Abstract
We provide sufficient conditions for the existence of periodic solutions for an idealized electrostatic actuator modeled by the Liénard-type equation with , , and , (called squeeze film damping force), or , (called linear damping force). If 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 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 . 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.
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 the Nathanson 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. If is the initial distance between the electrodes and both are biased by a voltage , 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 Figure 1, the electrostatic force acting on the movable electrode is expressed as where is an independent variable related to time, is the potential difference between the two plates, is the area of the plate that is facing the other, is the dielectric constant of the medium in between the plates, and is the position of the movable electrode with respect to . The other forces involved are the restoring and the damping force , where represents the induced velocity of the movable electrode. For the restoring force, we have considered a linear stiffness force, in such case with and for the damping force, two types were considered: with . The first comes from simplifying the problem to a moving sphere in fluid at a velocity . 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.
(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: where , represents the induced acceleration of the position . 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 with respect to and with respect to . Therefore, our nondimensional variables and satisfies and the corresponding nondimensional equation from (4) is with
Throughout this document, we consider a DC-AC voltage source of the form with (-voltage source) and with zero average. Voltage will also be nonnegative; therefore, from now on we assume where
When the damping force 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 -periodic solutions of (6) for the non-conservative () and for the conservative case (), 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 Section 2, 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:
for continuous function, where is an open connected set with . 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 is a -periodic function in the variable , then all the solutions of (10) would be also -periodic.
Theorem 1. Let be a lower and an upper solution of the boundary value problem (10) such that . Define
Assume that there exists such that for all in . Then for any solution of (10) such that on there exists such that for all in .
If the assumption in () holds. Let
and assume that there exists such that for all in for some where
Then, the boundary value problem (10) has at least one solution such that
Proof. The existence of such that and Theorem 13 (see Appendix) lead us to the conclusion that there exists such that for any solution of (10) and any solution , of respectively, with in also satisfies for all in and . 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 .
Now, we consider the family of boundary value problems
where , , , be continuous functions, . Define
Following the notation of (10), we have
Regarding the existence of periodic solutions of (18), we lead to the following results.
Theorem 2. Assume that , and is decreasing with isolated critical points in . Then, (18) admits unique constant lower and upper solution and , respectively, such that satisfying Moreover, there exists a solution of (18) such that
Proof. Since is decreasing with isolated critical points in and , then there exist unique solutions , , in for
respectively. Moreover, . Additionally, notice that
which implies
It is equivalent to
Therefore, and they are lower and upper solutions, respectively.
Let
Notice that for all it follows
Then, the positive function satisfies
Hence, satisfies the Nagumo condition and Theorem 12 (see Appendix); we can conclude that there exists a solution of (18) such that
Theorem 3. Let and , , be differentiable functions and continuous.
Assume that , and with isolated critical points in . Then, (18) admits unique constant reversed-ordered upper and lower solutions , respectively, such that and .
If the assumptions in holds, define
and let the unique positive solution of
Assume that there exists such that
where
and there exists such that
Then, the boundary value problem (18) has at least one solution such that
Proof. Suppose that and with isolated critical points in . This implies that there exists only one pair of values in such that Notice that because is monotone non-decreasing function and . Additionally are upper and lower solutions of (18) because which proves part . In order to prove part , first notice that with Then, by Theorem 1 part there exists such that for any solution of (18) such that on we have on and following Theorem 13 and its remark, the value is the unique positive solution of Notice that by the mean value theorem: Therefore, by Theorem 1, the boundary value problem (18) has at least one solution such that
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: where and , , a continuous function, -periodic with respect to and having a continuous partial derivative with respect to . The following notation will be used throughout the rest of the paper.
The positive part of a given a function is defined as (1)Given a pair of function , we write , if for almost every and in a subset of positive measure(2): -periodic function with the norm (3)For some and , denotes the best Sobolev constant in the following inequality:
Let us start with some results over the linear differential operator: where
is a positive constant and , with for some
Proposition 4. Suppose that for some . Then, we have the following conclusions: (1)Each possible -periodic solution of is either trivial or different from zero for each (2)If then cannot admit nontrivial -periodic solutions simultaneously(3) does not admit negative Floquet’s multipliers (i)If satisfies , then does not admit real Floquet’s multipliers, i.e., is elliptic and has no nontrivial -periodic solutionsThe trivial solution is locally exponentially asymptotically stable with rate of exponential decay .
The proof of Proposition 4 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 .
3. An Improvement for the Linear Damping Case
In this section, we consider the Duffing equation:
with This equation corresponds to the Nathanson model (6) with linear damping force , , and . The existence and stability of periodic solutions of (50) have been considered in [9] for the case and also in [7] for . 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)(II) and for all and some Then, Equation (50) has exactly two positive -periodic solutions and such that
with , , the corresponding solutions of
in , respectively. Moreover, is asymptotically stable, and is unstable.
Additionally, if
then is locally exponentially asymptotically stable with rate of exponential decay .
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
In such a case, is given by . Direct computations prove that is monotone increasing in and monotone decreasing in with isolated critical points at and . Therefore, by the assumption and Theorems 2 and 3, it follows directly that the solutions of
provide constant upper solutions , with and constant lower solutions , with , respectively. Moreover,
Step 2: Existence of periodic solutions
Applying Theorem 2, there exists at least one -periodic solution of (50) such that
In order to apply Theorem 15 (part A) on the set
it is necessary to study the condition
for some function such that and some , where . Then, a direct computation shows that
for all . Let , . The previous inequality along with the assumptions (I) and (II) imply the inequality (59) with and for some . This proves the existence of at least one -periodic solution of (50) which is asymptotically stable and such that
if the number of -periodic solutions of (50) between and is finite.
Step 3: Multiplicity of periodic solutions
Assume that and are two different nontrivial -periodic solutions of (50). Define , . Then, is a nontrivial -periodic solution of the equation:
with By the inequality (60), the condition (II) and part 1 of Proposition 4, we conclude that or for all . Therefore,
for all . Now, assume that there exists a third nontrivial -periodic solution of (50). The preceding arguments allow us to assume that
In consequence, the nontrivial -periodic functions , satisfy the equations:
where . Since
it follows that for all . Therefore, by part 2 of Proposition 4, we reach a contradiction. This proves that there is at most two positive -periodic solutions of (50).
To sum up, under the assumptions (I) and (II), Equation (50) has exactly two positive -periodic solutions which are precisely the functions and satisfying
Moreover, by Step 1, is asymptotically stable, and is unstable.
Step 4: Exponential stability
Finally, we want to apply Theorem 16. We need to find a lower bound of for all . Then, direct computations show that
Then, by (60) and the previous inequality, we have
for all . Define ; therefore, . From (53), we can deduce that . Then, by Theorem 16, the -periodic function is exponentially asymptotically stable with rate of exponential decay . This completes the proof.
Remark 6. Condition (53) can be replaced by
where
Additionally, respect to the results over the Nathanson model with constant damping given in [7, 9], the criteria that we illustrated over the function in Theorem 5 have the advantage that considers the norms , 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 of are , and . In order to test different combinations of parameters, let
If then . 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 with Theorem 5.
(a)
(b)
To test the exponentially asymptotical stability property of one of the combination of parameters, let , , and . Following the results depicted in Figure 3, with that combination, it is possible to prove the existence of and it is exponentially asymptotically stable.
Let the error between any other solution of (50) and the periodic solution be defined as
Following the results of Theorem 5, is locally exponentially asymptotically stable with rate of exponential decay . Then, there exists an adequate positive value such that
For this example, Figure 4 depicts in blue for 10 different solutions of (50) and in red with .
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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 with and .
Our approach to the existence of solutions of (74) is again throughout the Theorems 2 and 3. Therefore, from the notation of those theorems, with for all in . As before, the function is given by
Recall that is monotone non-decreasing in and monotone non-increasing in the interval with a unique local maximum at .
Theorem 7. Assume that . Then, there exists , with such that where and satisfy
Furthermore, (1)The problem (74) admits a -periodic solution such that(2)Letwith the unique solution of
Assume that the following conditions hold: with the unique value that satisfies which is equivalent to
Then, there exists a periodic solution of (74) such that
Proof. Under the existence of follows directly from the monotonicity properties of the function in each of the considered intervals.
For part 1 by Theorem 2 applied over , the problem (74) admits constant lower and upper solutions that correspond to and , respectively. Additionally, there exists a solution of (6) such that
For part 2 by Theorem 3 (part ()) applied over , the problem (74) admits constant lower and upper solutions that correspond to and , respectively. Furthermore, since we have
Define,
From here, we are able to find a unique positive constant that satisfies
Now consider the set
In order to fulfill the all the conditions in in Theorem 3, for the value it is necessary to show that
Indeed, notice that
The function is monotone increasing in the domain ; then
Additionally,
Since and , then
because . Hence,
Then, by Theorem 3, there exists a solution of (74) such that
Notice that it is fairly simply to prove the existence of . However, to prove the existence of requires intermediate computations of values that depend not only on parameters such as , and but also in other intermediate values. Therefore, here we present the steps that allow us to prove the existence of for a set of parameters . (1)Find and .(2)If , find , in such that (3)Compute and (4)Find the positive value such that (5)Compute the value given in Theorem 7(6)If , find such that (7)Finally, if we can conclude the existence of .
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 . Example 2 is based on these steps for different parameters .
4.1. Linear Stability
As a final contribution of this work, we provide some results about the linear stability of any periodic solution of (74) located in . 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
Proposition 8. Under the assumption of Theorem 7, then any possible periodic solution of (74) located is locally asymptotically stable if
Proof. Under the change of variables, Equation (100) can be written as with given by Notice that if any solution of (104) is bounded, then any solution of (100) converges to zero. Indeed, notice that with . Direct computations show that then, if Moreover, and also, for all . From here, we deduce the following: From the assumptions (101), (102) follows directly
From the previous computations and Theorem 18, we deduce that is locally asymptotically stable.
Remark 9. To arrive to the conclusion of Proposition 8, 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 of (74) by asking the following conditions:
We arrive to this conclusion by Theorem 18 with the conditions that resemble the Lyapunov-Borg’s criteria for stability.
Example 2. Following the same definition (71) from Example 1 for parameter , in Figure 5, we display the combination of parameters that allowed or did not allow to prove the existence of for Equation (74) by means of Theorem 7. If the existence is guaranteed, we attempt to prove with Proposition 8 that is asymptotically stable. The numerical results of Figure 5 indicate that if we take low values of it seems to be more possible to prove the existence of with Theorem 7.
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(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 under known as pull-in voltage, which is precisely the critical voltage associated with the pull-in phenomenon. The location and the -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 , providing new and significant knowledge of the dynamics of this model.
Appendix
A.1. The Upper and Lower Solution Method
Consider the boundary value problem
where is an open connected set with and , is a continuous function.We have that there exists a constant such that
Definition 10 Lower and upper solution. A function is called lower solution of (A.1) relative to the domain if
for all and .
A function is called upper solutions of (A.1) relative to the domain if all the previous conditions hold with the reverse inequalities.
The lower and upper solutions are well-ordered if
Meanwhile, the lower and upper solutions are in the reversed order if
Given such that , define the set
Definition 11 Nagumo condition. Let continuous. The function satisfies the Nagumo condition on if there exists a positive continuous function such that for all .
Theorem 12. Let and be lower and upper solution of (10) such that for all in . If satisfies the Nagumo condition in , then the problem (5) has at least one solution such that
Theorem 13. Let a continuous function with a continuous partial derivative on the third variable such that in where and is a connected domain. Consider where . Then, for any solution for all in of any of the following problems:
For sake of completeness, we will make the proof for the boundary value problem (A.9) following the ideas of the proof of Proposition I-4.5 in [12]. The procedures for (A.10) and (A.11) are similar.
Proof. Since , then by the mean value theorem, we have that with . Notice that Define such that Let be a solution of (A.9) such that for all in . Suppose that there exists such that . Let be the closest zero to ; then, for all in . Notice that since then, and hence, which is a contradiction.
Suppose that there exists such that . Let be the closest zero to ; then, for all in . Notice that since
then and hence which is a contradiction.
Remark 14. Notice that an alternative definition for the function is with . This allows us to compute as the positive real value that satisfies the inequality. In particular, we can select such that .
A.2. Multiplicity and Stability of Periodic Solution for Duffing Equations
We finish this section showing some results that provide a connection between lower and upper solution method and the multiplicity and stability of periodic solutions of the Duffing equation:
where and , , a continuous function, -periodic with respect to and having a continuous partial derivative with respect to . Consider the linear differential operator: where
is a positive constant and , with for some and . Here, is the best Sobolev constant in the following inequality:
Explicitly (see [17]), (A)If and for some , there exists with and verifying
Theorem 15. Let , a continuous function, -periodic with respect to and having a continuous partial derivative with respect to . Assume that and are a couple of lower and upper solutions, respectively, of the boundary value problem: with and for all .
Then, (A.31) has at least an asymptotically stable solution (which is -periodic) , such that provided that the number of -periodic solutions between and is finite. (B)If , then (A.31) has at least an unstable solution (which is -periodic) such thatprovided that the number of -periodic solutions between and is finite.
Proof. For part (A), the proof can be found in [15, 18], and for part (B) the proof can be found in [19].
Another and more accurate results about the stability of periodic solutions of Duffing equations like (A.31) are found in [14, 16] and indicate the following.
Theorem 16. Assume that exists and satisfies where and are -periodic functions such that and for some . Then (A.31) has a unique and locally exponentially asymptotical stable -periodic solution . And the rate of exponential decay of is .
Remark 17. Following the lines of the proof of Theorem 16 given in [14, 16] it is easy to deduce that the exponential decay of only requires that This follows if we are able to located for all and compute a lower bound of for all
Theorem 18 Stability test for Hill’s equations. Let be a -periodic function such that
Assume that for some , the Hill’s equation:
is stable (elliptic) when
or
Furthermore, the upper bounds for are the best possible.
If , (i.e., for all on a subset of positive measure), Theorem 18 establishes that
then, (A.38) is elliptic, which corresponds to the Lyapunov-Borg’s stability criteria (see [20]). If , Theorem 18 established that if (i.e., is -periodic and ) such that
then (A.38) is elliptic, which corresponds to the Lyapunov-Zukovskii’s stability criteria. The proof of Theorem 18 can be found in [21, 22].
Data Availability
No data is used.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors have been financially supported by the Pontificia Universidad Javeriana Cali, through the convocatoria interna por una universidad transformadora (2022-2023) project No. 2652.