11486


FACTA UNIVERSITATIS  
Series: Mechanical Engineering Vol. 21, No 2, 2023, pp. 191 - 200  

https://doi.org/10.22190/FUME230114007H 

© 2023 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND 

Original scientific paper 

PULL-DOWN INSTABILITY OF  

THE QUADRATIC NONLINEAR OSCILLATORS 

Ji-Huan He1,2,3, Qian Yang3, Chun-Hui He4,5,  

Abdulrahman Ali Alsolami5 

1National Engineering Laboratory for Modern Silk,  

College of Textile and Clothing Engineering, Soochow University, Suzhou, China 
2School of Mathematics and Information Science, Henan Polytechnic University,  

Jiaozuo, China  
3School of Science, Xi’an University of Architecture and Technology, Xi’an, China 

4School of Mathematics, China University of Mining and Technology, Xuzhou,  

Jiangsu, China  
5Department of Mathematics, Faculty of Science, King Abdulaziz University,  

Jeddah, Saudi Arabia 

Abstract. A nonlinear vibration system, over a span of convincing periodic motion, 

might break out abruptly a catastrophic instability, but the lack of a theoretical tool has 

obscured the prediction of the outbreak. This paper deploys the amplitude-frequency 

formulation for nonlinear oscillators to reveal the critically important mechanism of the 

pseudo-periodic motion, and finds the quadratic nonlinear force contributes to the 

pull-down phenomenon in each cycle of the periodic motion, when the force reaches a 

threshold value, the pull-down instability occurs. A criterion for prediction of the 

pull-down instability is proposed and verified numerically. 

Key words: Micro-electromechanical system, Eardrum oscillator, Pull-in instability, 

Asymmetrical oscillation, Vibration attenuation, Energy harvesting 

1. INTRODUCTION 

The nonlinear oscillators with even nonlinearities, especially the quadratic nonlinear 

oscillators [1-5], became a hot topic recently in both mathematics and engineering, because 

scientists and engineers cannot exactly differentiate between the periodic motion and the 

fake periodic motion with ease. Scientists have now found that a convincing periodic 

 
Received: January 14, 2023 / Accepted February 08, 2023  

Corresponding author: Ji-Huan He and Chun-Hui He 

Affiliation, National Engineering Laboratory for Modern Silk, College of Textile and Clothing Engineering, 
Soochow University, Suzhou, P. R. China 

E-mails: Hejihuan@suda.edu.cn  (J.H. He), mathew_he@yahoo.com (C.H. He) 

mailto:Hejihuan@suda.edu.cn
mailto:mathew_he@yahoo.com


192 J.H. HE, Q. YANG, C.H. HE, A.A. ALSOLAMI 

 

motion might be a fake one, which is called as the pseudo-periodic motion [6,7], after some 

cycles of seeming periodic motion, its motion property will be suddenly changed, and the 

instability occurs, and scientists are often unable to spot the pseudo-periodic motion. 

Though mathematicians can reveal easily the bifurcation property of such a nonlinear 

oscillator [8], the pseudo-periodic motion of the quadratic nonlinear oscillator has never 

been reported, and the critically important mechanism of the pseudo-periodic motion needs 

to be revealed, so that its instability can be completely avoided. 

We consider an important nonlinear oscillator which involves even nonlinearities, the 

Toda oscillator [9,10,11], which plays an important role in physics: 

 
2

2
( 1) 0+ − =

ud u
k e

dt
  (1) 

where k is constant. Eq. (1) can be written equivalently in the following form 

 
2

2 3 4

2
0

2 6 24
+ + + + + =

d u k k k
u u u u

dt
 (2) 

The Toda oscillator behaves periodically when k is small, however the amplitude might 

lead to either infinity or zero when the absolute value of k reaches a threshold value. 

As another example, we consider a microelectromechanical system, which can be 

modelled by the following differential equation [12]: 

 
2

2
0

1
+ − =

−

d u K
u

udt
 (3) 

where u is dimensionless displacement and K is a voltage-related parameter. Eq. (3) can be 

written as  

 
2

2 3 4

2
(1 ) 0+ − + + + + =

d u
u K u u u

dt
 (4) 

The even nonlinearities in Eq. (4) are the main contributor to the pull-in instability, 

which should be avoided in the practical applications. The pull-in instability has been 

widely studied in the micro electromechanical systems (MEMS) [13-17], when the voltage 

is larger than its threshold value, the pull-in instability occurs, otherwise the systems move 

periodically. In this paper we focus on a similar phenomenon in quadratic nonlinear 

oscillators [18,19], and we call it as the pull-down instability.  

2. THE QUADRATIC NONLINEAR OSCILLATOR 

We consider the following quadratic nonlinear oscillator 

 
2

2

2
0+ + =

d u
u u

dt
  (5) 

with the following initial conditions 



 Title of the Paper (all the main words start with a capital leter, but not connecting words) 193 

 

 (0) , (0) 0= =u A u   (6) 

where u is the displacement, β is a positive constant, A is the initial amplitude.  

Eq. (5) is called as the eardrum oscillator [19]. For small values of β, a periodic motion 

is predicted; however, when it increases to a threshold value, the kinetic energy will be 

gradually consumed, and finally the periodic motion is forbidden, this phenomenon is 

called as the pull-down instability.  

Fig. 1 shows the dynamical properties of Eq. (5) when A=0.1. When β<<1, a periodic 

solution is predicted as shown in Fig. 1(b). When β is larger than a threshold value, the 

pull-down instability occurs as shown in Fig. 1(d). For β near to the threshold value, the 

system will have a periodic motion first, and after some cycles, the pull-down instability 

happens suddenly. This phenomenon is called as the pseudo-periodic motion [6,7] as 

shown in Fig. 1(c). 

 
(a) (b) 

 
(c) (d) 

Fig. 1 Periodic motion and pull-down instability of the quadratic nonlinear oscillator when 

A=0.1.  (a) phase diagram; (b) periodic motion; (c) pseudo-periodic motion; and 

(d) pull-down instability  

A periodic motion can be generally expressed by a sinusoidal function u(t) = Bsin(ωt+θ), 

where B and θ are determined by the initial conditions. The displacement of the harmonic 

motion varies from B to negative B periodically.  



194 J.H. HE, Q. YANG, C.H. HE, A.A. ALSOLAMI 

 

In Fig. 1, we find that during the periodic motion (Fig. 1(b)) and the pseudo-periodic 

motion (Fig. 1(c)), the quadratic force pulls down the system in each cycle. The amplitude 

changes from 0.1 to negative 0.11 in Fig. 1(b) and negative 0.2 in Fig. 1(c), respectively. 

This unsymmetrical oscillation phenomenon is interesting and amazing because it cannot 

be seen in any a conservative nonlinear oscillator [20-22], and it can be concluded that the 

quadratic nonlinear oscillator cannot be always used for vibration attenuation and an 

energy harvesting system [23,24]. The unsymmetrical amplitude in each cycle certainly 

frustrates scientists and engineers, because even the van der Pol oscillator [25] leads to a 

sinusoidal motion when time tends to infinity.   

When β increases to a threshold value, the pull-down instability occurs, and a lack of a 

theoretical model to predict the time and the condition of the pull-down outbreak has 

greatly reduced the operation reliability, and hindered its applications.  

3. PERIODIC MOTION WITH ASYMMETRIC AMPLITUDES   

This section introduces the frequency formulation for nonlinear oscillators [26] and 

extends to the nonlinear oscillators with even nonlinearities. 

Considering a nonlinear equation in the form   

 
2

2
( ) 0+ =

d u
u

dt
 , (0) , (0) 0= =x A x  (7) 

where  λ is a nonlinear function of u, generally it can be expressed as  

 
3 2 1

1 3 2 1
( )

+

+
= + + +

n

n
u a u a u a u   (8) 

For a nonlinear oscillator, it requires 

 
( )

0
u

u


 (9) 

The frequency formulation is [26]  

 
3

2

2 ( )
=

=u A

u

u


  (10) 

The frequency formulation is simple but it gives a relatively high accuracy of the 

frequency-amplitude relationship, and it becomes a universal tool to various vibration 

systems [27-29].  

For λ(u)=a1u+a3u3, Eq. (7) is the Duffing equation, and Eq. (10) leads to the following 

result 

 
2 2

1 3

3
=

4
+a a A  (11) 

Eq. (11) is same as that obtained by the homotopy perturbation method [30]. Using the 

frequency formulation of Eq. (10), for Eqs. (5) and (6), we have  



 Title of the Paper (all the main words start with a capital leter, but not connecting words) 195 

 

 
3

= 1
2

+ A     (12) 

Table 1 shows Eq. (12) has good accuracy of the approximate period, T=2π/ω, for 

βA<<1, and its accuracy deteriorates greatly when the value of βA becomes large. The 

errors arise in the quadratic force, f=βu2. When the quadratic force is small, Eq. (5) can be 

solved by the perturbation method [19], but in practical applications, we do not know the 

criterion for the small value.  

Table 1 Comparison the approximate periods with the exact one  

A β Exact 

period 

Approximate 

Period Eq.(12) 

Error 

 

Approximate 

Period Eq.(20) 

Error 

(%) 

0.1 1    6.266 6.0276   3.8% 6.300 0.55% 

0.3 1    6.627 5.5979 15.5% 6.450 2.66% 

0.5 0.3 6.308 5.9109   6.2% 6.323 0.24% 

0.7 0.3 6.399 5.7795   9.6% 6.363 0.56% 

Eq. (10) is valid for nonlinear oscillators with only odd nonlinear terms, our problem 

has a quadratic nonlinear term, so the frequency formulation has to be modified.  

We re-write Eq. (5) in the form   

 
2

2
(1 ) 0, 0+ − = 

d u
u u u

dt
  (13) 

 
2

2
(1 ) 0, 0+ + = 

d u
u u u

dt
  (14) 

Eqs. (13) and (14) give, respectively, the following frequency formulations:  

 
2

1

3
=1

2
− A   (15) 

 
2

2

3
=1

2
+ A   (16) 

The solution can be approximately expressed as  

 
2

1

cos , 0
( )

( ) cos , 0

A t u
u t

A A t u






= 

+  
 (17) 

where ΔA in the increment of the amplitude when u<0.  

The periods of Eqs. (13) and (14) are, respectively, given as  

 
1

1

2 2
=

3
1

2

=

−

T

A

 




  (18) 



196 J.H. HE, Q. YANG, C.H. HE, A.A. ALSOLAMI 

 

 
2

2

2 2
=

3
1+

2

=T

A

 




 (19) 

So the period of Eq. (5) can be calculated as  

 
1 2 =

2 3 3
1 1+

2 2

+
= +

−

T T
T

A A

 

 

 (20) 

Eq. (20) shows much better accuracy for βA<0.5, see Table 1. When βA=1.1547, the 

period by Eq. (20) becomes infinite large. 

4. PULL-DOWN INSTABILITY 

When βA tends to the threshold value, the pull-down instability occurs. The motion of 

Eq. (5) can be decomposed into two parts as given in Eqs. (13) and (14). It is obvious that 

Eq. (13) has a tendance of a pull-down motion, and Eq. (14) gives an opposite motion. We 

consider an extreme case when u1=-2A for Eq. (13), and u2=A for Eq. (14), the combined 

displacement is u1+u2=-2A+A=-A, so Eq. (5) is still periodic.  However if u1<-2A, we have 

|u1+u2|>A, under such case, the motion is pulled down, and its periodic motion is forbidden.  

We consider the extreme condition of u1=-2A, Eq. (13) can be written approximately as  

  
2

2
(1 2 ) 0+ − =

d u
A u

dt
  (21) 

This predicts  

 
2

1
=1 2− A   (22) 

The pull-down instability occurs when 
2

1
0  , that is  

 0.5A  (23)  

So βA=0.5 is the threshold value.  

We give another approach to determination of the threshold value. Eq. (5) can be written in 

the form  

 
2 2 31 1 1

( )
2 2 3

+ + =
du

u u H
dt

   (24) 

where H is the Hamilton constant. By the initial conditions given in Eq. (6), Eq. (24) 

becomes 

 
2 2 2 3 31 1 1

( ) ( ) ( )
2 2 3

= − + −
du

A u A u
dt

   (25) 

When u = -2A, the kinetic energy tends to zero, so the pull-down instability occurs when  

 
2 2 2 3 31 1 1

( ) ( 2 ) ( ( 2 ) ) ( ( 2 ) ) 0
2 2 3

= − = − − + − − 
du

u A A A A A
dt

  (26) 



 Title of the Paper (all the main words start with a capital leter, but not connecting words) 197 

 

Eq. (26) results in βA>0.5 as shown in Eq. (23). When βA<0.5, a periodic motion or a 

pseudo-periodic motion is predicted. When βA>0.5, the pull-down instability occurs. To 

verify this criterion, we consider various values of A and β, and the results are illustrated in 

Figs. 2-4.   

When the value of βA is less than the threshold value, 0.5, but it is near to the value, the 

pseudo-periodic motion is observed, for example, βA=0.495 in Fig. 2(c); while it is far 

from the value, for example, βA=0.3 in Fig. 2(c), a periodic motion is seen. So far, we have 

no criterion for exactly predicting the value of βA for a pseudo-periodic motion, where an 

unsymmetrical oscillation deteriorates gradually in each cycle, and when u reaches 

negative 2A, the  convincing periodic motion stops suddenly. 

  
(a) (b) 

  
(b) (d) 

Fig. 2 The criterion of the pull-down instability when A=0.3.  (a) phase diagram; (b) periodic 

motion βA=0.3; (c) pseudo-periodic motion βA=0.495 and (d) pull-down instability 

βA=0.51 



198 J.H. HE, Q. YANG, C.H. HE, A.A. ALSOLAMI 

 

 
(a) (b) 

 
(b) (d) 

Fig. 3 The criterion of the pull-down instability when A=0.5.  (a) phase diagram; (b) periodic 

motion βA=0.15; (c) pseudo-periodic motion βA=0.495 and (d) pull-down instability 

βA=0.55  

In Figs. 2-4, we can see that when βA is small, a periodic motion is predicted. With 

gradually increasing βA, a pseudo-periodic motion might occur. Finally, when βA is larger 

than 0.5, the pull-in instability is observed.  

5. CONCLUSION 

This paper proposes an extremely important concept of the pull-down breakout for 

nonlinear oscillators with even nonlinearities, and it will become a powerful research tool in 

nonlinear vibration theory. A sophisticated modification of the frequency formulation is 

suggested, and a criterion is built for judging the pull-down instability, the unsymmetrical 

amplitude motion, and the pseudo-periodic motion. The amplitude increment, ΔA, in Eq. (17) 

will be discussed in a forthcoming paper, and when the absolute value of ΔA is larger than A, 

the pull-down instability occurs abruptly. Hence, the amplitude change is the best parameter 

to predict the pseudo-periodic motion and the time when the pull-down instability occurs.   



 Title of the Paper (all the main words start with a capital leter, but not connecting words) 199 

 

 
(a) (b) 

  
(c)  (d) 

Fig. 4 The criterion of the pull-down instability when A=0.7:  (a) phase diagram; (b) periodic 

motion βA=0.21; (c) pseudo-periodic motion βA=0.49 and (d) pull-down instability 

βA=0.56 

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