DYNAMICS OF A CANONICAL ELECTROSTATIC MEMS/NEMS ...

DYNAMICS OF A CANONICAL ELECTROSTATIC MEMS/NEMS SYSTEM SHANGBING AI∗ AND JOHN A. PELESKO

†

Abstract. The mass-spring model of electrostatically actuated microelectromechanical systems (MEMS) or nanoelectromechanical systems (NEMS) is pervasive in the MEMS and NEMS literature. Nonetheless a rigorous analysis of this model does not exist. Here periodic solutions of the canonical mass-spring model in the viscosity dominated time harmonic regime are studied. Ranges of the dimensionless average applied voltage and dimensionless frequency of voltage variation are delineated such that periodic solutions exist. Parameter ranges where such solutions fail to exist are identified; this provides a dynamic analog to the static “pull-in” instability well known to MEMS/NEMS researchers. Key words. MEMS, nanotechnology, shooting method, electrostatics AMS subject classifications. 34C15, 34C60, 70K40

1. Introduction. As the characteristic length of engineering systems approaches the micro or nanometer scale the role of electrostatics grows correspondingly. Often perceived as a nuisance in the macro-world, such as in the case of the destruction of sensitive electronic circuits due to electrostatic discharge (ESD),

1

electrostatic forces are increasingly being

used to provide accurate, controlled, stable locomotion for micro and nanoelectromechancical systems (MEMS or NEMS). In this approach voltage differences are applied between mechanical components of the system. This induces a Coulomb force between components which is varied in strength by varying the applied voltage. This technique is already employed in devices such as accelerometers, [5], optical switches, [6], microgrippers, [8], micro force gauges, [19], transducers, [4], micro pumps, [16] and nanotweezers, [12]. In order to understand the operation of such devices researchers in the MEMS and NEMS communities have relied upon idealized mathematical models. The typical approach, first introduced into the literature by Nathanson in 1967, [14], is to create a “lumped” massspring model. Here, the elastic behavior of the system is represented by a linear spring while ∗ School

of Mathematics and CDSNS, Georgia Institute of Technology, Atlanta, Georgia, 30332

([email protected]). † School of Mathematics and CDSNS, Georgia Institute of Technology, Atlanta, Georgia, 30332 ([email protected]). 1 ESD is no laughing matter. Before the advent of nonflammable anesthetics an errant spark from a doctor’s scalpel would sometimes ignite the ether in a patient’s lungs, rendering the operation a failure. 1

2

S. Ai and J.A. Pelesko

electrostatic forces are computed using a simple parallel plate capacitor approximation. This mass-spring model has persisted in the MEMS/NEMS literature and has been rediscovered and discussed by numerous authors, [12, 9, 15, 18, 7, 13, 11, 17]. Nevertheless, the mathematical analysis of this canonical model has remained primitive. Typically, authors have restricted their attention to steady-state solutions, [12], relied upon numerical simulation for dynamical information, [7], or have utilized perturbation methods to study approximate dynamics in some region of parameter space, [18]. In this paper we begin to remedy this situation by providing a rigorous analysis of the viscosity dominated time harmonically forced mass-spring MEMS/NEMS model. While this analysis does not capture the dynamics of every possible MEMS or NEMS device, it is relevant for the study of devices such as micropumps, [16], microgrippers, [8], or nanotweezers, [12], which operate in the viscosity dominated regime. In Section 2, for the convenience of the reader we provide a brief derivation of the model. In Section 3, we consider the situation where inertial forces are completely negligible. In this case, the model is reduced to a nonlinear first order non-autonomous ordinary differential equation. We study the existence of periodic solutions to this equation. We determine ranges of the dimensionless applied voltage and dimensionless forcing frequency for which such solutions exist. We show that outside of these ranges the model has no solution. This is the dynamic analog of the static instability well known to MEMS/NEMS researchers as the “pull-in” or “snap-down” instability. In this instability, when a constant applied voltage is increased beyond a certain critical voltage there is no longer a steady-state configuration of the device where mechanical members remain separate. Here in the dynamic situation, the device cannot be operated in an oscillatory mode if the mean applied voltage is too large or the forcing frequency is too small. In Section 4, we consider a situation where inertial forces are small, but non-negligible. In this case, the model becomes a nonlinear second order non-autonomous ordinary differential equation. Again we investigate periodic solutions of this equation and determine criteria necessary for such solutions to exist. Finally, in Section 5, we discuss the implications of our analysis for MEMS/NEMS device behavior. 2. Formulation of the Canonical Model. The system sketched in Figure 2.1 represents a “lumped” approximation of a typical electrostatically actuated MEMS/NEMS device. The governing equation for this system is m

d2 x = Fs + Fd + Fe . dt02

(2.1)

3

Canonical MEMS/NEMS Model

Spring Dashpot

u Mass, m

L

V

Fig. 2.1. Sketch of the damped mass-spring system.

Here, v is the displacement of the top plate from the top wall and m is the top plate’s mass. We assume that the bottom plate is held in place. The forces acting on our system are the spring force, Fs , a damping force represented by the dashpot in Figure 2.1, Fd , and the electrostatic force, Fe , due to the applied voltage difference between the plates. We assume that the spring is a linear spring and follows Hooke’s law Fs = −k(x − l)

(2.2)

where l is the rest length of the spring and k is the spring constant. We assume that damping is linearly proportional to the velocity, that is Fd = −a

dx dt0

(2.3)

and compute the electrostatic force by treating the plates in Figure 2.1 as infinite parallel plates. This yields Fe =

1 0 AV 2 cos2 (Ωt0 ). 2 (L − x)2

(2.4)

Here, 0 is the permittivity of free space, A is the area of the plates, V is the average applied voltage and Ω is the frequency at which the applied voltage is varied. Inserting equations

4


(2.2), (2.3) and (2.4) into equation (2.1) yields m

dx 1 0 AV 2 d2 x + a + k(x − l) = cos2 (Ωt0 ). dt02 dt0 2 (L − x)2

(2.5)

We recast this equation in dimensionless form by introducing a dimensionless length scale x=

y−l L−l

(2.6)

k 0 t. a

(2.7)

and dimensionless time scale t=

Introducing equations (2.6) and (2.7) into equation (2.5) yields λ 1 d2 y dy +y = + cos2 (γt). α2 dt2 dt (1 − y)2

(2.8)

The dimensionless parameter α may be interpreted as a damping coefficient which measures the relative strength of the viscous damping force as compared to the spring force. The dimensionless parameter λ measures the relative strength of electrostatic and elastic forces in our system. The dimensionless parameter ω is the ratio of damping and forcing time scales. When damping effects dominate over inertial effects, we expect the parameter α to be large. If inertial effects are completely negligible, we send α → ∞ and study the reduced model λ dy +y = cos2 (γt). dt (1 − y)2

(2.9)

This situation is studied in the next section. In Section 4, we return to the case where α is large, but inertial effects are not completely negligible and we study equation 2.8. 3. The Viscosity Dominated Case. In this section we study the periodic solutions of equation (2.9). We restrict our attention to solutions where y < 1 as y > 1 implies that the top plate in Figure 2.1 has passed through the bottom plate. It is convenient to introduce a factor of one-third via a simple rescaling and study u0 = We are interested in the

λ cos2 ( ωt 1 2 ) −u+ , 3 (1 − u)2

2π ω -periodic

0 0.

Let n 1 π

r arccos

4 2 , 27λ

λ − 4 2 o 27 . 1 1 + 12 λπ 3

If 0 < ω < ω0 , then (3.2)-(3.3) has no solution. Proof. We first show (i). Assume that (3.2)-(3.3) has a solution w with 0 < w < 1 on [0,

2π ω ].

2

4 27

It follows that w 3 − w ≤ 2π 0 = w( ) − w(0) = ω

Z 0

on [0, 2π ω ], and so

2π ω

2

(w 3 − w − λ cos2 ωt) dt
ωt − 16 ω 3 t3 for t > 0, we obtain w(t) ≤ 1 +

λ 1 4 1 4 t− t + sin ωt < 1 + ( − λ)t + λω 2 t3 . 27 2 ω 27 12

Notice by the definitions of t¯ and ω0 that false. Then we evaluate (3.4) at t = of T . Hence, T
0 whenever w(t) =

8 27

0 for t ∈ [0, 2π ω ] and w (t) < 0 whenever w(t) = 1

8 8 and t ∈ [0, ωπ ) ∪ ( ωπ , 2π ω ], it follows that (1 − δ, 1] ⊂ A and [ 27 , 27 + δ) ⊂ B for sufficiently

small δ > 0. The continuity of the solutions with respect to initial data yields that A and 8 , 1]. Clearly, A and B are disjoint. Therefore, the connectedness B are open sets of [ 27 8 8 , 1] implies that there exists at least a w ˆ0 ∈ ( 27 , 1) \ (A ∪ B) such that of the interval [ 27

ˆ0 ) = w ˆ0 . We also have w( 2π ω ,w on the lines w = 1 and w =

8 27

8 27

< w1 < 1 on [0, 2π ω ], for otherwise the vector fields of (3.2)

lead to either w( 2π ˆ0 ) > 1 or w( 2π ˆ0 ) < ω ,w ω ,w

8 27 ,

contradicting

the definition of w ˆ0 . Let w1 (t) := w(t, wˆ0 ), which gives the existence of w1 . To show the existence of w2 we define another two sets: o o n n 2π 2π 8 8 A0 = w0 ∈ [0, ] : w( , w0 ) > w0 , B 0 = w0 ∈ [0, ] : w( , w0 ) < w0 . 27 ω 27 ω 0 Notice that w0 (t) < 0 whenever w(t) = 0 for t ∈ [0, ωπ ) ∪ ( ωπ , 2π ω ]; it follows that [0, δ] ⊂ B 8 8 and [ 27 − δ, 27 ] ⊂ A0 for sufficiently small δ > 0. Therefore, as above we obtain that there 8 ) \ (A0 ∪ B 0 ) such that w( 2π ¯0 ) = w ¯0 and 0 < w(t, w¯0 ) < is a w ¯0 ∈ (0, 27 ω ,w

8 27

for t ∈ [0, 2π ω ].

Let w2 (t) := w(t, w¯0 ), which gives the existence of w2 . Next we show that w1 and w2 are the only solutions of (3.2)-(3.3). First we observe from the vector fields of (3.2) that any solution w of (3.2)-(3.3) satisfies either 0 < w < or

8 27

8 27

< w < 1 on [0, 2π ω ]. Hence, it suffices to show that w1 and w1 are the only solutions

of (3.2)-(3.3) satisfying

8 27

< w < 1 and 0 < w
0. (3.11) = 1 27 δ 3 3 w012 − 2

w(2π, w012 ) − w012 >

2π

h

w012 −

Then using a shooting argument on I1 as we did in the proof of Theorem 3.2 we find that there is a w0 ∈ I1 = (w01 − δ, w01 + δ) such that w(τ, w0 ) is a solution of (3.6)-(3.7), which we denote by w1 . The estimate (3.5) for i = 1 follows from (3.8). To show the existence of w2 we define I2 = [w021 , w022 ] = [w02 − δ, w02 + δ]. For w0 ∈ I2 , we can similarly show that w(·, w0 ) exists on [0, 2π] and has the following corresponding estimates to (3.8) and (3.9) |w(τ, w0 ) − w02 |
0, (3.14) = 27

w(2π, w021 ) − w021 >

2π

10


and

Z

h 8π 23 8π si − w022 − − λ cos2 ds w022 − 27 27 2 0 Z 2π h 2 8π si 3 − λ cos2 ds 0 2 2 2

for τ ∈ [0, 2π], which implies w(2π) − w(0) > 0, contradicting (3.7). Therefore, any solution w of (3.6)-(3.7) satisfies either w(0) < and hence for τ ∈ [0, 2π], w(τ ) < w(0) + w(τ ) > w(0) −

8 27π

>

8 27

8π 27

provided that w(0) >

< 8 27

8 27

8 1 27 − 2 δ

or w(0) >

provided that w(0)
0, let w1,ω and w2,ω be < w1,ω < 1 and 0 < w2,ω
0 is sufficiently small, then ωλ sin ωt 2π ]. ≤ M ω 2 on [0, w1,ω − W1,ω + − 13 2 ω 2( 3 W1,ω − 1)2

(3.16)

11


(ii) For any 0 < b < 2π, there exists a Mb > 0 such that if ω > 0 is sufficiently small, then w2,ω − W2,ω +

ωλ sin ωt − 13

2( 23 W2,ω

−

1)2

≤ Mb ω 2

on [0,

b ]. ω

(3.17)

Remark 3.1. (a) The higher order approximations to w1 and w2 can be similarly obtained. For simplicity we will not pursue them here. (b) The reason in (ii) we consider 2π 0 2π the compact intervals in [0, 2π ω ) is that W2 ( ω ) = 0 and W2 ( ω ) does not exists.

Proof. For convenience, let wi := Wi,ω and Wi := Wi,ω (i = 1, 2). Make the change of variable of t by τ = ωt and denote w˙ =

dw dτ .

Then wi (i = 1, 2) satisfy

2

ω w˙ = w 3 − w − λ cos2

τ 2

2

on [0, 2π],

and w(0) = w(2π) = 0, Wi (i = 1, 2) satisfy W 3 − W − λ cos2 |w1 − W1 − ωW11 | ≤ M ω 2 where W11 =

1 ˙ a0 W1

− 13

and a0 (τ ) = 23 W1

τ 2

(3.18) = 0, and (3.16) reduces to

on [0, 2π],

(3.19)

− 1.

¯ 1 = W1 + ωW11 + M ω 2 . We wish to show that if M > 0 is For given M > 0, let W sufficiently large and ω is sufficiently small, then 2 ¯ 3 −W ¯ 1 − λcos2 τ ¯˙ 1 > W ωW 1 2

on [0, 2π].

(3.20)

In fact, using the mean value theorem we have 2 ¯ 3 −W ¯˙ 1 ¯ 1 − λcos2 τ ) − ω W (W 1 2 2

2 ˙ 1 − ω2W ˙ 11 = (W1 + ωW11 + M ω 2 ) 3 − W13 − (ωW11 + M ω 2 ) − ω W a1 (τ ) ˙ 1 + M a1 (τ )ω 2 − ω 2 W ˙ 11 , −1 W =ω a0 (τ )

(3.21)

1

where a1 (τ ) = 23 (W1 + θ1 (ωW11 + M ω 2 ))− 3 − 1 and θ1 (τ ) ∈ (0, 1). Again by the mean value theorem, 2 1 a1 (τ ) −1 −1= [(W1 + θ1 (ωW11 + M ω 2 )− 3 − W1 3 ] a0 (τ ) 3a0 (τ ) 4 2 (W1 + θ2 θ1 (ωW11 + M ω 2 ))− 3 θ1 (ωW11 + M ω 2 ), =− 9a0 (τ ) where θ2 (τ ) ∈ (0, 1). Since

8 27

(3.22)

< W1 ≤ 1, it follows that there are constants c1 > 0

independent of ω such that −1 < a0 (τ ) < −c1 , −1 < a2 (τ ) < −c1 , and W1 + θ1 θ2 (ωW11 +

12


M ω2) >

8 27

if ω is sufficiently small. Then, by taking an M sufficiently large, (3.20) follows

from (3.21) and (3.22) immediately for ω sufficiently small. ¯ 1. Observe that (3.20) implies that no solution of (3.18) can exit from the curve w = W Similarly we can show that no solution of (3.18) can exit from the curve w = W1 + ωW11 − M ω 2 . Therefore, the strip |w − W1 − ωW11 | ≤ M ω 2 is a positive invariant set of (3.18). Hence any solution of (3.18) with w(0) ∈ [W1 + ωW11 − M ω 2 , W1 + ωW11 + M ω 2 ] will stay in this strip for all t > 0 and a shooting argument shows that there exists a periodic solution w∗ of (3.18) inside this strip with

8 27

< w∗ (0) ≤ W1 (0) + ωW11 + M ω 2 < 1 if ω is sufficiently 8 27

small. Since w˙ < 0 on the line w = 1, it follows that

< w∗ < 1 on [0, 2π]. By the

uniqueness we have proved in Theorem 3.2 it follows that w∗ ≡ w1 . This shows (3.19) and hence (i). To show (ii), under the above transformation we need to show that for given 0 < b < 2π, there is a Mb > 0 such that if ω is sufficiently small, then |w2 − W2 − ωW21 | ≤ Mb ω 2 − 13

where W21 = ( 23 W2

on [0, b]

˙ 2 . In order to do this, we first claim that for given 0 < − 1)−1 W

8 − W2 (0), W2 (b + δ0 )}, if ω is sufficiently small, then δ0 < 2π − b and 0 < δ1 < min{ 27

W2 (b + δ0 ) − δ1 < w2 < W2 (0) + δ1 on [0, b + δ0 ]. Assume that w2 ≥ W2 (0) + δ1 at some τˆ ∈ [0, b + δ0 ]. Since w˙ > 0 on the line w = W2 (0) + δ1 , it follows that w2 stays above this line for t ∈ (ˆ τ , 2π] and so W2 (0) + δ1 ≤ w2
c2 on [ˆ τ , 2π] where

2 3

2τ ˆ 8 c2 = min{W2 (0)+δ1 ≤w≤ 27 ,0≤τ ≤2π} (w − w − λcos 2 ). Hence, w2 (t +

contradicting that w2
w2 (tˆ) + 1 > 1,

This shows that w2 < W2 (0) + δ1 on [0, b + δ0] if ω is sufficiently

small. Similarly, we can show that w2 > W2 (b + δ0 ) − δ1 on [0, b + δ0 ] if ω is sufficiently small. This shows the claim. ¯ 2 = W2 + ωW21 + (M − 1)ω 2 . We can show Now we are ready to show (ii). Let W similarly that there exists M > 0 which might depend on b and δ0 such that for sufficiently 2 ¯ 3 −W ¯ 2 − λcos2 τ on [0, b + δ0 ]. The reason that the inequality here has ¯˙ 2 < W small ω, ω W 2 2 − 13

opposite sign to that in (3.20) is 1 − 23 W2

¯ 2 . Then by the mean value < 0. Let z = w2 − W

theorem and the above the claim we get for sufficiently small ω ω z˙ >

h2 3

i 1 (W2 + θ3 (ωW21 + (M − 1)ω 2 ))− 3 − 1 z ≥ c3 z

on [0, b + δ0 ]

(3.23)

for some constant c3 > 0 independent of M and ω. Assume that there exists a τ¯ ∈ [0, b] τ ) ≥ ω 2 . Integrating (3.23) over such that w2 ≥ W2 + ωW21 + M ω 2 at τ = τ¯. Then z(¯

13


[¯ τ , τ¯ +

3 c3 ω| ln ω|]

we obtain that z(¯ τ+

3 c3 ω| ln ω|)

>

1 ω

→ ∞ as ω → 0, contradicting the

¯ 2. boundedness of w2 and W To show w2 > W2 + ωW21 − M ω 2 , we let W 2 = W2 + ωW21 − (M − 1)ω 2 . Similarly, we 2

˙ 2 > W 3 − W 2 − λcos2 τ and so ω z˙ > c4 z on [0, b + δ0 ] for some constant c4 > 0 obtain ω W 2 2 independent of ω, where z = W 2 −w2 . If there is τ¯ ∈ [0, b] such that w2 ≤ W2 +ωW21 −M ω 2 at τ¯, then z(¯ τ ) ≥ ω 2 and then by integration we obtain z(¯ τ + c34 ω| ln ω|) >

1 ω

→ ∞ as ω → 0,

again contradicting the boundedness of w2 and W 2 . This completes the proof of Theorem 3.4. Theorem 3.5. Assume that 0 < λ
0 such that if ω is sufficiently large λ sin ωt M ≤ 2 wi (t) − w0i − 2ω ω

for t ∈ [0,

2π ]. ω

(3.24)

Proof. We only show (3.24) for i = 1. It suffices to show that λ sin τ ≤ M 2 w1 (τ ) − w01 − 2 where again τ = ωt and =

for τ ∈ [0, 2π],

(3.25)

1 ω.

Let w0 := w1 (0). Since |w1 (τ ) − w0 | ≤

8π 27

for τ ∈ [0, 2π], it follows that w1 (τ ) =

w0 + O() for τ ∈ [0, 2π] as → 0, where the constant in O() does not depend on τ . Then integrating (3.6) on [0, 2π] yields, as → 0, 2 1 0 = w1 (2π) − w0 = 2π(w03 − w0 − λ) + O(2 ), 2

and so 2 1 w03 − w0 − λ = O(). 2

Next, let e1 (τ ) = w1 (τ ) − w0 + 2

λ sin τ 2 .

e˙ 1 (τ ) = ( w13 − w1 −

(3.26)

Then e1 satisfies

2 λ 1 ) = [(w03 − w0 − λ) + O()]. 2 2

It follows from (3.26) that e˙ 1 (τ ) = O(2 ) for τ ∈ [0, π] if is sufficiently small. Then using e1 (0) = 0 we get e1 (τ ) = O(2 ) on [0, 2π] and so w1 (τ ) = w0 −

λ sin τ + O(2 ) 2

for τ ∈ [0, 2π].

(3.27)

14


To complete the proof, we are going to show that w0 = w01 + O(2 ) as → 0. First, it R 2π follows (3.27) and 0 sin τ dτ = 0 that 0 = w1 (2π) − w0 Z 2π λ sin 2τ λ sin 2τ 2 τ + O(2 )) 3 − (w0 − + O(2 )) − λ cos2 ] dτ = [(w0 − 4 4 2 0 h 2 i 1 = (w03 − w0 − λ)π + O(2 ) , 2 and so 2 1 w03 − w0 − λ = O(2 ). 2

(3.28)

Now, let w0 = w01 + w11 . It follows from (3.28) that w0 → w01 and so w11 = o(1) as 2

3

→ 0. It then follows that w03 = (w01 + w11 ) 2 = w01 + (3.28), we have

2

1

3 3w01

2 w11 + O(w11 ). Hence, from

2 2 2 1 2 2 3 O(2 ) = w01 − w01 − λ + 1 − 1 w11 + O(w11 ) = 1 − 1 w11 + O(w11 ). 2 3 3 3w 3w 01

Since

2

1

3 3w01

01

− 1 6= 0, it follows that w11 = O(2 ), and so w0 = w01 + O(2 ). Substituting this

into (3.27) gives (3.25). This completes the proof of Theorem 3.5.

4. Non-negligible Inertial Effects. In this section we return to our basic model, equation (2.5), ignore damping terms and scale the time variable with respect to the forcing frequency. That is, we introduce the scalings t = Ωt0 2

v=

L−u . L−l

(4.1)

This yields 2 v 00 = 1 − v −

λ cos2 2t , v2

v>0

(4.2)

where λ is as before and is the ratio of the forcing frequency to the natural frequency of the oscillator. We use vα to denote the solution of (4.2) satisfying v(0) = α,

v 0 (0) = 0.

(4.3)

It is easy to check that if vα0 (π) = 0, then vα is symmetric about t = π and so vα is a 2π-periodic solution of (4.2). In this section we shall show the existence of such solutions by shooting arguments similar to those used in [1, 2, 3, 10].

15


Before stating our main result, we introduce some notation. For given 0 < λ
V+ (0) be a number satisfying λ 1 λ 1 . b − b2 + = V− (0) − V−2 (0) + 2 b 2 V− (0) Note that b is the maximal value of the homoclinic orbit of the equation 2 v 00 = 1 − v −

λ v2

which satisfies 12 (v 0 )2 = v − V− (0) − 12 (v 2 − V− (0)2 ) + ( vλ2 − V−λ(0)2 ). The main result of the this section is: Theorem 4.1. Assume that 0 < λ
0 such that if 0 < < 0 , then for each positive integer n with 2 ≤ n ≤

K ,

(4.2) has at least one 2π-periodic solution vn, such that

vn, has exactly n maxima in (0, π] with t = π being the last one. (ii) For any given integer n ≥ 2 and 0 < < min{0 , K n }, let vn, be the solution given in (i). The following statements hold: (a) vn, (0) → 0 as → 0. (b) Let tˆj and tj with tj < tˆj (j = 1, · · · , n − 1) be the minima and the maxima in (0, π] of vn, respectively. Then tj , tˆj → π, vn, (tj ) → 0 and vn, (tˆj ) → 2 (j = 1, · · · , n) as → 0. (c) For any given interval [c, d] ⊂ (0, π), there exists an M := Mc,d > 0 (which is independent of n and ) such that if is sufficiently small, then |vn, − V− | ≤ M 2 on [c, d]. Remark 4.1. In above theorem, (i) asserts that the number of the 2π-periodic solutions

16


of (4.2) goes to infinity as → 0, while (ii) asserts that if is sufficiently small, then vn, has one spike near 0 and all other spikes lying in [0, π] near t = π. In order to show Theorem 4.1 we need a series of the lemmas. Lemma 4.2. Assume that v is a positive solution of (4.2) on [0, π] and t1 and t2 are two successive minima (or maxima) of v in [0, π]. Then v(t1 ) < v(t2 ). Proof. Assume that t1 and t2 are two successive minima of v in [0, π]. Let t0 ∈ (t1 , t2 ) be the maximum of v. Since function

1 v(t)

1 v(t)

is strictly monotone on [t1 , t0 ] and [t0 , t2 ], it follows that the

has an inverse on each of these two intervals. Let t− = t− (v) and t+ = t+ (v) be

the inverses of

1 v

on [t1 , t0 ] and [t0 , t2 ] respectively. Multiplying (4.2) by v 0 and integrating

over [t− (v), t+ (v)] we get Z

1 1 2 0 2 (v ) (t+ ) − 2 (v 0 )2 (t− ) = λ 2 2 Since cos2

t 2

1 v(t+ ) 1 v(t0 )

cos2

t+ (η) t− (η) − cos2 dη. 2 2

is decreasing on [0, π], it follows that the integral in the above formula is negative,

which implies that v 0 (t+ ) vanishes before v 0 (t− ) and so v(t1 ) < v(t2 ). The other case can be proved similarly. Lemma 4.3. Assume that 0 < λ
0.

(4.4)

and at τ = 0.

(4.5)

Let V0 be the solution of V¨ = 1 − V − Vλ2 with V (0) = α and V˙ (0) = 0. Then V0 is a periodic solution. Let T0 be the least positive period of V0 . It follows that the maximal value of V0 is α and the minimal value of V0 reaches at τ =

T0 2

with V0 ( T20 ) > V− (0). Take a δ1 > 0 satisfying

be the solution of V¨ = 1 − V − Vµλ2 V0 ( T20 ) − 3δ1 > V− (0) and V0 ( T20 ) + 3δ1 < V+ (0). Let Vα,µ ˜ satisfying V (0) = α ˜ and V˙ (0) = 0 with α ˜ ∈ [V0 ( T20 ) − δ1 , V0 ( T20 ) + δ1 ] and µ ∈ [1 − δ2 , 1], are all periodic. The existence of where δ2 > 0 is chosen sufficiently small such that Vα,µ ˜ δ2 follows from the phase-plane diagrams of the equations V¨ = 1 − V −

µλ V2.

Furthermore,

˙˜ | ≤ M ¯ , where ¯ > 1 such that Tα,µ ≤ T¯ and |Vα,µ there exist constants T¯ > 0 and M ˜ ˜ | + |Vα,µ is the least positive period of Vα,µ Tα,µ ˜ ˜ .

17


We use mathematical induction and perturbation methods to show the existence of the local minima τn ∈ (0, π) (1 ≤ n ≤ N ) of v such that 0 < τn+1 − τn ≤

3 Tn , 2

¯ 1 , |v(τn+1 ) − v(τn )| ≤ M

¯ 1 2 |v − Vn | ≤ M

on [τn , τn+1 ], (4.6)

where

δ¯ T0 T0 N = max n ≥ 1 : τn < , v(τn ) ∈ V0 ( ) − δ1 , V0 ( ) + δ1 2 2 √ π ¯ 1 > 0 is a constant independent of and n, Vn is with δ¯ = min{ 2 , 2 arccos 1 − δ2 } and M the solution of λ cos2 τ2n , V¨ = 1 − V − V2

V (τn ) = v(τn ),

V˙ (τn ) = 0,

and Tn is the least period of Vn . If τn exist, then it follows from the definition of δ¯ that ˜ = v(τn ) and µ = cos2 Vn = Vα,µ ˜ with α

τn 2

∈ (1 − δ2 , 1] for all 1 ≤ n ≤ N . Hence, Tn ≤ T¯

¯. and |Vn | + |V˙ n | ≤ M We first show the existence of τ1 . Let w0 = v − V0 . Then w0 satisfies w0 (0) = w˙ 0 (0) = 0 and

λ sin2 τ λ (2V0 + w0 ) 2 w + =: a0 (τ )w0 + b0 (τ ). w ¨0 = − 1 − 0 (w0 + V0 )2 V02 (w0 + V0 )2

Define T00 = sup{τ ∈ (0, 23 T0 ) : |w0 | + |w˙ 0 | < δ1 on (0, τ )}. Then on [0, T00 ), we have |a0 (τ )| ≤ M0 and |b0 (τ )| ≤ M0 2 for some constant M0 > 0 independent of . For τ ∈ (0, T00 ), we use the equivalent integral equations for w0 and w˙ 0 to get Z τ Z τ |w0 (τ )| + |w˙ 0 (τ )| ≤ (|a0 (s)| + 1)(|w0 (s)| + |w˙ 0 (s)|) ds + |b0 (s)| ds 0 0 Z τ ≤ (2M0 + 1) (|w0 (s)| + |w˙ 0 (s)|) ds + M0 T0 2 . 0

From the Gronwall’s inequality we obtain

q Hence if

Kα

for some constant Kα > 0 independent of . This completes the proof

of the lemma. Lemma 4.4. √ t t V− (t) = λ cos 1 + O cos 2 2

Proof. since V−2 (t) − V−3 (t) − λ cos2

t 2

as t → π.

(4.8)

= 0 and V− (π) = 0, it follows that V− (t) → 0 as

t → π, and so √ √ λ| cos 2t | t = λ| cos | 1 + O(V− (t)) V− (t) = p 2 1 − V− (t)

(4.9)

which implies that V− (t) = O(| cos 2t |) as t → π. Inserting this estimate into (4.9) gives (4.8). Lemma 4.5. Let vα be a positive solution of (4.2)-(4.3) with α > 1. (i) There exists α ¯ such that if α ≥ α, ¯ then vα0 < 0 as long as v > 0 in (0, π]. (ii) There exists a positive constant M independent of and vα such that if tˆ is a local maximum of v with 0 < tˆ ≤ π, then vα (tˆ) ≤ M . Proof. Let v = vα . Assume that there is a t0 ∈ (0, π] which is the first time where 0

v (t0 ) = 0. Since v(0) = α > 1, we have v 0 < 0 in (0, t0 ) and v(t0 ) > V− (t0 ). Multiply v 0 on

19


both sides of (4.2) and integrate over [0, t0 ] to obtain λ cos2 t21 λ cos2 t20 λ 1 1 + − v(0) − v 2 (0) = v(t0 ) − v 2 (t0 ) − 2 2 v(t0 ) v(0) 2

Z

t0

0

sin t dt. v

(4.10)

Using v(t0 ) > V− (t0 ) and Lemma 4.4 we easily see that each term in the right-hand side of (4.10) is bounded with a bound independent og t0 , v and , which implies v(0) = α is uniformly bounded independent of v, t0 and . This shows (i). t) be the closest minimum of v to tˆ. Again multiply v 0 on To show (ii), we let t ∈ (0, ˆ both sides of (4.2) and integrate over [t, tˆ] to obtain λ cos2 1 1 v(tˆ) − v 2 (tˆ) = v(t) − v 2 (t) + 2 2 v(tˆ)

tˆ 2

−

λ cos2 v(t)

t 2

+

λ 2

Z

tˆ t

sin t dt. v

(4.11)

By the similar reason to the above we know that the right-hand side of (4.11) is uniformly bounded with respect to and v. The assertion (ii) follows. Proof of (i) of Theorem 4.1. Take a α0 ∈ (1, b). Then Lemma 4.3 implies that if < α0 , then vα0 has at least

K α0

many maxima 0 < tˆ1 < tˆ2 < · · · < π. Lemma 4.2 shows

that all maxima in (0, π) lie above the line v = 1 and so vα000 (tˆn ) < 0. Now we fix an < 0 . We consider the change in these maxima as α increases from α0 . Again Lemma 4.2 shows that all maxima of vα lying in (0, π) lie above the line v = 1 and so vα00 (tˆn ) < 0. Hence, by (ii) of Lemma 4.5 and the implicit function theorem as we increase α from α0 , each tˆn varies continuously with α as long as it lies in (0, π]. However, if α > α ¯ , then from (i) of Lemma 4.5 we know that vα does not have any maximum at all. Therefore, a simple shooting argument shows that there exists at least an ¯ ) for each n ≤ N such that tˆn (αn ) = π. The corresponding solution vαn has αn ∈ (α0 , α exactly n maxima in (0, π], which gives the existence of vn, as described in (i) of Theorem 4.1. To show (ii) of Theorem 4.1 we need another lemma: Lemma 4.6.

For any closed interval [c0 , d0 ] ⊂ (0, π), there exist constants M :=

Mc0 ,d0 > 0 and γ = γc0 ,d0 > 0 such that if v is a solution of (4.2) satisfying v
0 as small as we want and t¯ ∈ I such that w ≥ V− + M 2 at t = t¯. Let w = v − v1 . We first assume that w0 (t¯) ≥ 0. Since w(t¯) ≥ 2 , it follows easily that there exist a constant γ > 0 which may depends on c0 and d0 but is independent of and v such that 2 w00 ≥ γ 2 w and w0 > 0 on (t¯, d0 ]. Multiplying 2 w00 ≥ γ 2 w by w0 and integrate on [t¯, d0 ] we obtain

s

γ w > 0

w2 (t) − w2 (t¯) +

2 0 2 ¯ (w ) (t). γ2

Integrating the above inequality one more time on [t¯, d0 ] and using d0 − t¯ ≥ γ3 | ln |, w(t¯) ≥ 2 we get

s w(d0 ) +

h i 2 1 ¯ w2 (d0 ) − w2 (t¯) + 2 (w0 )2 (t¯) > w(t¯) + w0 (t¯) e γ (d0 −t) > , γ γ

which contradicts v (d0 )

1

whcih again

gives a contradiction. This shows (4.13). To show the other side inequality of (4.12), we use fact that there exists a constant M > 0 such that if is sufficiently small and v2 = V− − (M − 1)2 , then 2 v200 > 1 − v2 −

λ cos2 v22

t 2

on

[c0 , d0 ]. By letting w = v2 − z, the similar contradictions can be obtained. This completes the proof of Lemma 4.6. Proof of (ii) of Theorem 4.1. We denote v := vn, . We first show that v (0) → b as → 0. Assume that it is false. Then there is a subsequence j such that j → 0 and ˜ 6= b as j → ∞. From the equation (4.4) we see that on compact τ interval, as vj (0) → α j → ∞, vj tends uniformly to the solution V of λ V¨ = 1 − V − 2 , v

V (0) = α ˜ , V˙ (0) = 0.

(4.14)

Then, the phase-plane diagrams of (4.14) show that that if j is sufficiently large, then vj either has more than n oscillations in an O(j )-neighborhood of t = 0 provided that α ˜ < b or ˜ > b. Those contradictions vj does not oscillate at all as long as it is positive provided that α show α ˜ = b, so do (a). Using the similar phase plane arguments as above, we conclude that any minima of v must be close to V− if is sufficiently small. For otherwise, it would result in that v


21

oscillate more than n times as → 0. Since, by Lemma 4.2, the successive minima increase and V− decreases, it follows that all minima must occur near t = π. Hence, lim→0 tj = 0, lim→0 tˆj = 0 and lim→0 v (tj ) = V− (π) = 0 (j = 1, 2, · · · , n). To show that for each j = 1, 2, · · · , n, v (tˆj ) → 2 as → 0 , we replace in (4.11) tˆ by tˆj and t by tj and use Lemma 4.4 together with v > V− on [tj , tˆj ] and tj → 0 as → 0 to obtain that each term in the right-hand side of (4.11) goes to zero as → 0. Hence, v (tˆj ) − 12 v2 (tˆj ) → 0 as → 0, which implies lim→0 v (tˆj ) = 2. This shows (b). Finally, by the phase plane arguments we know that v → V− as → 0 uniformly on [c0 , d0 ] with c0 =

c 2

and d0 =

π+d 2 ,

which combining with lim→0 t1 = π and Lemma 4.6

yields the assertion in (c). This completes the proof of Theorem 4.1. 5. Discussion. We began by presenting the canonical model of electrostatically actuated MEMS/NEMS devices. We studied the model in two limits. First, in Section 3, we considered the case where inertial terms were negligible relative to damping terms in this system. This led to the study of a non-autonomous nonlinear first order differential equation. For this system, in the unforced case, it is well known that there are no steady-state solutions to the problem if λ > 4/27. That is, if the applied voltage is to large, the “pull-in” phenomena occurs. The principle result of Section 3 is the dynamic analog to the static pull-in phenomena. That is, we have shown that in the forced case, if λ < 4/27 there are exactly two solutions, while if 8/27 > λ > 4/27 there are two solutions for a certain range of forcing frequencies. If λ > 8/27, there are no solutions. Physically, we have shown that if the system is forced with the correct frequency solutions can exist beyond the static pull-in voltage. Next, in Section 4, we considered the case where inertial terms were small but nonnegligible and damping terms could be ignored. This led to the study of a non-autonomous nonlinear second order differential equation. A small parameter, , appeared in front of the inertial terms. The surprising result of this section was that as −→ 0 the number of periodic solutions tended to infinity. This indicated that in the presence of inertial terms, the canonical MEMS/NEMS model has much richer dynamics than in the viscosity dominated case. Acknowledgements Both authors thank the Center for Dynamical Systems and Nonlinear Studies in the School of Mathematics at the Georgia Institute of Technology. The second author was supported under NSF #0071474 and thanks the National Science Foundation.

22

S. Ai and J.A. Pelesko REFERENCES

[1] S. Ai, Multi-pulse orbits for a singularly perturbed nearly-integrable system, to appear in J. Diff. Eq.. [2] S. Ai, On a BVP of a singularly perturbed o.d.e., submitted. [3] S. Ai and S.P. Hastings, A shooting approach to layers and chaos in a forced Duffing equation, I, to appear in J. Diff. Eq.. [4] M.J. Anderson, J.A. Hill, C.M. Fortunko, N.S. Dogan, and R.D. Moore, BroadBand Electrostatic Transducers: Modeling and Experiments, J. Acoust. Soc. Am., 97 (1995), pp. 262–272. [5] M. Bao and W. Wang, Future of Microelectromechanical Systems (MEMS), Sensors and Actuators A, 56 (1996), pp. 135-141. [6] H. Camon, F. Larnaudie, F. Rivoirard and B. Jammes, Analytical Simulation of a 1D Single Crystal Electrostatic Micromirror, Proceedings of Modeling and Simulation of Microsystems, (1999). [7] E.K. Chan, E.C. Kan, R.W. Dutton and P.M. Pinsky, Nonlinear Dynamic Modeling of Micromachined Microwave Switches, IEEE MTT-S Digest, (1997), pp. 1511-1514. [8] P.B. Chu and K.S.J. Pister, Analysis of Closed-loop Control of Parallel-Plate Electrostatic Microgrippers, Proc. IEEE Int. Conf. Robotics and Automation, (1994), pp. 820-825. [9] R. Gao, Z.L. Wang, Z. Bai, W.A. de Heer, L. Dai and M. Gao, Nanomechanics of Individual Carbon Nanotubes from Pyrolytically Grown Arrays, Physical Review Letters, v. 85, no. 3, (2000), pp. 622-625. [10] S.P. Hastings and J.B. McLeod, On the periodic solutions of a forced second-order equation, J. Non. Sci., 1 (1991), pp. 225-245. [11] M. Horenstein, T. Bifano, S. Pappas, J. Perreault and R. Krishnamoorthy-Mali, Real Time Optical Correction using Electrostatically Actuated MEMS Device, J. Electrostatics, v. 46, (1999), pp. 91-101. [12] P. Kim and C.M. Lieber, Nanotube Nanotweezers, Science, v. 286, (1999), p. 2148-2150. [13] J.B. Muldavin and G.M. Rebeiz, 30 GHz Tuned MEMS Switches, IEEE MTT-S Digest, (1999), pp. 1511-1518. [14] H.C. Nathanson, W.E. Newell, R.A. Wickstrom and J.R. Davis, The Resonant Gate Transistor, IEEE Tran. on Elec. Dev., v. 14, no. 3, (1967), pp. 117-133. [15] P. Poncharal, Z.L. Wang, D. Ugarte and W.A. de Heer, Electrostatic Deflections and Electromechanical Resonances of Carbon Nanotubes, Science, v. 283, (1999),pp. 1513-1516. [16] M.T.A. Saif, B.E. Alaca and H. Sehitoglu, Analytical Modeling of Electrostatic Membrane Actuator Micro Pumps, IEEE Journal of Microelectromechanical Systems, 8 (1999), pp. 335-344. [17] J.I. Seeger and B.E. Boser, Dynamics and Control of Parallel-Plate Actuators Beyond the Electrostatic Instability, Transducers ’99, (1999), pp. 474-477. [18] F. Shi, P. Ramesh and S. Murkherjee, Simulation Methods for Micro-Electro-Mechanical Structures (MEMS) with Application to a Microtweezer, Computers and Structures, v. 56, no. 5, (1995), pp. 769-783. [19] H.A.C. Tilmans, M. Elwenspoek and J.H.J. Fluitman, Micro Resonant Force Gauges, Sensors and Actuators A, 30 (1992), pp. 35-53.