm sequence for the case of zero initial conditions

Table 2 .
Sturm's polynomials for non-zero initial conditions