# Periodic Oscillations in MEMS under Squeeze Film Damping Force

Authors: Juan Beron; Andrés Rivera
Journal: Journal of Applied Mathematics
 (2022)

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## Abstract

We provide sufficient conditions for the existence of periodic solutions for an idealized electrostatic actuator modeled by the Liénard-type equationx¨+FDx,ẋ+x=βV2t/1−x2,x∈−∞,1 with β∈ℝ+, V∈Cℝ/Tℤ, and FDx,ẋ=κẋ/1−x3, κ∈ℝ+ (called squeeze film damping force), or FDx,ẋ=cẋ, c∈ℝ+ (called linear damping force). If FD is of squeeze film type, we have proven that there exists at least two positive periodic solutions, one of them locally asymptotically stable. Meanwhile, if FD is a linear damping force, we have proven that there are only two positive periodic solutions. One is unstable, and the other is locally exponentially asymptotically stable with rate of decay of c/2. Our technique can be applied to a class of Liénard equations that model several microelectromechanical system devices, including the comb-drive finger model and torsional actuators.


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## Body

## 1. Introduction

This paper presents a mathematical study of the existence and stability of periodic solutions of a Liénard-type equation that describes the motions of the movable plate (movable electrode) in an idealized parallel-plate electrostatic actuator, nowadays known as theNathanson model. This actuator is an example of a large set of devices composed of microscale (one or more of their dimensions are in the micrometers range) mechanical and electronic elements integrated in a common silicon substrate. This recent technology, known in the literature as microsystem technology (MST) or microelectromechanical systems (MEMS), has become very successful in the commercial front (a complete and recent survey of MEMS literature with applications can be found in [1]). Due to their size, they can fit in several devices such as TVs, microwaves circuits, cardiac pacemakers, pressure sensors, accelerometers and gyroscopes for automobiles, and wearable electronic devices [2]. Examples of MEMS are the acceleration sensor and scanner developed and produced by Bosch [3, 4]. The mathematical formulation of the Nathanson model was initially presented in 1967 by the American electrical engineer H. C. Nathanson et al. [5]. The study focuses on the structural instability phenomenon which results from the variations in voltage load and leads to a saddle-node bifurcation, called pull-in. After more than 50 years, the Nathanson model continues to draw a lot of attention. Many researchers have been devoted to its analytical and numerical study, mainly to understanding and characterizing the pull-in phenomenon through different techniques and mathematical formulations (see for instance [1, 2, 6–10]).The fundamental configuration of the Nathanson model assumes rectangular electrodes. One stationary and the other are allowed to move. The electrodes can have any shape, but for simplicity, rectangular electrodes are more commonly used. Ifd is the initial distance between the electrodes and both are biased by a voltage V, then an electrostatic force will be generated which pulls the movable electrode.If the schematic diagram of the Nathanson model is like the one shown in Figure1, the electrostatic force FEτ,x^ acting on the movable electrode is expressed as
(1)FEτ,x^=εAV2τ2d−x∧2,x^∈−∞,d,where τ is an independent variable related to time, Vτ is the potential difference between the two plates, A is the area of the plate that is facing the other, ε is the dielectric constant of the medium in between the plates, and x^ is the position of the movable electrode with respect to d. The other forces involved are the restoring FRx^ and the damping force F^Dx^,x∧′, where x∧′ represents the induced velocity of the movable electrode. For the restoring force, we have considered a linear stiffness force, in such case FRx^=kx^ with k>0, and for the damping force, two types were considered:
(2)F^Dx^,x∧′=c~x∧′,lineardamping,(3)F^Dx^,x∧′=γd−x∧3x∧′,squeezefilmdamping,with c~,γ∈ℝ+. The first comes from simplifying the problem to a moving sphere in fluid at a velocity x∧′. The second corresponds to the most common and dominant dissipation mechanism in MEMS, which is related to the study of the damping force on a microstructure with a big surface that traps a fluid in a small space. When the plates separate, the inner pressure is smaller than the outer pressure as shown in Figure 2(a). When the plates get closer, the opposite occurs, as seen in Figure 2(b). This effect is called squeeze film damping, and it is significantly present in parallel plate actuators, which have a proportionally bigger surface area in comparison with the distance in between the plates. For more details on damping in MEMS, see [2] and the references therein.Figure 1
Parallel plate capacitor with one movable plate.Figure 2
Squeeze film damping on parallel plate capacitors.
(a)(b)Under the previous considerations and from Newton’s second law, the equation of motion of the movable electrode is given by the following second order nonlinear differential equation:(4)mx∧″+F^Dx^,x∧′+kx^=ϵAV2τ2d−x∧2,where x∧″=x∧″τ, represents the induced acceleration of the position x^=x^τ. The gravitational force is not considered because it is too small compared to the electrostatic force in microstructures. In order to reduce the number of parameters, we can normalize x^ with respect to d and τ with respect to T=m/k. Therefore, our nondimensional variables x and t satisfies
(5)x^=xd,τ=tT,and the corresponding nondimensional equation from (4) is
(6)x¨+FDx,ẋ+x=βV2t1−x2,x∈−∞,1,with
(7)FDx,ẋ=T2mdF^Dxd,ẋdT,Vt=VtT,andβ=εA2kd3.Throughout this document, we consider a DC-AC voltage sourceVt of the form
(8)Vt=v0+δvt,with v0∈ℝ+ (DC-voltage source) and vt∈Cℝ/Tℤ with zero average. Voltage V will also be nonnegative; therefore, from now on we assume δ∈0,−v0/vmin where
(9)vmax≔maxt∈0,Tvt,vmin≔mint∈0,Tvt.When the damping forceF^D is given by (2) (namely, the linear damping force), the authors in [7, 9] present a rigorous analysis of the existence and stability of exactly two positive T-periodic solutions of (6) for the non-conservative (c~≠0) and for the conservative case (c~=0), respectively. In both papers, classical functional and topological techniques were employed such as the upper and lower solution method, Leray-Schauder degree, and the topological index of a periodic solution. As far as we are aware, no papers have been published regarding the study of periodic solutions of (6) when the damping force is given by (3). Hence, this paper pursues two goals: firstly, to provide an alternative and accurate stability criteria for the two periodic solutions of (6) with linear damping force, and secondly, to present sufficient conditions for the existence and linear stability of periodic solutions of (6) with under squeeze film damping force. We remark that the techniques and ideas in this document can be applied to study periodic motions in other MEMS devices and microstructures, for example, torsional actuators, comb-drive devices, atomic force microscope microcantilevers (see [2]), and the recent graphene-based Nathanson model (see [11]).We have divided the document into four sections: following the introduction, in Section2, we developed the main tools for the proofs. Sections 3 and 4 are devoted to state and prove the main results. In addition, numerical validations are provided to illustrate the results applied to (4) using explicit values of the parameters taken from the specialized literature [2]. Finally, to provide a self-contained manuscript, we included an appendix in which we established well-known results about the method of lower and upper solutions for second order differential equations and multiplicity and stability of periodic solutions of Liérnard equations.


## 2. Preliminary Results

Different approaches can be used to study the existence of solutions of the boundary value problem:(10)x¨+ft,x,ẋ=0,x0=xT,ẋ0=ẋT,forf:D⟶ℝ continuous function, where D⊆ℝ×l1,l2×ℝ is an open connected set with −∞≤l1≤l2≤∞. Topological degree, averaging method, and lower and upper solutions are perhaps the most common used tools from nonlinear analysis to address this problem. In this section, we use the lower and upper solution method to obtain existence results for Liénard type families of (10). It is worth to mention that if f is a T-periodic function in the variable t, then all the solutions of (10) would be also T-periodic.Theorem 1.
Letζ,η∈C20,T be a lower and an upper solution of the boundary value problem (10) such that η≤ζ. Define
(11)E≔t,x,y∈Dt∈0,T,ηt≤x≤ζt.
† Assume that there exists N≥0 such that ∂yft,x,y≤N for all t,x,y in E. Then for any solution ut of (10) such that ηt≤ut≤ζt on 0,T, there exists R≔RN>0 such that −R≤u̇t≤R for all t in 0,T.
†† If the assumption in (†) holds. Let
(12)W≔t,x,y∈Et∈0,T,−R≤y≤R,
and assume that there existsM≥0 such that for all t,x,y in W(13)∂xft,x,y≤M≤πT2,σt≤L,N≤HL,
for someL∈M,π/T2 where
(14)σt≔ft,ζt,ζ̇t−ft,ηt,η̇tζt−ηt,HL=L−MLcotTL2.Then, the boundary value problem (10) has at least one solution ψ such that
(15)ηt≤ψt≤ζt,∀t∈0,T.Proof.
The existence ofN≥0 such that ∂yft,x,y≤N and Theorem 13 (see Appendix) lead us to the conclusion that there exists R>0 such that for any solution u1 of (10) and any solution u2, u3 of
(16)x¨≥ft,x,ẋ,x0=xT,ẋ0=ẋT,(17)x¨≤ft,x,ẋ,x0=xT,ẋ0=ẋT,respectively, with t,uit,u̇it in E also satisfies u̇it<R for all t in 0,T and i=1,2,3. This proves the first statement †. The second statement †† follows the same lines of the proof of Theorem 3.2 of Chapter 5 in [12] relative to W.Now, we consider the family of boundary value problems(18)x¨+ct,xẋ+Kx=FtGx,x0=xT,ẋ0=ẋT.wherec:0,T×I⟶R, K:I⟶R, G:I⟶R, F:0,T⟶R be continuous functions, I⊂R. Define
(19)ϕx=KxGx,andFmax=maxt∈0,TFt,Fmin=mint∈0,TFt.Following the notation of (10), we have
(20)ft,x,ẋ=ct,xẋ+Kx−Ft/Gx.Regarding the existence of periodic solutions of (18), we lead to the following results.Theorem 2.
Assume thatFmin,Fmax∈ϕI, Gx>0 and ϕx is decreasing with isolated critical points in I. Then, (18) admits unique constant lower and upper solution ζ and η, respectively, such that ζ≤η satisfying
(21)ϕζ=Fmaxandϕη=Fmin.
Moreover, there exists a solutionφt of (18) such that
(22)ζ≤φt≤η,∀t∈0,T.Proof.
Sinceϕ is decreasing with isolated critical points in I and Fmin,Fmax∈ϕI, then there exist unique solutions ζ, η, in I for
(23)ϕζ=Fmax,andϕη=Fmin,respectively. Moreover, ζ≤η. Additionally, notice that
(24)KηGη=Fmin≤Ft≤Fmax=KζGζ,∀t∈0,T,which implies
(25)Kη−Ft/Gη≤0andKζ−Ft/Gζ≥0.
It is equivalent to(26)ft,η,0≤0andft,ζ,0≥0.
Therefore,ζ≤η, and they are lower and upper solutions, respectively.
Let(27)E≔0,T×ζ,η×ℝandN=max0,T×ζ,ηct,x.
Notice that for allt,x,v1,t,x,v2∈E, it follows
(28)ft,x,v1−ft,x,v2=ct,xv1−v2≤Nv1−v2.
Then, the positive functionρs=Ns+maxEft,x,0 satisfies
(29)∫0∞sρsds=∞andft,x,v≤ρv,∀t,x,v∈E.Hence,f satisfies the Nagumo condition and Theorem 12 (see Appendix); we can conclude that there exists a solution φ of (18) such that
(30)ζ≤φt≤η,∀t∈0,T.Theorem 3.
LetI^⊂ℝ and c:0,T×I^⟶ℝ, K:I^⟶ℝ, G:I^⟶ℝ be differentiable functions and F:0,T⟶ℝ continuous.
‡ Assume that Fmin,Fmax∈ϕI^, Gx>0, and ϕ′x≥0 with isolated critical points in I^. Then, (18) admits unique constant reversed-ordered upper and lower solutions η,ζ, respectively, such that ϕη=Fmin and ϕζ=Fmax.
‡‡ If the assumptions in ‡ holds, define
(31)E^=0,T×η,ζ×ℝ,N=maxE^ct,x,a^=maxN,maxE^Kx−Ft/Gx,and let R the unique positive solution of
(32)R−lnR+1=a^ζ−η.
Assume that there existsM>0 such that
(33)maxW∂∂xct,xẋ+Kx−FtGx≤M≤πT2,where
(34)W=t,x,y∈E^y∈−R,R,and there exists L∈M,π/T2 such that
(35)N≤HL,withHL=L−MLcotTL2.Then, the boundary value problem (18) has at least one solution ψ such that
(36)η≤ψt≤ζ,∀t∈0,T.Proof.
Suppose thatGx>0 and ϕ′x≥0 with isolated critical points in I^. This implies that there exists only one pair of values ζ,η in I^ such that
(37)ϕζ=Fmax,ϕη=Fmin.
Notice thatη≤ζ because ϕ is monotone non-decreasing function and Fmin≤Fmax. Additionally η,ζ are upper and lower solutions of (18) because
(38)ft,ζ,0=KζGζ−FtGζ=Fmax−FtGζ≥0,∀t∈R,ft,η,0=KηGη−FtGη=Fmin−FtGη≤0,∀t∈R,which proves part ‡. In order to prove part ‡‡, first notice that
(39)∂yft,x,y=ct,x≤maxE^ct,x=N,∀t,x,y∈E^.with
(40)E^=0,T×η,ζ×R.
Then, by Theorem1 part †, there exists R>0 such that for any solution ut of (18) such that η≤ut≤ζ on 0,T, we have −R≤u̇t≤R on 0,T, and following Theorem 13 and its remark, the value R is the unique positive solution of
(41)R−lnR+1=a^ζ−η.
Notice that by the mean value theorem:(42)σt≔ft,ζ,0−ft,η,0ζ−η≤maxW∂xft,x,y≤M≤L,∀t∈0,T.
Therefore, by Theorem1, the boundary value problem (18) has at least one solution ψ such that
(43)η≤ψt≤ζ,∀t∈0,T.


### 2.1. Multiplicity and Stability of Periodic Solution for Duffing Equations

We end this section by showing some results about multiplicity and stability of periodic solutions for the Duffing-type equation:(44)x¨+cẋ+gt,x=0,where c>0 and g:ℝ×l1,l2⟶ℝ, −∞≤l1<l2≤∞, a continuous function, T-periodic with respect to t, and having a continuous partial derivative with respect to x. The following notation will be used throughout the rest of the paper.The positive part of a given a functionf is defined as f+t≔maxft,0.(1)
Given a pair of functionf,g∈Lp0,T, we write f≻g, if f≥g for almost every t and f>g in a subset of positive measure(2)
Lpℝ/Tℤ: T-periodic function f∈Lp0,T with the norm
(45)fLp≔fLp0,T,p∈1,∞.(3)
For somep∈1,∞ and p∗=p/p−1, Kq denotes the best Sobolev constant in the following inequality:
(46)CuLq2≤u̇L22,∀u∈H010,T.Let us start with some results over the linear differential operator:(47)La:W⟶L10,Tω⟶Laω=ω¨+cω̇+atω,where
(48)W=ω∈W2,10,T:ω0=ωT,ω̇0=ω̇T,c is a positive constant and a∈Γp,c, with
(49)Γp,c=a∈Lp0,T:a−c2/4+Lp<K2p∗,for some p∈1,∞.Proposition 4.
Suppose thata,a1,a2∈Γp,c for some p∈0,∞. Then, we have the following conclusions:
(1)
Each possibleT-periodic solution ω=ωt of Laω=0 is either trivial or different from zero for each t∈0,T(2)
Ifa2≻a1, then Laiω=0i=1,2 cannot admit nontrivial T-periodic solutions simultaneously(3)
Laω=0 does not admit negative Floquet’s multipliers
(i)
Ifa¯=1/T∫0Tatdt satisfies a¯>c2/4, thenLaω=0 does not admit real Floquet’s multipliers, i.e., Law=0 is elliptic and has no nontrivial T-periodic solutions
The trivial solutionω≡0 is locally exponentially asymptotically stable with rate of exponential decay c/2.The proof of Proposition4 can be found in several papers, see, for example, [13–16]. It is worth pointing out that the arguments in [14–16] are not precise but easy to fix by choosing correctly the set Γp,c.


## 2.1. Multiplicity and Stability of Periodic Solution for Duffing Equations

We end this section by showing some results about multiplicity and stability of periodic solutions for the Duffing-type equation:(44)x¨+cẋ+gt,x=0,where c>0 and g:ℝ×l1,l2⟶ℝ, −∞≤l1<l2≤∞, a continuous function, T-periodic with respect to t, and having a continuous partial derivative with respect to x. The following notation will be used throughout the rest of the paper.The positive part of a given a functionf is defined as f+t≔maxft,0.(1)
Given a pair of functionf,g∈Lp0,T, we write f≻g, if f≥g for almost every t and f>g in a subset of positive measure(2)
Lpℝ/Tℤ: T-periodic function f∈Lp0,T with the norm
(45)fLp≔fLp0,T,p∈1,∞.(3)
For somep∈1,∞ and p∗=p/p−1, Kq denotes the best Sobolev constant in the following inequality:
(46)CuLq2≤u̇L22,∀u∈H010,T.Let us start with some results over the linear differential operator:(47)La:W⟶L10,Tω⟶Laω=ω¨+cω̇+atω,where
(48)W=ω∈W2,10,T:ω0=ωT,ω̇0=ω̇T,c is a positive constant and a∈Γp,c, with
(49)Γp,c=a∈Lp0,T:a−c2/4+Lp<K2p∗,for some p∈1,∞.Proposition 4.
Suppose thata,a1,a2∈Γp,c for some p∈0,∞. Then, we have the following conclusions:
(1)
Each possibleT-periodic solution ω=ωt of Laω=0 is either trivial or different from zero for each t∈0,T(2)
Ifa2≻a1, then Laiω=0i=1,2 cannot admit nontrivial T-periodic solutions simultaneously(3)
Laω=0 does not admit negative Floquet’s multipliers
(i)
Ifa¯=1/T∫0Tatdt satisfies a¯>c2/4, thenLaω=0 does not admit real Floquet’s multipliers, i.e., Law=0 is elliptic and has no nontrivial T-periodic solutions
The trivial solutionω≡0 is locally exponentially asymptotically stable with rate of exponential decay c/2.The proof of Proposition4 can be found in several papers, see, for example, [13–16]. It is worth pointing out that the arguments in [14–16] are not precise but easy to fix by choosing correctly the set Γp,c.


## 3. An Improvement for the Linear Damping Case

In this section, we consider the Duffing equation:(50)x¨+cẋ+x=βV2t1−x2,withx∈−∞,−1. This equation corresponds to the Nathanson model (6) with linear damping force FDx,ẋ=cẋ, c≥0, and c∈R. The existence and stability of periodic solutions of (50) have been considered in [9] for the case c=0 and also in [7] for c>0. The results exposed here respect to (50) have the purpose to combine the ideas found in the mentioned papers and the results of Theorems 15 and 16 in the Appendix.Theorem 5.
Assume the following conditions:(I)
0<Vmin2<Vmax2≤4/27β(II)
1−2βV2t∈Γp,c and c2/4+2βV2t<1 for all t∈0,T and some p∈1,∞
Then, Equation (50) has exactly two positive T-periodic solutions ψ1 and ψ2 such that
(51)η1≤ψ1t≤ζ1≤13≤ζ2≤ψ2t≤η2,∀t∈0,T,with ηi, ζi, i=1,2 the corresponding solutions of
(52)x1−x2=βVmin2andx1−x2=βVmax2,in 0,1, respectively. Moreover, ψ1 is asymptotically stable, and ψ2 is unstable.
Additionally, if(53)c24<1−3ζ11−ζ1,then ψ1 is locally exponentially asymptotically stable with rate of exponential decay c/2.Proof.
We divide the proof in 4 steps following the ideas in [9].
Step 1: Constant lower and upper solutions
Equation (50) is Equation (18) with
(54)ct,x=c,Kx=x,Gx=1−x2,Ft=βV2t.
In such a case,ϕx=KxGx is given by ϕx=x1−x2. Direct computations prove that ϕx is monotone increasing in I^=0,1/3 and monotone decreasing in I=1/3,1 with isolated critical points at x=1/3 and x=1. Therefore, by the assumption I and Theorems 2 and 3, it follows directly that the solutions of
(55)x1−x2=Fmin,x1−x2=Fmax,provide constant upper solutions ηi, i=1,2 with 0<η1<1/3<η2<1 and constant lower solutions ζi, i=1,2 with 0<ζ1<1/3<ζ2<1, respectively. Moreover,
(56)η1<ζ1≤13≤ζ2<η2<1.
Step 2: Existence of periodic solutions
Applying Theorem2, there exists at least one T-periodic solution ψ2 of (50) such that
(57)ζ2≤ψ2t≤η2,∀t∈0,T.
In order to apply Theorem15 (part A) on the set
(58)E~=t,x∈ℝ×0,1/3η1≤x≤ζ1,it is necessary to study the condition
(59)∂xgt,x≤at,∀t,x∈E~,
for some functiona such that a≻c2/4 and a∈Γp,c, some p∈1,∞, where gt,x=x−βV2t/1−x2. Then, a direct computation shows that
(60)∂xgt,x=1−2βV2t1−x3<1−2βV2t,
for allt,x∈ℝ×0,1. Let at≔1−2βV2t, t∈ℝ. The previous inequality along with the assumptions (I) and (II) imply the inequality (59) with a≻c2/4, and a∈Γp,c for some p∈1,∞. This proves the existence of at least one T-periodic solution ψ1 of (50) which is asymptotically stable and such that
(61)η1≤ψ1t≤ζ1,∀t∈0,T,if the number of T-periodic solutions of (50) between η1 and ζ1 is finite.
Step 3: Multiplicity of periodic solutions
Assume thatφ1 and φ2 are two different nontrivial T-periodic solutions of (50). Define vt=φ2t−φ1t, t∈ℝ. Then, v is a nontrivial T-periodic solution of the equation:
(62)v¨+cv̇+a~tv=0,with a~t=∫01∂xgt,φ1t+mφ2t−φ1tdm. By the inequality (60), the condition (II) and part 1 of Proposition 4, we conclude that vt>0 or vt<0 for all t∈ℝ. Therefore,
(63)φ2t<φ1torφ1t<φ2t,for all t∈R. Now, assume that there exists a third nontrivial T-periodic solution φ3 of (50). The preceding arguments allow us to assume that
(64)φ1t<φ2t<φ3t∀t∈0,T.
In consequence, the nontrivialT-periodic functions vit=φi+1t−φit, i=1,2 satisfy the equations:
(65)v¨t+cv̇+a~itv=0,i=1,2,where a~it=∫01∂xgt,φit+mφi+1t−φitdm. Since
(66)∂x2gt,x=−6βV2t1−x4,∀t,x∈R×−∞,1,it follows that a2t<a1t for all t∈ℝ. Therefore, by part 2 of Proposition 4, we reach a contradiction. This proves that there is at most two positive T-periodic solutions of (50).
To sum up, under the assumptions (I) and (II), Equation (50) has exactly two positive T-periodic solutions which are precisely the functions ψ1 and ψ2 satisfying
(67)η1≤ψ1t≤ζ1≤13≤ζ2≤ψ2t≤η2,∀t∈0,T.
Moreover, by Step 1,ψ1 is asymptotically stable, and ψ2 is unstable.
Step 4: Exponential stability
Finally, we want to apply Theorem16. We need to find a lower bound of ∂xgt,x for all t,x∈E~. Then, direct computations show that
(68)∂xgt,x>1−2βV2t1−ζ13>1−2βVmax21−ζ13=1−3ζ11−ζ1>0,∀t,x∈E~.
Then, by (60) and the previous inequality, we have
(69)0<1−3ζ11−ζ1<∂xgt,x<1−2βV2t,for all t,x∈E~. Define lt≔1−2βV2t/1−ζ13; therefore, l∈Cℝ/Tℤ. From (53), we can deduce that l¯>c2/4. Then, by Theorem 16, the T-periodic function ψ1 is exponentially asymptotically stable with rate of exponential decay c/2. This completes the proof.Remark 6.
Condition (53) can be replaced by
(70)V¯2<1−ζ232β1−c24,where V¯2=1/T∫0TV2tdt.
Additionally, respect to the results over the Nathanson model with constant damping given in [7, 9], the criteria that we illustrated over the function ∂xgt,x−c2/4 in Theorem 5 have the advantage that considers the Lp norms p∈1,∞, and not over the supremum of its range. In consequence, Theorem 5 leads to a refinement of the results founded in [7, 9].Example 1.
The values required to determine the existence ofψ1 are Fmax, Fmin, and T. In order to test different combinations of parameters, let
(71)227=Fmax+Fmin2A=Fmax−Fmin2.IfA≤2/27, then Fmax≤4/27. Figure 3 shows the combination of parameters that allowed to prove the existence and exponentially asymptotical stability, only existence, or did not allow to prove the existence of ψ1 with Theorem 5.Figure 3
(a) Withc=0.9. The values of AandT that satisfied the conditions of Theorem 7 for existence and exponentially asymptotical stability are plotted in the green area. In blue, only where the conditions of Theorem 7 for the existence are satisfied and in red where they are not. (b) Boundary between satisfying and not satisfying the conditions for existence of Theorem 7 for different values of c.
(a)(b)To test the exponentially asymptotical stability property of one of the combination of parameters, letc=0.9, A=0.07, T=2, and Vt=0.1579cos2πt/T+0.2217. Following the results depicted in Figure 3, with that combination, it is possible to prove the existence of ψ1, and it is exponentially asymptotically stable.Let the error between any other solutionφ of (50) and the periodic solution ψ1 be defined as
(72)εt=φt−ψ1t+φ̇t−ψ̇1t.Following the results of Theorem5, ψ1 is locally exponentially asymptotically stable with rate of exponential decay c/2. Then, there exists an adequate positive value d such that
(73)ε^n≔lnεnT≤lnd+cT2n≕υ^n.For this example, Figure4 depicts υ^n in blue for 10 different solutions of (50) and υ^ in red with lnd=−2.Figure 4
(a) Poincaré map of nine different solutions of (50). The periodic solution is painted in black, while the others are in red with their initial point in green. (b) Plot in blue of ε^ for nine different solutions of the differential equation (50). Plot in red of (73) which shows that it is an upper bound of ε^.
(a)(b)


## 4. On the Squeeze Film Damping Case

In this section, we present some analytic and numerical validation on the existence and linear stability of periodic solutions for the Nathanson model under the squeeze film damping effect. Therefore, the boundary value problem(74)x¨+κ1−x3ẋ+x=βV2t1−x2,with κ=γ/d3>0 and x∈−∞,1.Our approach to the existence of solutions of (74) is again throughout the Theorems 2 and 3. Therefore, from the notation of those theorems,
(75)ct,x=κ1−x3,Gx=1−x2,Kx=x,Ft=βV2t,with Gx>0 for all x in −∞,1. As before, the function ϕ is given by
(76)ϕx=x1−x2,x∈R.Recall thatϕ is monotone non-decreasing in −∞,1/3 and monotone non-increasing in the interval 1/3,1 with a unique local maximum at x=1/3.Theorem 7.
Assume that0<Vmin2≤Vmax2≤4/27β. Then, there exists ζi, ηi with i=1,2 such that
(77)0<η1<ζ1≤1/3,and1/3≤ζ2<η2<1,
whereζi and ηi satisfy
(78)ϕηi=βVmin2,ϕζi=βVmax2,i=1,2.Furthermore,(1)
The problem (74) admits a T-periodic solution ψ2 such that(79)ζ2≤ψ2t≤η2∀t∈0,T,(2)
Let(80)N=κ1−ζ13,a^=maxN,ζ1−η1,M=3κR1−ζ14+1−3η11−η1,with R>0 the unique solution of
(81)R−lnR+1=a^ζ1−η1.Assume that the following conditions hold:(82)M≤π/T2andN≤HL∗=L∗−ML∗cotTL∗2,with L∗∈M,π/T2 the unique value that satisfies H′L=0 which is equivalent to
(83)sinTL∗=TL∗L∗−ML∗+M.Then, there exists a periodic solutionψ1 of (74) such that
(84)η1≤ψ1t≤ζ1,∀t∈0,T.Proof.
Under the existence ofζi,ηi follows directly from the monotonicity properties of the function ϕ in each of the considered intervals.
For part1 by Theorem 2 applied over I=1/3,1, the problem (74) admits constant lower and upper solutions that correspond to ζ2 and η2, respectively. Additionally, there exists a solution ψ2 of (6) such that
(85)ζ2≤ψ2t≤η2,∀t∈0,T.
For part 2 by Theorem3 (part (‡)) applied over I^=0,1/3, the problem (74) admits constant lower and upper solutions that correspond to ζ1 and η1, respectively. Furthermore, since ϕη1=βVmin2, we have
(86)x−βV2t1−x2≤ζ1−βVmin21−η12=ζ1−η1,∀t,x∈0,T×η1,ζ1.
Define,(87)N≔maxR×η1,ζ1κ1−x3=κ1−ζ13,a^≔maxN,ζ1−η1.
From here, we are able to find a unique positive constantR that satisfies
(88)R−lnR+1=a^ζ1−η1.
Now consider the set(89)W=t,x,y:t∈0,T,x∈η1,ζ1,y∈−R,R.
In order to fulfill the all the conditions in‡‡ in Theorem 3, for the value M>0, it is necessary to show that
(90)maxW∂∂xκ1−x3ẋ+x−βV2t1−x2≤3κ1−ζ14R+1−3η11−η1=M.
Indeed, notice that(91)maxW∂∂xκ1−x3ẋ+x−βV2t1−x2=maxW3κ1−x4ẋ+1−2βV2t1−x3≤maxW3κ1−x4R+maxW1−2βV2t1−x3.
The function3κ/1−x4 is monotone increasing in the domain −∞,1; then
(92)3κ1−x4≤3κ1−ζ14,∀x∈η1,ζ1.
Additionally,(93)1−2Fmax1−ζ13≤1−2βV2t1−x3≤1−2Fmin1−η13.
Sinceϕζ1=Fmax and ϕη1=Fmin, then
(94)0≤1−3ζ11−ζ1=1−2ζ11−ζ1≤1−2βV2t1−x3≤1−η11−η1=1−3η11−η1,
becauseζ1∈0,1/3. Hence,
(95)maxW∂∂xκ1−x3ẋ+x−βV2t1−x2≤M,
Then, by Theorem3, there exists a solution ψ1 of (74) such that
(96)η1≤ψ1t≤ζ1,∀t∈0,T.Notice that it is fairly simply to prove the existence ofψ2. However, to prove the existence of ψ1 requires intermediate computations of values that depend not only on parameters such as κ, β and T but also in other intermediate values. Therefore, here we present the steps that allow us to prove the existence of ψ1 for a set of parameters κ,T,Ft,T.
(1)
FindFmin and Fmax.(2)
If0<Fmin≤Fmax≤4/27, find ζ1, η1 in 0,1/3 such that
(97)ϕζ1=Fmax,ϕη1=Fmin.(3)
ComputeN and a^.(4)
Find the positive valueR>0 such that
(98)R−lnR+1=a^ζ1−η1.(5)
Compute the valueM given in Theorem 7(6)
IfM≤π/T2, find L∗ such that
(99)sinTL∗=TL∗L∗−ML∗+M.(7)
Finally, ifN≤HL∗ we can conclude the existence of ψ1.If any of the conditions given in the Steps (3), (12) and (16) are not satisfied, then we cannot use Theorem 7 to conclude the existence of ψ1. Example 2 is based on these steps for different parameters κ,T,Ft,T.


### 4.1. Linear Stability

As a final contribution of this work, we provide some results about the linear stability of any periodic solutionψ1 of (74) located in η1,ζ1. Our approach is based on the analysis of the linear equation corresponding to the given periodic solution. Direct computations shows that the associated Hill’s equation is given by
(100)w¨+κ1−ψ1t3ẇ+qtw=0,withqt=3κ1−ψ1t4ψ̇1t+1−2βV2t1−ψ1t3.Proposition 8.
Under the assumption of Theorem7, then any possible periodic solution ψ1 of (74) located η1,ζ1 is locally asymptotically stable if
(101)1−3ζ11−ζ1>κ21−ζ143R+κ21−ζ12,(102)πT2≥1−3η11−η1+κ21−ζ143R−κ1−ζ1421−η16.Proof.
Under the change of variables,(103)wt=utvt,vt=e−Pt,Pt=12∫κ1−ψ1t3dt,Equation (100) can be written as
(104)u¨+Qtu=0,
withQt given by
(105)Qt=3κ21−ψ1t4ψ̇1t−κ241−ψ1t6+1−2βV2t1−ψ1t3.
Notice that if any solutionut of (104) is bounded, then any solution wt of (100) converges to zero. Indeed, notice that
(106)η1≤ψ1t≤ζ1andψ̇1t<R,∀t∈R,with 0<η1<ζ1≤1/3. Direct computations show that
(107)κt21−η13≤Pt≤κt21−ζ13,∀t∈R,then, vt⟶0 if t⟶∞. Moreover,
(108)−3κR21−ζ14≤3κψ̇1t21−ψ1t4≤3κR21−ζ14,−κ241−ζ16≤−κ241−ψ1t6≤−κ241−η16,and also,
(109)0<1−3ζ11−ζ1≤1−2βV2t1−ψ1t3≤1−3η11−η1,for all t∈R. From here, we deduce the following:
(110)1−3ζ11−ζ1−κ21−ζ143R+κ21−ζ13≤Qt,∀t∈R,Qt≤1−3η11−η1+κ21−ζ143R−κ1−ζ1421−η16,∀t∈R.
From the assumptions (101), (102) follows directly
(111)0<Qt≤πT2,∀t∈R.From the previous computations and Theorem18, we deduce that ψ1 is locally asymptotically stable.Remark 9.
To arrive to the conclusion of Proposition8, we used the results of Theorem 18 that resemble the Lyapunov-Zukovskii’s criteria for stability. We could also arrive to the linear stability of the periodic solution ψ1 of (74) by asking the following conditions:
(112)1−3ζ11−ζ1>γ21−ζ143R+γ21−ζ13,2T2≥1−3η11−η1−γ241−η16.We arrive to this conclusion by Theorem18 with the conditions that resemble the Lyapunov-Borg’s criteria for stability.Example 2.
Following the same definition (71) from Example 1 for parameter A, in Figure 5, we display the combination of parameters that allowed or did not allow to prove the existence of ψ1 for Equation (74) by means of Theorem 7. If the existence is guaranteed, we attempt to prove with Proposition 8 that ψ1 is asymptotically stable. The numerical results of Figure 5 indicate that if we take low values of κ, it seems to be more possible to prove the existence of ψ1 with Theorem 7.Figure 5
(a) Areas in the planeAandT where the conditions for the existence are satisfied (blue) and where they are not (red) with κ=0.3674. (b) Boundary between satisfying and not satisfying the conditions of Theorem 7 for different values of κ.
(a)(b)


## 4.1. Linear Stability

As a final contribution of this work, we provide some results about the linear stability of any periodic solutionψ1 of (74) located in η1,ζ1. Our approach is based on the analysis of the linear equation corresponding to the given periodic solution. Direct computations shows that the associated Hill’s equation is given by
(100)w¨+κ1−ψ1t3ẇ+qtw=0,withqt=3κ1−ψ1t4ψ̇1t+1−2βV2t1−ψ1t3.Proposition 8.
Under the assumption of Theorem7, then any possible periodic solution ψ1 of (74) located η1,ζ1 is locally asymptotically stable if
(101)1−3ζ11−ζ1>κ21−ζ143R+κ21−ζ12,(102)πT2≥1−3η11−η1+κ21−ζ143R−κ1−ζ1421−η16.Proof.
Under the change of variables,(103)wt=utvt,vt=e−Pt,Pt=12∫κ1−ψ1t3dt,Equation (100) can be written as
(104)u¨+Qtu=0,
withQt given by
(105)Qt=3κ21−ψ1t4ψ̇1t−κ241−ψ1t6+1−2βV2t1−ψ1t3.
Notice that if any solutionut of (104) is bounded, then any solution wt of (100) converges to zero. Indeed, notice that
(106)η1≤ψ1t≤ζ1andψ̇1t<R,∀t∈R,with 0<η1<ζ1≤1/3. Direct computations show that
(107)κt21−η13≤Pt≤κt21−ζ13,∀t∈R,then, vt⟶0 if t⟶∞. Moreover,
(108)−3κR21−ζ14≤3κψ̇1t21−ψ1t4≤3κR21−ζ14,−κ241−ζ16≤−κ241−ψ1t6≤−κ241−η16,and also,
(109)0<1−3ζ11−ζ1≤1−2βV2t1−ψ1t3≤1−3η11−η1,for all t∈R. From here, we deduce the following:
(110)1−3ζ11−ζ1−κ21−ζ143R+κ21−ζ13≤Qt,∀t∈R,Qt≤1−3η11−η1+κ21−ζ143R−κ1−ζ1421−η16,∀t∈R.
From the assumptions (101), (102) follows directly
(111)0<Qt≤πT2,∀t∈R.From the previous computations and Theorem18, we deduce that ψ1 is locally asymptotically stable.Remark 9.
To arrive to the conclusion of Proposition8, we used the results of Theorem 18 that resemble the Lyapunov-Zukovskii’s criteria for stability. We could also arrive to the linear stability of the periodic solution ψ1 of (74) by asking the following conditions:
(112)1−3ζ11−ζ1>γ21−ζ143R+γ21−ζ13,2T2≥1−3η11−η1−γ241−η16.We arrive to this conclusion by Theorem18 with the conditions that resemble the Lyapunov-Borg’s criteria for stability.Example 2.
Following the same definition (71) from Example 1 for parameter A, in Figure 5, we display the combination of parameters that allowed or did not allow to prove the existence of ψ1 for Equation (74) by means of Theorem 7. If the existence is guaranteed, we attempt to prove with Proposition 8 that ψ1 is asymptotically stable. The numerical results of Figure 5 indicate that if we take low values of κ, it seems to be more possible to prove the existence of ψ1 with Theorem 7.Figure 5
(a) Areas in the planeAandT where the conditions for the existence are satisfied (blue) and where they are not (red) with κ=0.3674. (b) Boundary between satisfying and not satisfying the conditions of Theorem 7 for different values of κ.
(a)(b)


## 5. Conclusions and Outlook

In this work, we have rigorously shown the existence of at least two positive periodic solutions for the Nathanson model under squeeze damping forces, as a direct consequence of a periodic voltage load with a maximum value underV0=4/27β known as pull-in voltage, which is precisely the critical voltage associated with the pull-in phenomenon. The location and the L∞-norm of these solutions are provided, and we have also been able to give an algorithm to show the numerical conditions of Theorem 7. We also note that these analytical and numerical computations could be reproduced for other types of MEMS devices and new algorithms could be developed to show the existence and linear stability. Future work could explore the upper boundaries of the number of positive periodic solutions, and more interestingly, the possibility of periodic oscillations with a negative or nonconstant sign, for example, for the comb-drive model under squeeze damping force. We also revisited the Nathanson model under linear damping. Although this problem is considered in [7–9], Theorem 5 improves the existence and stability results as a result of appropriate conditions over the voltage load and the viscous damping coefficient c, providing new and significant knowledge of the dynamics of this model.



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