13732


FACTA UNIVERSITATIS  
Series: Mechanical Engineering Vol. 23, No 4, 2025, pp. 649 - 665  

https://doi.org/10.22190/FUME250528023H 

© 2025 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND 

Original scientific paper 

VARIATIONAL APPROACH TO  

MICRO-ELECTRO-MECHANICAL SYSTEMS 

Ji-Huan He1,2,3,4, Jinfeng Ma5, Abdulrahman Ali Alsolami6,  

Chun-Hui He7,8 

1School of Information Engineering, Yango University, Fuzhou, China 
2Department of Mathematical Sciences, Saveetha School of Engineering, SIMATS, 

Chennai, Tamil Nadu, India 
3School of Jia Yang; Zhejiang Shuren University, Hangzhou, China 

4School of Mathematics and Big Data, Hohhot Minzu College, Hohhot, China 
5College of Civil Engineering and Architecture, Jiaxing University, Jiaxing, China 

6Department of Mathematics, Faculty of Science, King Abdulaziz University, Saudi Arabia 
7School of Civil Engineering & Transportation, Foshan University, Foshan, China 

8School of Mathematics, China University of Mining and Technology, Xuzhou China   

ORCID iDs: Ji-Huan He  https://orcid.org/0000-0002-1636-0559  

 Jinfeng Ma  https://orcid.org/0009-0000-2348-9142    

 Abdulrahman Ali Alsolami  https://orcid.org/0000-0002-8189-4987 
 Chun-Hui He  https://orcid.org/0000-0003-0810-5248    

Abstract. This paper presents a novel approach for formulating a variational principle 

tailored to microelectromechanical systems (MEMS) through the utilization of the semi-

inverse method. The resulting variational principle is that of least action, which is of 

considerable significance. The newly presented variational principle has the potential to be 

applied in a number of advantageous ways. One of the primary applications of this approach 

is the determination of the pull-in voltage. The application of this principle allows for a more 

accurate and efficient determination of the pull-in voltage. This is of paramount importance 

for the optimal functioning and optimization of MEMS devices. The enhanced accuracy in 

determining the pull-in voltage enables more precise design and greater reliability of MEMS-

based systems. Additionally, the enhanced computational efficiency allows for the saving of 

valuable time and resources during the design process. Furthermore, the paper addresses the 

topic of fractal MEMS and puts forth a novel approach to fractional differentiation based on 

two-scale fractal differentiation, which is anticipated to facilitate the discovery of new insights 

and optimization strategies for MEMS devices. 

Key words: Semi-inverse method, Pull-in instability, Fractal MEMS, Two-scale fractal 

derivative based fractional derivative, Wearable textiles 

 
Received: May 28, 2025 / Accepted July 08, 2025    
Corresponding author: Ji-Huan He, Jinfeng Ma, Chun-Hui He 

School of Jia Yang, Zhejiang Shuren University, Hangzhou, China 

College of Civil Engineering and Architecture, Jiaxing University, Jiaxing, China 
School of Civil Engineering & Transportation, Foshan University, Foshan, China 

Email: hejihuan@ygu.edu.cn (J.H. He), majinfenglz@163.com (J. M), mathew_he@yahoo.com (C.H. He) 



650 J.H. HE, J. MA, A.A. ALSOLAM, C.H. HE 

1. INTRODUCTION 

Micro-Electro-Mechanical Systems (MEMS) have been instrumental in the modern 

technological landscape, primarily due to their ability to facilitate the development of 

miniature integrated devices that effectively combine mechanical and electrical 

functionalities [1]. Their applications are found in critical fields, such as wearable textiles 

[2]. MEMS sensors have been developed for the monitoring of physiological parameters, 

including heart rate and body temperature, with exceptional precision. Flexible actuators 

have been incorporated into adaptive clothing, thereby empowering it to respond 

dynamically to environmental stimuli. In the domain of biomedical engineering, MEMS 

devices play a pivotal role in facilitating targeted drug delivery and real-time diagnostics. 

These devices leverage their microscale size to interact seamlessly with biological systems, 

a property that has been demonstrated to support inner ear theragnostics [4]. Furthermore, 

MEMS oscillators function as essential components in communication systems, providing 

stable timing signals that are crucial for 6G and subsequent generations [5]. 

Roy et al. [6] developed a generative AI-assisted piezo-MEMS ultrasound device 

tailored for plant dehydration monitoring. This device merges piezo-MEMS technology 

with generative AI to enable efficient tracking of plant water status. Liang and Lin [7] 

introduced a MEMS electrothermal actuator (ETA) featuring highly linear actuation and 

stable characteristics. The actuator under consideration consists of a bilayer cantilever 

array of Au and Si₃N₄, and it is compatible with standard CMOS manufacturing processes. 

The device's surface roughness engineering prevents stiction, exhibits exceptional linearity 

between displacement and squared driving voltage, achieves an ultra-low power 

consumption of 7.5 mW, and maintains stable performance over 100 cycles. Potential 

applications include optical modulation and medical imaging, among others. 

These diverse applications necessitate accurate models to predict dynamic behaviors, 

particularly pull-in instability – a phenomenon where electrostatic forces surpass 

mechanical restoring forces, leading to device failure and compromising operational 

reliability [8]. 

The variational principle has long served as a foundational tenet within the realm of 

theoretical physics, offering a unifying framework for the analysis of system dynamics 

through the minimization of energy. From Hamilton's principle, which governs classical 

mechanics, to its extensions in field theory and quantum mechanics, variational methods 

integrate conservation laws and dynamic constraints into a single formulation [9]. In the 

field of MEMS research, early applications of variational principles centered on linear 

oscillations and elementary geometries [10], drawing on techniques such as the variational 

iteration method [11] and finite element analysis [12]. However, these approaches 

encounter challenges due to nonlinear electrostatic-mechanical coupling, microscale 

effects (e.g., surface tension), and complex fractal geometries. These limitations hinder the 

accuracy of predictions of critical parameters like pull-in voltage [13, 14]. Concurrently, 

Niu has substantiated the robust minimum condition of a variational formulation applicable 

to MEMS systems [15]. 

Against this backdrop, the need for a tailored variational principle for MEMS becomes 

evident [16]. Traditional models fail to capture singularities in displacement fields and 

overlook fractal-induced dynamics, leading to design inefficiencies. This paper addresses 

these gaps by developing a novel variational formulation via the semi-inverse method, with 

applications to pull-in instability analysis and fractal MEMS modeling.   



 Variational Approach to Micro-Electro-Mechanical Systems 651 

2. MEMS OSCILLATOR 

Following the notable developments of the Duffing oscillator and van der Pol oscillator, 

the MEMS oscillator is identified as the subsequent most crucial component in the domain 

of nanotechnology. A MEMS oscillator represents a revolutionary timing device that has 

gained significant prominence within the field of electronics. It employs 

microelectromechanical systems technology to facilitate precise and reliable oscillatory 

behavior. The micromachining technology that emerged in the late 1980s enables the 

fabrication of sensors and actuators with a micron-scale resolution. Such micro transducers 

can be integrated with signal conditioning and processing circuitry to form micro-

electromechanical-systems, which are capable of performing real-time distributed control. 

The system operates periodically in general, but the pseudo-pull-in stability is a subject of 

research study [17, 18]. 

The dimensionless MEMS oscillator can be written as [17]: 

 0
1

m
w w

w
 + − =

−
 (1) 

where w is the dimensionless displacement, 0<w<1, and m is voltage-related parameter.  

This equation arises from the dynamic balance between mechanical restoring forces 

and electrostatic forces in a parallel-plate MEMS structure. When m is small, Eq. (1) has 

periodic solution, however, when m is larger than a threshold value, m*, the pull-in motion 

occurs, and m* is often called as the pull-in voltage. This equation possesses distinctive 

characteristics, namely: 1) zero initial conditions, i.e., w(0)=0 and w’(0)=0; and 2) the 

singularity at w=1. Nevertheless, it can be effectively solved by the homotopy perturbation 

method [19, 20, 21], the variational iteration method [22, 23], and He-Liu’s modified 

frequency formulation [24], albeit with the aid of certain special techniques. The variational 

principle in MEMS systems constitutes a fundamental concept that facilitates 

comprehension and analysis of the behavior of these complex systems (see Ref. [10]). 

To explain Eq. (1) in a mechanical view, we consider the motion for a particle with unit 

mass in a potential field V:  

 
21

( ) ln(1 )
2

V w w m w= + −  (2) 

The variational principle of MEMS oscillator given in Eq. (1) can be written in the form 

 
2 2 21 1 1

( ) ( ) ln(1 )
2 2 2

J w w V w dt w w m w dt
   

 = − = − − −   
   
   (3) 

According to the variational formulation given in Eq. (3), the following identity   

 
2 21 1

ln(1 )
2 2

w w m w H + + − =  (4) 

where H is the Hamilton invariant, it can be calculated by the zero initial conditions, and it 

results in H=0, so we have  

 
2 21 1

ln(1 ) 0
2 2

w w m w + + − =  (5) 



652 J.H. HE, J. MA, A.A. ALSOLAM, C.H. HE 

We introduce a new variable, K, defined as: 

 
21

2
K w= . (6) 

The new variable is the kinetic energy, from Eq. (5), we have:  

 
21

ln(1 )
2

K w m w= − − − . (7) 

For small displacement, Eq. (7) can be approximately expressed as  

 
2 2 2 21 1 1 1

( )
2 2 2 2

K w m w w w mw mw= − − − − = − + + , (8) 

or 

 

22 4 8 (1 )

2(1 )

m m K m
w

m

+ − −
=

−
. (9) 

Eq. (9) implies that the displacement (w) is a function of m and K. 

3. CRITERION FOR PULL-IN INSTABILITY 

Pull-in instability in MEMS refers to a phenomenon where an electrostatically actuated 

MEMS device collapses or snaps to a new stable state. This occurs due to the interaction 

between electrostatic forces and mechanical restoring forces. As the applied voltage 

increases, the electrostatic force overcomes the mechanical restoring force, leading to 

instability. It can cause failure in MEMS devices and is an important consideration in their 

design. Understanding and predicting pull-in instability is crucial for ensuring the reliable 

operation of MEMS. 

The criterion for pull-in instability in MEMS is a key aspect in understanding and 

designing these devices. As mentioned earlier, the criterion for pull-in instability is that 

when the kinetic energy and the accelerated speed become zero simultaneously [16]. This 

means that at this point, the device is in a state of unstable equilibrium and is likely to 

collapse or snap to a new stable state. This criterion provides a quantitative measure for 

engineers to determine when pull-in instability may occur. For example, in the design of a 

MEMS actuator, engineers can use this criterion to calculate the maximum allowable 

voltage that can be applied before pull-in instability occurs. By understanding this criterion, 

engineers can design MEMS devices that are more stable and reliable, and avoid potential 

failures due to pull-in instability. 

The criterion for pull-in instability is that when the kinetic energy and the accelerated 

speed become zero simultaneously. According to Eq. (1) and Eq. (7), we have:  

 0
1

m
w

w
− =

−
 (10) 



 Variational Approach to Micro-Electro-Mechanical Systems 653 

 
21

ln(1 ) 0
2

w m w− − − =  (11) 

Solving Eqs. (10) and (11) simultaneously, we find the threshold values of the 

displacement and the voltage,  

 
* 0.7153

* 0.2036

w

m

=


=
 (12) 

That implies that in the event that the value of w is greater than w* or the value of m is 

greater than m*, the phenomenon of pull-in instability takes place, w* is called as the pull-

in displacement, and m* is the pull-in voltage. This criterion for pull-in instability is highly 

accurate and extremely reliable. The accurate determination of the pull-in voltage provides 

a valuable benchmark for researchers and engineers working in the field of electrostatically 

actuated MEMS/NEMS. It allows for a more precise understanding and prediction of the 

occurrence of pull-in instability, enabling better design and optimization of these micro 

and nano devices. With such a highly accurate criterion, it becomes possible to make more 

informed decisions regarding the design parameters and operating conditions to avoid or 

manage pull-in instability effectively. 

4. SEMI-INVERSE METHOD AND A NEW VARIATIONAL FORMULATION  

FOR MEMS OSCILLATOR 

The semi-inverse method [25] is a powerful approach for establishing a variational 

principle. It offers a systematic way to solve complex problems in various fields, including 

mechanics and engineering. Though the semi-inverse method, many new variational 

formulations were appeared in literature for incompressible fluids [26], lubrication 

problems [27], fractal solitary waves [28] and the simplified Navier-Stokes equations [29]. 

We re-write Eq. (1) in the following form: 

 
2 0w w w w w m + − − − =  (13) 

The semi-inverse method commences with the formulation of an informed hypothesis 

regarding the form of the variational formulation. Based on physical intuition or prior 

knowledge, a trial functional is selected that contains some unknown parameters. 

Subsequently, the variational problem is formulated by applying specific constraints and 

principles. 

The fundamental concept is to partially invert the problem by selecting a trial functional 

that satisfies some of the essential properties of the solution. Subsequently, the functional 

is varied with respect to the unknown parameters, thereby obtaining the Euler-Lagrange 

equations. 

The semi-inverse method is advantageous in that it reduces the complexity of the 

problem by narrowing down the search space. It offers a methodology for the acquisition 

of approximate solutions when the identification of exact solutions is challenging. 

Moreover, it is capable of handling nonlinear problems and can be extended to multi-

dimensional and time-dependent systems. 



654 J.H. HE, J. MA, A.A. ALSOLAM, C.H. HE 

In accordance with the semi-inverse method, the following assumption is made: 

 
2 2 2 31 1 1 1

( , )
2 2 4 3

J w K w w w w w mw F dt
 

 = − + + + + 
 
  (14) 

where F is unknown yet.  

Equation (14) represents a trial variational principle in our study. In this context, the 

independent variables play a crucial role in defining the problem. The kinetic energy (K) 

is considered an independent variable as it reflects the main dynamical property of the 

MEMS oscillator. When K = 0, the pull-in motion occurs. Additionally, the displacement 

(w) is another independent variable. When w=1, the pull-in instability occurs. Thus, K and 

w are the main parameters for studying the pull-in property. 

The variational principle allows us to understand the significance of these variables and 

how they interact. A comprehensive understanding of these independent variables is 

essential for a detailed analysis of the variational principle and its applications in our study. 

The Euler-Lagrange equation with respect to w is:  

 2 21 1
( ) 0

2 4

F
w w w w w w m

w




  − − + + + + + = , (15) 

or 

 2 21
0

2

F
w w w w w w m

w




  − − + + + + + = , (16) 

where δF/δw is the fractional derivative [30], defined as: 

 
2

2
( ) ( )

F F d F d F

w w dt w wdt





  
= − +

   
. (17) 

In view of Eqs. (13) and (6), we have:  

 2 2 21 1

2 2

F
w w w w w w m w K

w




   = + − − − − = − = − . (18) 

From Eq. (18), F can be identified as: 

 1F Kw F= − + , (19) 

where F1 is an unknown function of K. So the trial-functional can be updated as:  

 
2 2 2 3

1

1 1 1 1
( , )

2 2 4 3
J w K w w w w w mw Kw F dt

 
 = − + + + − + 

 
  (20) 

The Euler-Lagrange equation with respect to K is:  

 1 0
F

w
K




− + = . (21) 

That is:  



 Variational Approach to Micro-Electro-Mechanical Systems 655 

 

2

1
2 4 8 (1 )

2(1 )

m m K mF
w

K m





+ − −
= =

−
. (22) 

Now F1 can be identified as:  

 
3/2

2

1 2

2 1
4 8 (1 )

2(1 ) 24(1 )

mK
F m K m

m m
 = − − − − −

. (23) 

Finally, the following variational formulation is obtained:  

 

2 2 2 3

3/2
2

2

1 1 1 1
( , )

2 2 4 3

2 1
4 8 (1 )

2(1 ) 24(1 )

J w K w w w w w mw Kw dt

mK
m K m dt

m m

 
 = − + + + − 

 

 
 + − − −  − − 





 (24) 

The new obtained variational formulation given in Eq. (24) gives deeper insight into 

the physical property of the MEMS oscillator in an energy sense, but it is valid for the small 

displacement. When w is not small enough, the derivation of the variational principle is 

much more complex, and we obtain the following variational formulation  

 

2 2 2 3

3

3

2 2 3

1 1 1 1
( , )

2 2 4 3

1
ln(1 ) ln(1 )

6

1 1
ln(1 ) ( )

2 2

J w K w w w w w mw dt

w mw w mw m w dt

a K w m w b K w dt

 
 = − + + + 

 

 
− + − − − − 

 

   
+ + + − + −  

   







 (25) 

where a and b are nonzero constants.  

Now we use the established variational formulation to investigate the pull-in voltage. 

Before pull-in inability, the kinetic energy and accelerated velocity tend to zero, under this 

extreme case, we have: 

 
2 31 1

( )
2 3

J w w w mw dt
 

= − + + 
 
 . (26) 

The Euler-Lagrange equation becomes: 

 
2 0w w m− + + =  (27) 

Solving m from Eq. (27), and submitting the result into Eq. (7), we have:  

 
2 21

( ) ln(1 ) 0
2

w w w w+ − − =  (28) 

This equation can be solved by Newton iteration method, here introduces an ancient 

Chinese algorithm [31, 32]. We assume two trial solutions w1=0.7 and w2=0.75, Eq. (28) 



656 J.H. HE, J. MA, A.A. ALSOLAM, C.H. HE 

produces the residuals R1= -0.007834 and R2=0.06249. The ancient Chinese algorithm 

results in the following approximate solution: 

 1 2 2 1

1 2

0.71
R w R w

w
R R

−
= =

−
. (29) 

We choose w3=0.71, and the residual is R3=-0.001737, which leads to:  

 1 3 3 1

1 3

0.7128
R w R w

w
R R

−
= =

−
. (30) 

The pull-in voltage can be calculated as: 

 
2* 0.2045m w w= − = . (31) 

The precise value is 0.203632188, as demonstrated in Fig. 1, which illustrates the 

phase-space dynamics of the MEMS oscillator by plotting dimensionless velocity against 

displacement for varying voltage-related parameter. 

 

Fig. 1 Phase trajectories for m=0.05, 0.1, 0.15, and  0.203632189  

Sub-critical voltage behavior manifests when m < m*. For values of m ranging from 

0.05 to 0.15, the trajectories exhibit closed, elliptical loops, thereby directly corresponding 

to periodic solutions where electrostatic and mechanical forces maintain stable equilibrium. 

These loops remain constrained within the range of values 0 < w < w*, thereby confirming 

that kinetic energy and acceleration never simultaneously reach zero—a finding that is 

consistent with the established pull-in criterion. 

The critical voltage behavior emerges when the parameter m is equal to m*. The 

innermost trajectory, collapsing to a point near w = w*, visualizes the threshold at which the 

system teeters on instability. The loop contracts towards a singularity, thereby satisfying the 

pull-in condition: when m=m*, the kinetic energy and acceleration simultaneously approach 

zero. This phase transition serves to substantiate the variational principle's forecast of a well-

defined boundary between stable oscillation and pull-in motion. 



 Variational Approach to Micro-Electro-Mechanical Systems 657 

Equation (31) demonstrates a notable level of precision. It is noteworthy that the ancient 

Chinese algorithm attains higher accuracy with a single iteration, underscoring its property 

of rapid convergence to a substantial level of precision. In addition, the variational principle 

facilitates precise voltage-controlled manipulation, thereby providing a robust framework 

for highly accurate voltage regulation. The synergy between the specified pull-in voltage 

value, the efficacy of the algorithm, and the precision of the variational principle 

underscores their collective potential for facilitating precise and regulated operations in the 

domains of MEMS design and optimization. 

Fig. 2 demonstrates that when  m < m* or  w < w*, the system exhibits periodic 

solutions. However, as the voltage parameter m increases, the displacement, w, approaches 

its threshold value, w*.  

 

Fig. 2 Periodic solution for m<m* 

Once w exceeds w*, pull-in motion occurs, as shown in Fig. 3. In contrast to the periodic 

behavior observed in Fig. 2, where the dimensionless displacement oscillates within the 

bounded range, 0<w <w*, under sub-critical voltages (m<m*), the dynamics depicted in 

Fig. 3 reveal a striking shift when m exceeds the critical threshold, m*. Here, w abandons 

all oscillatory characteristics and instead increases monotonically, advancing relentlessly 

toward the singularity at w=1. This unidirectional trajectory signifies an irreversible pull-

in motion, a catastrophic transition driven by the breakdown of the delicate balance 

between electrostatic and mechanical forces that governs stable operation.   

In the sub-critical regime (Fig. 2), mechanical restoring forces—rooted in the elastic 

properties of the MEMS structure—successfully counteract the electrostatic attraction 

between electrodes, limiting w to periodic oscillations. However, as m surpasses m*, the 

electrostatic force undergoes a rapid amplification. This force exceeds the mechanical 

restoring capacity of the system, which weakens as the gap between electrodes narrows, as 

a result, the electrodes make contact, rendering the device inoperable.   



658 J.H. HE, J. MA, A.A. ALSOLAM, C.H. HE 

 

Fig. 3 Pull-in motion for m>m* 

Notably, Fig. 3 illustrates that the steepness of the pull-in curve intensifies with 

increasing m beyond m*. This phenomenon arises because a higher m corresponds to a 

stronger initial electrostatic force, which accelerates the narrowing of the electrode gap. As 

the gap shrinks, the electrostatic force increases further, creating a positive feedback loop 

that steepens the trajectory. This behavior aligns with the predictions of the variational 

principle, which emphasizes that beyond m*, the system’s energy balance (governed by Eq. 

(5) is violated, with the potential energy dominated by electrostatic contributions that drive 

the irreversible collapse.   

This monotonic, gap-closing motion captured in Fig. 3 thus serves as a critical 

experimental validation of the theoretical pull-in criterion: the simultaneous decay of 

kinetic energy and acceleration at w = w* marks the point of no return, beyond which the 

system is pulled inexorably toward the singularity—a hallmark of pull-in instability in 

electrostatically actuated MEMS devices. 

5. TWO-SCALE FRACTAL DERIVE BASED FRACTIONAL DERIVATIVE  

AND ITS APPLICATION TO MEMS SYSTEM  

The core variational principle, Eq. (24), is derived from the classical MEMS oscillator 

model, which presupposes idealized, smooth geometries. For fractional MEMS featuring 

fractal structures (e.g., rough surfaces or porous microstructures), the electrostatic-

mechanical coupling is inherently modulated by multi-scale geometric irregularities. To 

address this disparity, the variational functional ought to be extended to incorporate fractal 

dimensions, thereby explicitly establishing a connection between the semi-inverse method 

and fractal dynamics.   



 Variational Approach to Micro-Electro-Mechanical Systems 659 

The two-scale fractal, as a geometric concept, is tailored to describe porous media or 

unsmooth boundaries characterized by two distinct scales [33]. Within the domain of 

fractal MEMS systems, the two-scale fractal concept finds several pivotal applications: 

1) Modeling Complex Geometries: Fractal MEMS devices often have complex 

geometries that cannot be accurately described by traditional geometric models. The two-

scale fractal geometry provides a more realistic representation of these structures. For 

example, the surface of a fractal MEMS device may have irregularities and self - similar 

patterns at different scales, which can be modeled using two-scale fractals. This allows for 

a better understanding of how the geometry affects the device's physical properties such as 

capacitance, stiffness, and thermal conductivity. 

2) Understanding Physical Processes: The two-scale fractal derivative helps in 

understanding the physical processes that occur within fractal MEMS devices. It can be 

used to analyze how mechanical and electrical properties change with respect to the fractal 

geometry. For instance, in the study of pull - in instability in MEMS, the two-scale fractal 

derivative can provide a more detailed analysis of how the instability is affected by the 

fractal structure of the device. This can lead to better design and optimization of MEMS 

devices to avoid or control the pull - in instability. 

3) Novel Device Design: The concept of two-scale fractal can inspire novel designs for 

MEMS devices. By incorporating two-scale fractal geometries, it may be possible to 

achieve enhanced performance in terms of sensitivity, selectivity, or power consumption. 

For example, a fractal MEMS sensor with a two-scale fractal structure may have a higher 

sensitivity to certain physical quantities compared to a traditional MEMS sensor. This can 

open up new avenues for the development of advanced MEMS devices for various 

applications such as sensing, actuation, and signal processing. 

The study of MEMS oscillators often requires a comprehensive understanding of their 

behavior under different geometric and physical conditions. In this regard, the 

generalization of the governing equations to account for more complex geometries, such 

as the two-scale fractal, is of great importance. This section focuses on detailing how the 

Eq. (1) for the MEMS oscillator can be extended to incorporate the two-scale fractal 

concept, providing a more accurate and detailed description of the oscillator's behavior. 

Fractal MEMS systems [34, 35] hold the potential to be a truly revolutionary 

advancement in the extensive and ever-evolving field of microelectromechanical systems. 

These remarkable systems draw their inspiration from the captivating concepts such as the 

gecko effect [36], with its astonishing ability to allow geckos to adhere to various surfaces, 

and the lotus effect [37], which endows lotus leaves with their remarkable self-cleaning 

properties. The fractal surface at the nano scale is truly a wonder, as it possesses an 

extremely high surface energy, known as geometric potential [38, 39], which is widely 

used to design nano scale materials, demonstrating a creative and forward-thinking 

approach that could potentially reshape the landscape.  

Tian and her colleagues, in their remarkable research efforts as documented in Ref. 

[40], put forward the highly innovative concept of fractal MEMS systems. This concept is 

aimed at overcoming the pull-in instability that often plagues such systems. By introducing 

this concept, they strive to make the systems significantly more reliable. The governing 

equation becomes: 

 
2

2
0

1

D

D

d w m
w

wdt
+ − =

−
. (32) 



660 J.H. HE, J. MA, A.A. ALSOLAM, C.H. HE 

In this equation, D represents the two-fractal dimensions, which are essential in defining 

the fractal nature of the geometry. The two-scale fractal derivative is given as follows:  

 
0

0
0

0

( , ) ( , )
( ) (1 ) lim

( )

D

D Dt t t

w x t w x tw
t D

t t t− →

−
=  +

 −
. (33) 

This definition captures the essence of the fractal derivative by considering the limit of 

the difference quotient as the time interval approaches zero, weighted by the gamma 

function of the fractal dimension. 

The two-scale fractal derivative has found extensive application in modeling complex 

problems. As evidenced by numerous studies [41, 42], it has demonstrated its remarkable 

effectiveness in this regard. This derivative possesses the following properties: 

 
1

0( ) (1 )( )
D

D

D

w w
t D t

tt

− 
  + 


 (34) 

 
2

2
( )

D D D

D D D

w w

t t t

  
=

  
 (35) 

 

2

2

, 0

, 1

, 2

D

D

w D

w w
D

tt

w
D

t


 =


 
= =

 


=


 (36) 

The generalized equation has several important implications for the study of MEMS 

oscillators. Firstly, it provides a more accurate model for predicting the behavior of MEMS 

oscillators with fractal geometries. This is particularly useful in applications where the 

fractal nature of the MEMS structure plays a significant role. For example, in certain sensor 

and actuator designs, the fractal geometry can enhance the sensitivity and performance of the 

device. The generalized equation allows us to analyze how the fractal structure affects the 

oscillator's response to different inputs, such as voltage changes or external perturbations. 

Secondly, the generalized equation allows for a deeper understanding of the physical 

processes involved in the operation of MEMS oscillators. By considering the fractal 

geometry, we can better analyze the effects of various factors on the oscillator's 

performance. For instance, the surface roughness of the MEMS device, which is often 

related to its fractal structure, can influence the dissipation of energy and the damping 

characteristics of the oscillator. Additionally, the distribution of material properties within 

the fractal structure can affect the mechanical and electrical properties of the oscillator, 

such as its stiffness and capacitance. The generalized equation enables us to study these 

effects in more detail, providing insights into how to optimize the design and performance 

of MEMS oscillators. 

In conclusion, the generalization of Eq. (1) for the MEMS oscillator to the two-scale 

fractal case is a significant step forward in understanding the behavior of MEMS oscillators 

in the context of fractal geometry. This generalization not only allows for a more accurate 

description of the oscillator's behavior but also provides insights into the physical processes 



 Variational Approach to Micro-Electro-Mechanical Systems 661 

involved. Future research can further explore the applications and implications of this 

generalized equation for the development of more advanced MEMS devices. By continuing 

to study the behavior of MEMS oscillators under different geometric and physical 

conditions, we can improve the design and performance of these devices, leading to more 

efficient and reliable MEMS - based systems. 

The fractional MEMS is indeed another highly captivating hot topic in the field of 

technology. In this domain, the fractional derivative, as demonstrated in references [43], 

plays an absolutely crucial role. Fractional MEMS systems truly represent a remarkable 

technological innovation. These systems are an enthralling combination of advanced 

engineering and profound mathematical concepts. Through the incorporation of fractional 

calculus principles, they manage to achieve not only enhanced performance but also 

expanded functionality. The utilization of fractional derivatives enables a more accurate 

description of the intricate behaviors and dynamic processes exhibited by these miniature 

mechanical and electrical systems. For instance, in the area of stochastic processes [44] 

and adaptive chaos control [45], the fractional derivative has shown great potential in 

providing a deeper understanding and better control of these complex phenomena. As 

research in this field continues to progress, fractional MEMS systems are expected to bring 

about even more significant technological breakthroughs and open up new avenues for 

applications in various industries. 

Other definitions of the fractional derivatives can be modified in a similar way, for 

example, the Caputo-Fabrizio derivative and its generalization [46] can be updated as: 

 
0

1
exp( )

1 1

D
t

D

w w
ds

t t







 

 
= −

− − 
  (37) 

 
0

1 ( , )

1 ( , )

D
t

D

w k s w
ds

k tt t







 

 
=

− 
  (38) 

We give a generalized definition of the two-scale fractal derivative based fractional 

derivative in the forms: 

 
0

( )
( , ) ( , )

D
t

D

w w s
D t s ds

t s




  

 
=

 
  (39) 

 
0

( , ) ( , ) ( )
D

t

D

w
D t s w s ds

t t




  

 
=

 
  (40) 

where  is the normalization function and  is the kern function. By choosing suitably the 

normalization function and the kern function, the above definitions can be converted to a 

known fractional derivative, e.g., ABC fractional derivative [47, 48].  For example. if we 

choose: 

 
( )

( , )
1 ( )

D
t


 




=

−
, (41) 

 ( )

( )
( , ) ( )

( ) 1
t

t
t s E t s

t







 
= − 

− 
, (42) 



662 J.H. HE, J. MA, A.A. ALSOLAM, C.H. HE 

where Ξ(ω) is the normalization function, satisfying  Ξ(0)= Ξ(1)=1, Eω(t) is the ML function, 

we can convert Eq. (39) and Eq. (40) to a modified ABC fractional derivative [47, 48].  

Here, we introduce the two-scale fractal derivative based fractional derivative. In this 

context, within the known fractional derivative, the traditional derivatives are replaced by 

the two-scale fractal derivative. For example, the Caputo fractional derivative and 

Riemann-Liouville fractional derivative are updated respectively in the following manner: 

 1

0

1 ( )
( )

( )

D
t D

D

w w s
t s ds

Dt s




 

− − 
= −
 − 

 , (43) 

 1

0

1
( ) ( )

( )

D
t D

D

w
t s w s ds

Dt t




 

− − 
= −
 − 

 . (44) 

When D=1, Eq. (43) and Eq. (44) turn back to Caputo fractional derivative and Riemann-

Liouville fractional derivative, respectively.  

6. CONCLUSIONS 

In conclusion, this study has yielded several significant new findings, with particular 

emphasis on demonstrating the practical utility of the proposed variational principle 

through a specific MEMS system.   

Firstly, the application of the semi-inverse method to establish a new variational 

principle for MEMS systems has proven to be a groundbreaking advancement. It surpasses 

certain limitations of traditional variational approaches by offering a more precise portrayal 

of the physical behavior of MEMS devices, facilitating enhanced prediction of their 

performance, and presenting improved computational efficiency, which is vital for design 

and optimization procedures demanding multiple simulations. This underscores the semi-

inverse method as a potent tool for handling the intricate nature of MEMS systems.   

To illustrate concretely the applicability of this new variational principle, we focused 

on a specific case: MEMS pressure sensors integrated into wearable textiles [49]. These 

sensors are critical for real-time physiological monitoring (e.g., heart rate, respiratory rate, 

and body temperature) due to their miniaturization, flexibility, and low power consumption. 

However, their reliability is heavily constrained by pull-in instability, which can lead to 

sudden performance degradation or failure under dynamic physiological conditions.   

The proposed variational principle directly addresses this challenge. For the MEMS 

pressure sensor, we applied the new formulation to model the electrostatic-mechanical 

coupling behavior, where the sensor’s diaphragm displacement, w, and voltage-related 

parameter, m, are the core variables. Using the derived criterion for pull-in instability 

(simultaneous zero kinetic energy and accelerated speed), we computed the pull-in voltage 

with higher precision compared to traditional methods. Specifically, the variational 

principle enabled us to incorporate microscale effects (e.g., surface tension and fractal 

surface roughness) that are critical for wearable applications but often overlooked in 

conventional models.  

Furthermore, the two-scale fractal derivative-based fractional modeling, integrated 

with the variational principle, enhanced the accuracy of predicting the sensor’s dynamic 

response under varying fractal surface geometries, which is common in wearable textiles 

https://zh.wikipedia.org/wiki/%CE%9E
https://baike.baidu.com/item/%E5%B8%8C%E8%85%8A%E5%AD%97%E6%AF%8D/4428067#3-25
https://zh.wikipedia.org/wiki/%CE%9E
https://baike.baidu.com/item/%E5%B8%8C%E8%85%8A%E5%AD%97%E6%AF%8D/4428067#3-25
https://zh.wikipedia.org/wiki/%CE%9E
https://baike.baidu.com/item/%E5%B8%8C%E8%85%8A%E5%AD%97%E6%AF%8D/4428067#3-25
https://baike.baidu.com/item/%E5%B8%8C%E8%85%8A%E5%AD%97%E6%AF%8D/4428067#3-25


 Variational Approach to Micro-Electro-Mechanical Systems 663 

due to fabric texture. This integration allowed us to quantify how surface roughness 

(characterized by fractal dimension D) influences pull-in behavior.   

Secondly, this specific demonstration validates the potential of the new variational 

principle for future research. It serves as a foundation for further refinement, potentially 

leading to more inventive and dependable MEMS designs, particularly for wearable 

technologies where robustness and miniaturization are paramount. It also opens avenues 

for deeper investigations into the relationship between variational principles and 

microscale electromechanical performance.   

Finally, the application of MEMS in wearable textiles is reaffirmed as highly promising. 

The specific MEMS pressure sensor case demonstrates that our variational approach not only 

improves design efficiency, but also enhances the reliability of wearable systems.   

In summary, the proposed variational principle, validated through the MEMS pressure 

sensor in wearable textiles, advances both theoretical modeling and practical design of 

MEMS, paving the way for more robust and versatile microscale devices in emerging 

applications. Specifically, the modeling framework and analytical approach presented 

herein can be further extended to the study of MEMS graphene resonators [50], where 

precise characterization of dynamic behaviors and instability thresholds is equally critical 

for optimizing performance in high-frequency communication and sensing scenarios. This 

cross-application potential underscores the generalizability of our variational principle in 

addressing the core challenges of diverse MEMS configurations. 

Acknowledgement: The work is supported by Guangdong Basic and Applied Basic Research 

Foundation (No.2022A1515111135 & No.2025A1515010407). 

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