12816


FACTA UNIVERSITATIS  
Series: Mechanical Engineering Vol. 22, No 2, 2024, pp. 187 - 198  

https://doi.org/10.22190/FUME240603034H 

© 2024 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND 

Original scientific paper 

PERIODIC SOLUTION OF A  

MICRO-ELECTROMECHANICAL SYSTEM 

Ji-Huan He1,2 

1School of Jia Yang, Zhejiang Shuren University, Hangzhou, Zhejiang, China 
2National Engineering Laboratory for Modern Silk,  

College of Textile and Clothing Engineering, Soochow University, Suzhou, China 

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

Abstract. This paper examines the periodic motion of the micro-electro-mechanical 

system (MEMS), which is governed by a singularity that makes it challenging to find an 

exact solution and to understand its dynamical properties. This paper applies the 

frequency formulation to gain insight into the frequency-amplitude relationship of the 

system. It is found that when the amplitude reaches a threshold value, the periodic motion 

becomes pull-in instability. This finding simplifies the warning system for the system’s 

unsafe operating conditions, and the frequency-amplitude relationship can be used for 

optimal design of the system with high accuracy and high reliability. 

Key words: MEMS systems, Variational theory, Nonlinear oscillator 

 1. INTRODUCTION   

Micro-electro-mechanical systems (MEMS) have been a driving force in the technological 

revolution. They are employed in a multitude of cutting-edge applications, including fifth-

generation (5G) mobile networks [1], chips [2], ultra-sensitive sensors [3], robots [4], 

tsunami monitoring [5] and wearable smart fabrics [6,7]. This is due to their extremely 

simple structure, ultra-finely diminutive size, ultra-light weight, minimal energy consumption, 

tremendously high reliability, and enormously low cost. 

The MEMS system is generally open periodically. However, when the applied voltage 

exceeds a specified threshold value, the pull-in instability occurs, resulting in system failure. 

A substantial body of literature exists on the subject of pull-in instability. Mikhasev, et al. 

studied pull-in instability of carbon nanotube nano-tweezers [8], Tian and her colleagues 

[9] suggested a new mathematics concept that can control the pull-in instability, and Yang 

found the pseudo-pull-in stability [10].  

 
Received: June 03, 2024 / Accepted July 07, 2024 

Corresponding author: Ji-Huan He  
School of Jia Yang, Zhejiang Shuren University, Hangzhou, Zhejiang  

E-mail: hejihuan@suda.edu.cn 

https://orcid.org/0000-0002-1636-0559


188 J.-H. HE 

When the system opens near the pull-in voltage, any minor environmental change may 

result in a significant shift from normal periodic motion to pull-in motion. This property is 

widely used for ultrasensitive MEMS-based sensors, which are sensitive to even a small 

number of nanoparticles in air, or viruses in a room, or even minor environmental changes 

in, e.g., air velocity, air pressure, poisonous gases, and gravity [3]. It can monitor precisely 

the propagation of a nano-crack, ground subsidence, the terrestrial plate motion before an 

earthquake, and other advanced applications, for examples, gas sensors [11], microphones 

[12], biosensors [13], micro-coils [14], odorant sensors [15], ocean physics [16] and monitoring 

systems [17]. It is therefore crucial to elucidate its dynamical properties prior to the onset 

of pull-in motion. This paper will now turn its attention to their periodic motion. 

A plethora of analytical techniques exists for the analysis of nonlinear oscillators. These 

include the variational iteration method [18,19], the homotopy perturbation method[20], 

the Hamiltonian-based frequency formulation [21], the energy balance method [22], and 

others. This paper applies the frequency formulation [23] to investigate the periodic 

properties of the MEMS system. 

2. MATHEMATICAL MODEL   

Consider a MEMS system as illustrated in Fig. 1, where the micro/nano beam can be a 

polyvinylidene fluoride (PVDF) nanofiber [24] or metal wire, it locals at the middle of the 

two symmetrical current-carrying wires. According to the Biot–Savart law [25], the 

micro/nano beam is under a magnetic force 

 0 1 2 1 1
( )

2

i i
f

H w H w




= +

− +
 (1) 

where f represents the magnetic force per unit length, while 𝜇0 denotes the magnetic 

constant. The variables i1 and i2 correspond to the direct currents flowing through the wires, 

while H denotes the distance between them. 

 

Fig. 1  MEMS system with a simple structure 



 Periodic Solution of a Micro-Electromechanical System 189 

The governing equation can be expressed as  

 
2

0 1 2

2

1 1
( )

2

i i Ld w
m kw

H w H wds




+ = +

− +
 (2) 

where m is the mass of the micro/nano beam, k is its elastic coefficient, s is time. Eq. (2) 

can be written in a dimensionless form: 

 
2

2 2

1 1 2
( )
1 1 1

d u Ku
u K

u udt u
+ = − =

− + −
 (3) 

where u=w/H, t=sω0, ω0=√(k/m), K= 𝜇0 Li1 i2/(2πH2k).  

The initial conditions are  

 (0)u p= , (0)u q =  (4) 

where p and q are constants. 

The variational formulation for Eq. (3) is  

  2 21 1
( ) ( ) ln(1 ) ln(1 )

2 2

du
J u u K u u dt

dt

 
= − − − + + 

 
  (5) 

The variational principle for a MEMS system provides an energy approach to insight 

into the energy conservation during its operation [26]. It is also a useful mathematical tool 

for complex systems, such as nano-lubrication [27], solitary waves [28], and singular 

waves [29]. 

According to the variational formulation of Eq. (5), the following Hamilton function 

can be obtained: 

  2 21 1
( ) ln(1 ) ln(1 )

2 2

du
u K u u H

dt
+ + − + + =  (6) 

where H is Hamilton constant, which depends on the initial conditions of Eq. (4). After 

identification of H, Eq. (6) becomes 

    2 2 2 21 1 1 1
( ) ln(1 ) ln(1 ) ln(1 ) ln(1 )

2 2 2 2

du
u K u u q p K p p

dt
+ + − + + = + + − + +  (7) 

Taking the derivative of this equation with respect to t, we have  

 
2

2

1 1
0

1 1

du d u du du
u K

dt dt u u dtdt

− 
+ + + = − + 

 (8) 

This equation leads to Eq. (3). Eq. (7) can be used for analysis of the dynamical 

properties by the  Hamiltonian-based frequency formulation [21]. 



190 J.-H. HE 

3. FREQUENCY FORMULATION    

Considering the following general nonlinear oscillator 

 ( , , ) 0u h u u u  + =  (9) 

For a periodic solution of Eq. (9), it requires [23] 

 (0, , ) 0h u u  =  (10) 

and 

 ( , , ) / 0h u u u u    (11) 

We assume that approximate solution is given  

 ( ) cos( )u t A t = +  (12) 

where ω is the frequency, A and ϕ can be determined from the initial conditions: 

 
2

2

2

q
A p


= +  (13) 

According to the frequency formulation [23], we have  

 
2

2

2

1

( , , )
u NA

u A N

u NA

h u u u

u 


 =

=− −
=−

 
=  (14) 

where A is the amplitude, 0<N<1, in this paper we set N=√3/2 =0.8660. Chun-Hui He and 

Chao Liu suggested a modification [30]  

 

/4

2 0

/4

0

T

T

hdt

udt





=



 (15) 

where ξ is the weighting function. The frequency formulation is the simplest approach to 

nonlinear vibration systems, it has been successfully applied to various complex oscillators, 

for examples, fractal vibration systems [31], fractal–fractional oscillators [32], and 

nonlinear oscillators [33].  

For Eq. (3), h can be expressed as  

 
2

2

1

Ku
h u

u
= −

−
 (16) 

According to the frequency formulation given in Eq. (14), we have  

 

23
2 4

2 2 23 /2 3 3
4 4

1 22 2
(1 ) 1

1 1 1u A

A KK K

u A A


=

− −
= − = − =

− − −
 (17) 

For a periodic motion, ω must be positive. When ω2<0, the pull-in motion occurs. So 

ω=0 leads to the following critical relationship between the voltage and the amplitude.  



 Periodic Solution of a Micro-Electromechanical System 191 

 23

4
1 * 2 * 0A K− − =  (18) 

where A* and K* are, respectively, the critical amplitude and the critical voltage.  

 
4 8 *

*
3

K
A

−
=  (19) 

We can also write Eq. (3) in the form  

 2 3 2 0u u u u u Ku − + − − =  (20) 

where h can be expressed as  

 
2 3 2h u u u u Ku= − + − −  (21) 

According to the frequency formulation given in Eq. (14), we have  

 
2

2 2 2 2 2

3 /2

3 /2

3 3
1 2 1 2

4 4
u A

u A

u u u K A A K


 =

=−

 = − + − − = + − −   (22) 

Eq. (22) is equivalent to Eq. (17). In view of Eq. (13), we have the following frequency-

amplitude relationship  

 
2

2 23
234

4 2

2 2
1 1

1
1 ( )

K K

A q
p





= − = −
−

− +

 (23) 

or  

 
2 2 2 2 2 2 2

2

(4 3 3 8 ) (4 3 3 8 ) 12(4 3 )

2(4 3 )

q p K q p K p q

p


+ − − + + − − − −
=

−
 (24) 

The relationship between the critical voltage K* and the initial conditions is  

 2 2 2 2 2(4 3 3 8 *) 12(4 3 ) 0q p K p q+ − − − − =  (25) 

In order to verify the results, we consider two cases when p=A=0.1, q=0 and p=A=0.5, 

q=0.  For the first case, according to Eq. (18), the pull-in voltage can be calculated as  

 
23

4

1
* (1 0. 541 )

2
0. 962K = −  =  (26) 

While the exact value is K*=0.4951 (see Fig.2a) with a relative error of 0.23%. 

For the case of A=0.3, we have  

 
23

4

1
* (1 0. 543 )

2
662 0.K = −  =  (27) 

While the exact value is K*=0.3751 (see Fig.2b) with a relative error of 6.75%. For the 

case of A=0.5, we have  

 
23

4

1
* (1 0.5 )  0.40625

2
K = −  =  (28) 

While the exact value is K*=0.5 (see Fig. 2c) with a relative error of 8.3%.  



192 J.-H. HE 

a)   

b)  

c)   

Fig. 2 Phase diagram. (a) A=0.1; (b) A=0.3 (c) A=0.5 



 Periodic Solution of a Micro-Electromechanical System 193 

To improve the prediction accuracy for K*, we can use Eq.(7) to give an exact 

relationship between the critical voltage (K*) and the  critical amplitude (A*): 

  21
* * ln(1 *) ln(1 *)

2
A K A A H+ − + + =  (29) 

where H is determined by the initial conditions 

Fig.3, Fig.4 and Fig.5 compare the approximate solutions with the numerical solutions 

for A=0.1, 0.3 and 0.5, respectively,, and a good agreement is found, showing the method 

is reliable.  

It can be found that the errors become larger for a larger K. This is because when K> 

K*, the pull-in instability occurs, so when K tends to K*, the periodic motion tends 

gradually to the pull-in instability. 

a)  

b)  

Fig. 3 Exact solution (continnued line) vs approximate solution (discontinous line) for A=0.1. 

(a) K=0.4; (b) K=0.49 with K*=0.4951 



194 J.-H. HE 

a)  

b)  

c)  

Fig. 4 Exact solution (continued line)  vs approximate solution (discontinous line) for A=0.3. 

(a) K=0.2; (b) K=0.3; (c) K=0.32 with K*=0.50 



 Periodic Solution of a Micro-Electromechanical System 195 

a)  

b)  

c)  

Fig. 5 Exact solution (continued line)  vs approximate solution (discontinuous line) for 

A=0.5. (a) K=0.2; (b) K=0.3; (c) K=0.32 with K*=0.3751 



196 J.-H. HE 

3. A GENERALIZED MEMS OSCILLATOR    

In Fig.1, the wires can be thin plates, in that case, the governing equation can be 

expressed a s[22] 

 
4 2 2 2

2

4 2 2 2 20

1 1
( )

2 2 ( ) ( )

Lw w ES w w b
EI A N dx

L xx t x H w H w

 


     
+ = + + +     − +   

  (29) 

where w is the displacement, EI  the stiffness, ρ density, A area, N applied force, L length, 

S cross-section area, b width, ε dielectric constant, and ν the Poisson ratio. In Eq.(29), the 

last term is the electrostatic excitation.  

For the fixed-fixed micro/nano thin plate, the boundary conditions are  

 (0) ( ) 0w w L= =  and (0) ( ) 0
dw dw

L
dx dx

= =  (30) 

Considering the boundary conditions of Eq. (30), we assume that w can be expressed as  

 2 2 2

1 2( , ) ( ) (1 ) (1 )w t x u t x x c x c x= − + + +  (31) 

where ci (i=1,2,3,…) are constants. Submitting Eq. (31) into Eq. (29) and integrating the 

resultant equation with x from 0 to L, we finally have  

 
2

8 6 4 2 3 5 7 9

8 6 4 2 0 1 3 5 7 92
( ) 0

d u
a u a u a u a u a a u a u a u a u a u

dt
+ + + + + + + + + =  (32) 

where ai (i=0~9) are constants.  

We re-write Eq. (32) in the form  

 
3 5 7 92

1 3 5 7 9

2 8 6 4 2

8 6 4 2 0

0
a u a u a u a u a ud u

dt a u a u a u a u a

+ + + +
+ =

+ + + +
 (33) 

The frequency formulation can be modified as  

 

/4
3 5 7 9

1 3 5 7 9
2

/4
8 6 4 2

8 6 4 2 0

( ) cos( )

( ) cos( )

T

T

a u a u a u a u a u t dt

a u a u a u a u a u t dt









 


 

+

+

+ + + + +
=

+ + + + +




 (34) 

Submitting Eq. (12) into Eq. (34), after simple calculation, we have  

 

2 4 6 83 5 3 7 5 3 9 7 5 3
1 3 5 7 92 4 6 4 8 6 4 10 8 6 4

6 6 4 29 7 5 3 7 5 3 5 3 3
8 6 4 2 010 8 6 4 8 6 4 6 4 4

a a A a A a A a A

a A a A a A a A a


+ + + +
=

+ + + +
 (35) 

Additionally, we can write Eq. (32) in the form 

  3 5 7 9 8 6 4 2

1 3 5 7 9 8 6 4 2

0

1
( ) 0u a u a u a u a u a u a u u a u u a u u a u u

a
    + + + + + − + + + =  (36) 



 Periodic Solution of a Micro-Electromechanical System 197 

By Eq. (14) we have 

2

2 4 6 8 7 5 3
2 1 3 5 7 9 8 6 4 2

3 /20
3 /2

2 4 6 8 8 6 4 2 23 9 27 81 81 27 9 3
1 3 5 7 9 8 6 4 24 16 64 256 256 64 16 4

0

( )

( )

u A

u A

a a u a u a u a u a u u a u u a u u a uu

a

a a A a A a A a A a A a A a A a A

a







=

=−

   + + + + − + + +
=

+ + + + − + + +
=

 (37) 

After simple operation, Eq. (37) leads to the following frequency-amplitude relationship  

 

2 4 6 83 9 27 81
1 3 5 7 92 4 16 64 256

8 6 4 281 27 9 3
0 8 6 4 2256 64 16 4

a a A a A a A a A

a a A a A a A a A


+ + + +
=

+ + + +
 (38) 

Eq. (35) or Eq. (38) unlocks the frequency-amplitude relationship, and in view of 

Eq.(13), we can also reveal the effect of the initial conditions on the  frequency property. 

4. CONCLUSION  

This paper studies the frequency property of MEMS systems, its frequency-amplitude 

relationship is revealed, and the effect of the initial conditions on the frequency property is 

also elucidated. When the obtained frequency is zero, we can easily obtain the threshold 

value for K in Eq. (3), beyond which the pull-in instability occurs. The threshold value for 

the amplitude is also found, this makes it easy to monitor MEMS systems, and it can set 

off an alarm when the amplitude approaches to its threshold value.  

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