13730


FACTA UNIVERSITATIS  
Series: Mechanical Engineering Vol. 23, No 2, 2025, pp. 197 - 210   

https://doi.org/10.22190/FUME250526022H 

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

Original scientific paper 

A MODIFIED FREQUENCY FORMULATION FOR  

NONLINEAR MECHANICAL VIBRATIONS 

Chun-Hui He1,3, Ji-Huan He2,4,5,6, Jinfeng Ma7,  

Abdulrahman Ali Alsolami8, Xiao-Jun Yang3 

1 School of Civil Engineering & Transportation, Foshan University, Foshan, China 
2Department of Mathematical Sciences, Saveetha School of Engineering, SIMATS, 

Chennai, Tamil Nadu, India 
3School of Mathematics, China University of Mining and Technology, Xuzhou China   

4School of Jia Yang; Zhejiang Shuren University, Hangzhou, China 
5School of Information Engineering, Yango University, Fuzhou, China 

6School of Mathematics and Big Data, Hohhot Minzu College, Hohhot, China 
7College of Civil Engineering and Architecture, Jiaxing University, Jiaxing, China 

8Department of Mathematics, Faculty of Science, King Abdulaziz University,  

Jeddah, Saudi Arabia 

ORCID iDs: Chun-Hui He  https://orcid.org/0000-0003-0810-5248  
 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    
 Xiao-Jun Yang  https://orcid.org/0000-0003-0009-4599    

Abstract. The paper elucidates the characteristics of nonlinear oscillators that contain 

quasi-linear terms arising in mechanical engineering. Analytical solutions for this type 

of nonlinear oscillator are often difficult to obtain; they may involve singular and 

discontinuous terms, which can make them significantly more difficult to obtain. In this 

paper, a modified frequency formulation for this type of nonlinear oscillator is presented. 

While the mathematical proof is not provided, illustrative examples are included to 

demonstrate the remarkable simplicity and reliability of the approach. This paper 

challenges the conventional wisdom in analytical methods and offers a promising new 

direction for further investigation. 

Key words: Nonlinear vibration, Non-perturbative approach, Quasi-linear oscillator, 

Discontinuous oscillator, Fractional potential, Zigzag oscillation 

 
Received: May 26, 2025 / Accepted July 03, 2025  

Corresponding author: Ji-Huan He, Jinfeng Ma 

School of Jia Yang, Zhejiang Shuren University, Hangzhou, Zhejiang, China 
College of Civil Engineering and Architecture, Jiaxing University, Jiaxing, China 

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

https://orcid.org/0000-0003-0810-5248
https://orcid.org/0000-0002-1636-0559
https://orcid.org/0009-0000-2348-9142
https://orcid.org/0000-0002-8189-4987
https://orcid.org/0000-0003-0009-4599


198 C.H. HE, J.H. HE, J. MA, A.A. ALSOLAM, X.J. YANG 

 

1. INTRODUCTION 

In the ever-evolving landscape of contemporary science and technology, nonlinear 

vibration systems [1, 2] have risen to prominence as a foundational building block with 

far-reaching consequences across a diverse array of disciplines. Specifically, they have 

provided substantial support to the domain of mechanical engineering [3], particularly the 

field of micro-electromechanical systems (MEMS) [4]. It is imperative to comprehend the 

frequency-amplitude relationship to identify novel prospects and propel the advancement 

of related technologies. For instance, the capacity to discern the slightest tremors of 

starquakes, a capability of paramount importance in the realm of astronomy [5], exemplifies 

this phenomenon. These seismic events offer invaluable insights into the internal structure and 

composition of stars. Additionally, high-frequency vibrations have been observed in the 

coupling of excitons in π-conjugated molecules [6]. 

The nonlinearity exhibited by oscillators can have various origins. Cubic-quintic 

nonlinearity is a frequently employed phenomenon in the domain of mechanical engineering 

[7-9]. However, there are also nonlinearities of the form un), with n < 1. For instance, the 

integer n could be either 1/3 or -1. These nonlinear oscillators consist of both singular 

oscillators and discontinuous oscillators. The procurement of approximate solutions for these 

oscillatory phenomena poses a substantial challenge in the realms of mathematics and 

mechanical engineering. The complexity inherent to this phenomenon stems from the fact 

that singular oscillators frequently possess unique properties, thereby rendering traditional 

solution methods ineffective. Conversely, discontinuous oscillators pose significant challenges 

due to their sudden fluctuations in behavior. Confronting these challenges necessitates the 

development of innovative methodologies and a profound comprehension of both mathematical 

principles and mechanical systems. 

A paucity of known periodic functions exists for the approximation of nonlinear 

oscillators. In the domain of nonlinear vibration theory, the cosine function and the sine 

function are the most frequently employed. While it is possible to theoretically model a 

zigzag periodic motion through an infinite series of cosine or sine functions, this approach 

is mathematically intricate. Engineers generally prioritize the key factors that influence the 

period or frequency of these oscillators. However, contemporary analytical methodologies 

predominantly center on the pursuit of an approximate cosine or sine solution for a complex 

nonlinear vibration system, with the objective of deriving an expression for the frequency. 

In order to make a more precise prediction of the impact of parameters on the frequency, it 

is necessary to undertake a cumbersome derivation process. However, this approach 

frequently falls short in providing highly accurate results due to the inherent limitations of 

using cosine and sine functions. Alternative methods, such as numerical simulations or 

advanced experiment techniques [10], have the potential to serve as an auxiliary means of 

vibration analysis. However, these approaches are costly and cannot provide more accurate 

predictions or overcome the complexity associated with traditional methods. 

Another periodic function that must be considered is the Jacobi elliptic functions [7]. 

These functions are particularly well-suited for Duffing-like oscillators due to their specific 

mathematical properties, which can accurately capture the behavior of such oscillators. 

However, the differential operations of these mechanisms are intricate, posing a challenge 

for engineers to comprehend and utilize. The primary concern of an engineer is not the 

approximate solution per se; rather, the emphasis is placed on the precision of the frequency-

amplitude relationship. This relationship is of paramount importance, as it constitutes the 



 A Modified Frequency Formulation for Nonlinear Mechanical Vibrations 199 

 

foundation for the optimal design of a nonlinear vibration system. A precise understanding 

of the frequency-amplitude relationship is essential for engineers to fine-tune the parameters of 

the system, ensuring its optimal performance and reliability. Absent a reliable estimate of this 

relationship, the design process becomes a speculative endeavor, with the potential to yield 

suboptimal results and inefficiencies. 

2. PROBLEM FORMULATION 

In a previous publication, Ji-Huan He [11] considered the following nonlinear oscillator: 

 
2 1

1 2 1
1

0
N

n

n
n

u a u a u +

+
=

 + + =  (1) 

and proposed the following frequency formulation:  

 
2 2

1 2 1
1

3

2 2

N
n

n
n

a a A
n

+
=

= +
+

  (2) 

where a2n+1 (n=0~N) are constants, A is the amplitude. The frequency formulation is 

effective for nonlinear oscillators with high order nonlinearities, for example, the cubic-

quintic-septic Duffing oscillator [12]: 

 
2 2

1 2 1 2 1
1

( )
N

n

n n
n

a a u+ +
=

= +  (3) 

This is to choose a location point for each nonlinear term, u2n+1: 

 1/2

2 1

3
( )
2 2

n

nu A
n

+ =
+

 (4) 

This frequency formulation holds immense importance in the realms of both mathematics 

and nonlinear vibration theory. It offers a swift and effective means of understanding the two 

main factors of oscillators and their intricate relationship. This knowledge is essential for 

delving deep into the vibration properties and enabling advanced applications. Take, for 

example, for the most energy harvesting devices or vibration dampers, their low-frequency 

property implies a large amplitude motion. As an illustration, we now consider the following 

nonlinear oscillator [13]. 

 2 2 2(1 ) (1 ) 0au u auu u u + + − − =  (5) 

where α is a constant. It can be re-written as 

 
3 2 2 0u u u auu au u  − + + + =  (6) 

By the frequency formulation of Eq. (3), we have 

 
2 2 21 u au auu = − + + +  (7) 

The location point is chosen as 



200 C.H. HE, J.H. HE, J. MA, A.A. ALSOLAM, X.J. YANG 

 

 
3

2
u A=  (8) 

Its derivatives can be approximately calculated as   

 
2 2 1

2
u A u A = − =   (9) 

 
2 23

2
u u A = − = −   (10) 

So we have 

 
2 2 2 2 2 2 2 2 23 1 3 3 1

1 1
4 4 4 4 2

A a A a A A aA= − + + − = − + +     (11) 

After a simple calculation, we have 

 

3 2

4

1 2

2

1

1

A

aA

− +
=

+
  (12) 

which is exactly same as that in Ref. [14]. There are an abundant number of publications 

on the applications of the original frequency-amplitude formulation and its various 

modifications. He and Liu [15] gave a rigorous mathematical perspective for He’s 

frequency formulation. Ismail et al. [16] and Hashemi [17] found that this formulation is 

extremely well-suited for strongly nonlinear oscillators. Their studies have demonstrated 

its effectiveness in analyzing and understanding the complex periodic properties. Kawser 

et al. conducted a comparison between the frequency formulation and numerical simulation 

for the analysis of the jet engine vibration system. They discovered that the former is not 

only simple to use but also highly effective [18]. Tsaltas [19] concluded that the one-step 

frequency formulation holds great promise as a new approach for nonlinear oscillators. 

Additionally, other applications of the frequency formulation are equally captivating 

[20,21]. These applications span different fields and highlight the versatility and importance of 

the frequency formulation in understanding and solving a wide range of problems related to 

nonlinear oscillators. 

Although significant achievements have been attained, there is still ample room for 

further improvement, particularly for nonlinear oscillators with a nonlinear term of the 

form un where n<1. This type of oscillator encompasses singular oscillators and zigzag 

oscillators, which are extremely challenging to solve analytically. 

3. FREQUENCY FORMULATION FOR QUASI-LINEAR OSCILLATOR AND SINGULAR OSCILLATORS 

The aforementioned frequency formulation is applicable to n>1. It is evident that: 

 2 1 , for 1/ 2nu A n+   . (13) 

However, this is incompatible with the characteristics of a conservative oscillator, 

which exhibits vibrations from u=A to u=-A. There exists a large set of nonlinear oscillators 



 A Modified Frequency Formulation for Nonlinear Mechanical Vibrations 201 

 

with an irrational term [22] or a term of the form u1/(2n + 1) [23]. Though there were some 

effective modifications of the frequency formulation [24, 25], here a more effective 

formulation will be recommended.    

The following nonlinear oscillator is now considered. 

 
1

0, 2i

N
n

i i
i

u b u n
=

 + =   (14) 

where bi (i=1~N) are constants. For periodic solutions, it requires 

 
( 1)

1

0i

N
n

i
i

b u
−

=

  (15) 

The previous approaches to this type of nonlinear oscillators are some famous analytical 

methods, e.g., the variational iteration method [26] and the homotopy perturbation method 

[27, 28], all these methods require sophisticated technology with complex calculation, 

however, in the practical applications, the frequency-amplitude relationship is the most 

important factor. A simple yet reliable approach is always highly welcome. Now, this paper 

presents the following frequency formulation: 

 
( 1)2

1

6

5
i

N
n

i
i i

b A
n

−

=

=
+

  (16) 

It can be also written in the form  

 
( 1)2

1

( ) i

i

N
n

i n
i

b u
−

=

=  (17) 

where the location point for each nonlinear term is given as 

 
( 1)1/6

( )
5

i

i

n

n

i

u A
n

−
=

+
 (18) 

The new frequency formulation is extremely simple. A more straightforward approach 

is always advantageous for engineers. The frequency formulation demonstrates considerable 

promise for a diverse array of prospective applications. This novel approach has the potential to 

transform analytical techniques within the domain of vibration theory. The method provides a 

rapid yet reliable comprehension of intricate vibration systems through a straightforward step 

that is more efficient than traditional analytical methods, which entail complex and detailed 

calculations. It can be utilized for the optimization of complex vibration systems without the 

necessity for onerous computations. The remarkable versatility of this new formulation 

represents a significant advancement in the field of nonlinear vibration theory. It is adaptable to 

a variety of situations and requirements, thereby facilitating the realization of novel possibilities 

and opportunities. As research and development continue, the potential applications of this 

approach are likely to expand, with the prospect of substantial benefits for the field of 

nonlinear science. The following examples demonstrate the efficacy and simplicity of this 

approach. 



202 C.H. HE, J.H. HE, J. MA, A.A. ALSOLAM, X.J. YANG 

 

4. EXAMPLES 

4.1. Example 1. Nonlinear Oscillator with Fractional Potential 

Nonlinear oscillators with fractional potential are also called as quasi-linear oscillators, 

the nonlinear term is always in the form, u1/(2n+1), where n is a natural number. In this 

example, we consider the following oscillator. 

 1/3 0u u + =  (19) 

where β is a positive constant. Its variational formulation is 

 
2 4/31 3

( )
2 4

J u u u dt
 

= − 
 
   (20) 

The potential is represented by a fractional exponent, and thus Eq. (20) is designated as 

a nonlinear oscillator with fractional potential. This nonlinear oscillator has been the subject of 

extensive attention in the open literature, as evidenced by references [29, 30]. The nonlinear 

nature of the system presents a considerable challenge in attempting to find a general 

closed-form solution through analytical means. The nonlinear characteristics introduce 

intricate interactions and behaviors that render the derivation of an approximate solution 

an exceptionally challenging endeavor. 

Following meticulous and comprehensive calculations, a notable accomplishment is the 

derivation of the frequency-amplitude relationship for Eq. (20), as documented in [29, 30], 

which is: 

 
1/3 1.0768A−= . (21) 

This relationship, with a relative error of 0.59%, provides a crucial link between two 

fundamental properties of the oscillator and serves as a valuable resource for further studies 

and applications. This provides a basis for analyzing and predicting the behavior of the 

oscillator in different scenarios, thereby opening up new avenues for research and practical 

applications in a variety of fields. 

Notwithstanding the significant advancements achieved through meticulous and intricate 

computations, engineers have been keenly awaiting a straightforward yet efficacious 

methodology to swiftly and dependably comprehend the vibration characteristics. The time has 

now come to seize this opportunity. By employing the frequency formulation of Equation (17), 

the desired results can be obtained with immediate effect. 

 
2 2/3 2/3

1
3

6 18

5 16
A A− −= =

+
    (22) 

or 

 
1/2 1/3 1.060660 A−=   (23) 

It is noteworthy that it exhibits the same form as that of Eq. (22). Given its simplicity, 
the frequency formulation represents a significant accomplishment in the field.  

The accuracy of this method is comparable to that attained by the variational iteration 
method or the homotopy perturbation method. The one-step approach is a highly attractive 
proposition. The simplicity and efficiency of this approach render it an attractive option, 
particularly in situations where rapid insights and the frequency-amplitude relationship are 
required without the intricacy of more elaborate methods. 

https://baike.baidu.com/item/%E5%B8%8C%E8%85%8A%E5%AD%97%E6%AF%8D/4428067#3-2


 A Modified Frequency Formulation for Nonlinear Mechanical Vibrations 203 

 

The exact frequency is [29, 30]  

 1/2 1/31.070451exact A−=   (24) 

The relative error is 0.91%, which is deemed to be within an acceptable range for 

engineering applications.  

4.2. Example 2. Discontinuous Oscillator with Sign Function 

The study of nonlinear oscillators with discontinuous terms represents a fascinating and 

rich area of inquiry within the broader fields of physics and mechanical engineering. 

Discontinuous terms may originate from impacts, switches, or sudden alterations in system 

parameters. The aforementioned discontinuities present considerable difficulties in the 

analysis and comprehension of the oscillator's behavior. For example, a nonlinear term with 

the sign function frequently plays a pivotal role in such systems, as it can represent abrupt 

changes in direction or state. The study of nonlinear oscillators with discontinuous terms 

represents a vibrant and multifaceted research domain, with implications that extend far 

beyond the immediate field of inquiry and have the potential to inform a vast array of 

practical applications. An investigation of the frequency-amplitude relationship may 

facilitate the discovery of new possibilities for engineering and scientific advancement. 

Researchers are engaged in ongoing efforts to develop new methods for the analysis and 

control of these oscillators. This includes the creation of novel theoretical models and 

experimental techniques, with the aim of gaining deeper insights into their behavior and 

the design of more efficient and reliable systems. 

In this example, we consider a nonlinear oscillator subjected to a constant force, as 

illustrated in Fig. 1.   

 

Fig. 1 Discontinuous oscillator with discontinuous restoring force 

The restoring force follows: when u>0, F(u)=1; when u<0, F(u)=−1. The governing 

equation can be expressed as 

 sgn( ) 0u u + =  (25) 

where β=1/m, m is the mass of the oscillator, and sgn(u) is defined as 

 

0

0

, 0 1, 0

sgn( ) 0, 0 0, 0

1, 0, 0

u u u

u u u

uu u

  
 

= = = = 
 − −  

. (26) 



204 C.H. HE, J.H. HE, J. MA, A.A. ALSOLAM, X.J. YANG 

 

Considerable effort has been dedicated to seeking an accurate analytical approximation 

for Equation (26), as referenced in [31-33]. The primary mathematical tool utilized is the 

homotopy perturbation method. In the application of this method to nonlinear vibration 

systems, the construction of the homotopy equation, the selection of the initial guess, and 

the elimination of secular terms during the lengthy calculation process represent the primary 

focal points [34]. These aspects are of paramount importance in guaranteeing the efficacy 

and precision of the method in analyzing and approximating nonlinear vibration phenomena.  

After a complex calculation, Belendez et al. [30] obtained high-accuracy analytical 

approximations by the homotopy perturbation method, resulting in the following frequencies 

 1/2

1 1.128379= A− , (27) 

and 

 1/2

2 1.107452= A− . (28) 

The relative errors are 1.59% and 0.29%, respectively. Liu obtained the same result of 

Eq. (28) by the modified Lindstedt–Poincaré method [33]. Although the achievement has 

been attained, engineers are currently seeking a straightforward yet impactful methodology. 

By employing the frequency formulation, we have promptly derived the following outcome. 

 2 1 16 6

0 5 5
A A− −= =

+
   , (29) 

or 

 1/2 1/21.0954 A−=  . (30) 

This result is comparable to those produced by the homotopy perturbation method [31] 

and the modified Lindstedt–Poincaré method [33]. The exact frequency is: 

 1/2

ex 1.1107= A− . (31) 

The relative error is 1.37%, which is comparable to the results presented in Refs. [37, 

39], where a length calculation and specialized skills are required. In contrast, the present 

approach is notable for its simplicity. In nonlinear theoretical analysis, a minimal 

calculation with one or two lines can have maximal influence in the academic community, 

as Ludwig Mies van der Rohe (1886–1969) said, "Less is more". 

4.3. Example 3. Discontinuous Oscillator with Absolute Value Symbols 

The study of discontinuous oscillators with absolute value symbols represents a fascinating 

area of inquiry within the field of nonlinear dynamics. The introduction of absolute value 

symbols can result in the emergence of intricate dynamics, which in turn presents significant 

analytical challenges.  

Researchers employ a range of analytical techniques to examine these oscillators, 

undertaking significant efforts in this regard. Such techniques may include numerical 

simulations and analytical approximations. In this example, we consider a discontinuous 

oscillator with absolute value terms, as illustrated in Fig. 2. 



 A Modified Frequency Formulation for Nonlinear Mechanical Vibrations 205 

 

 

Fig. 2 Schematic of a discontinuous oscillator with absolute value terms 

It is assumed that the equivalent elastic coefficient is proportional to the magnitude of 

displacement, and the oscillator can be expressed as:  

 0u u u + = . (32) 

By the frequency formulation, we have: 

 2 6 6

2 5 7
A A= =

+
 , (33) 

or 

 0.92582 A= , (34) 

while the exact one is [35]: 

 0.914681exact A= . (35) 

The relative error is 1.21%, which is an acceptable level of precision given the simplicity 

of the method employed. 

Wang and He [36] achieved a notable result through a combination of sophisticated 

techniques and meticulous calculations, which is 

 
8

0.9214
3

A
A= =


. (36) 

This achievement was of tremendous significance because, prior to this, there was no 

analytical method that could effectively solve Eq. (33). Reference [36] made significant 

advancements in nonlinear science and nonlinear vibration theory. It provides an excellent 

illustration of the efficacy of meticulous analysis and innovative thinking in addressing 

complex problems and makes a valuable contribution to the expanding body of knowledge 

in the relevant discipline.  

The current approach allows for a straightforward and effective solution, demonstrating 

the value of simplicity in achieving comparable outcomes. The new frequency formulation 

provides a novel avenue for future research in nonlinear vibration theory and nonlinear studies. 

The optimal approach is to employ a method that is both straightforward and efficacious. 

4.4. Example 4. Singular Oscillator 

A singular oscillator represents a distinctive category of dynamical systems that display 

a range of intriguing and frequently intricate behaviors. These oscillators possess singularities, 

which give rise to distinctive characteristics that are not observed in traditional oscillators. 

The study of singular oscillators is of significant importance in a number of fields, 

including physics, engineering, and mathematics. It facilitates comprehension of complex 



206 C.H. HE, J.H. HE, J. MA, A.A. ALSOLAM, X.J. YANG 

 

systems and may yield novel insights and applications. Researchers are engaged in ongoing 

investigations into the properties and behaviors of singular oscillators, with the objective 

of unlocking their potential for innovative technologies and scientific discoveries [37]. The 

following example considers a singular oscillator: 

 1 0u u− + =  (37) 

According to the frequency formulation , we have: 

 2 2 2.
6

1 5
1 5A A− −= =

− +
   , (38) 

or 

 1/2 11.2247 A−=  . (39) 

The exact one is [37]: 

 exact

1.2533
=

A
 . (40) 

The relative error is 2.28 %. The singularity presents a challenge for both analytical and 

numerical methods. However, the one-step frequency formulation offers a solution that 

produces optimal results with minimal calculation. 

4.5. Example 5. Nonlinear Oscillator with Fraction Term 

The study of nonlinear oscillators with fraction terms represents a fascinating and 

promising area of mathematical inquiry. It presents a challenge to traditional analytical 

methods, as it may not conform to the standard mathematical models that are typically 

employed. Researchers employ sophisticated techniques to comprehend and anticipate the 

behavior of these oscillators. 

Consider the following nonlinear oscillator with a fraction term: 

 
2

0 2
1

0
N

n

n
n

u
u

c c u
=

 + =

+
. (41) 

In order to utilize the aforementioned frequency formulation, we assume that A tends 

to infinity. Under this assumption, Eq. (41) becomes: 

 
1 2

2
1

0
N

n

n
n

u c u −

=

 + = . (42) 

Its frequency-amplitude relationship is 

 
2 2 2

2 2
1 1

6 3

1 2 5 3

N N
n n

n n
n n

b A b A
n n

− −

= =

= =
− + −

   (43) 

For Eq. (42) the frequency formulation should be modified as: 

 
2

2

0 2
1

1

3

3

N
n

n
n

n
c b A

=

=
−

+
 . (44) 



 A Modified Frequency Formulation for Nonlinear Mechanical Vibrations 207 

 

In this modification, the case of c0=0 and N=1  can be transformed into the form of Eq. 

(44). So it is suitable for large amplitude oscillation.  

In order to show its effectiveness, we consider a simple case: 

 
2

0
1

u
u

u
 + =

+
. (45) 

By the frequency formulation of Eq. (45), the following result is obtained: 

 
2

1

2
1

3
A

=

+

 . (46) 

The exact period can be calculated as: 

 
0 2 2

1
4

ln(1 ) ln(1 )

A

T du
A u

=
+ − +

 . (47) 

For the case A tends to infinity, we have: 

 

0 2 2

0 2 2

1
lim 4 lim

ln(1 ) ln(1 )

1
4 2 2

ln( ) ln( )

A

A A

A

T du
A u

du A
A u

→ →
=

+ − +

= =
−



 

 (48) 

So we have: 

 
1 12

lim lim 1.2533
2

exact
A A

A A
T

− −

→ →
= = =

 
 , (49) 

while from Eq. (28), we have: 

 1

2

1
l 4im 1.22

2

3

7
A

A

A

−

→
= = . (50) 

The relative error is 2.28%. It should be note that Eq. (48) is valid for large amplitude. 

For the small values of A, Eq. (46) can be approximated as: 

 
3 0u u u + − = . (51) 

By the frequency formulation given in Eq. (2), we have: 

 
2 2 23 3

1 1 ( )
2 4

A A O A= − = − + . (52) 

Considering Eqs. (48) and (49), we recommend the following matching one: 

 
2

1

2 3(1 )
1

3 2(1 )

m m
A

m m

=
+ −

+
+ −

 , (53) 



208 C.H. HE, J.H. HE, J. MA, A.A. ALSOLAM, X.J. YANG 

 

where m is the weighting factor, m=0 for A<1 and m=1 for A>1, and the value of m can be 

determined by matching Eq. (53) with the exact one when A=1. 

Now this paper challenges the following nonlinear oscillators: 

 1/3 1/5 3 5

1 2 3 4 5sgn( ) 0u u b u b u b u b u b u + + + + + + = , (54) 

with arbitrary initial conditions: 

 (0) , (0)u u= =  . (55) 

and this paper gives the following result without proof: 

 

2 2/3 4/5 1 2 4

1 2 3 4 51 1
3 5

2/3 4/5 1 2 4

1 2 3 4 5

6 6 6 3 3
1

5 5 0 5 2 2 2 4

9 15 6 3 1
1

8 13 5 4 2

b A b u b A b A b A

b A b u b A b A b A

− − −

− − −

= + + + + +
+ + + + +

= + + + + +



 (56) 

and 

 
2

2 2

2
A+ =





. (57) 

5. DISCUSSION AND CONCLUSION  

Dozens of research groups around the world are engaged in the development of analytical 
methods for the study of the frequency-amplitude relationship of complex nonlinear 
oscillators [38].  

The nonlinear vibration theory is cheaper and greener than other artificial methods, 
including sophisticated experiments, precise measurements, and time-consuming simulations. 
The nonlinear vibration theory has the potential to enhance the efficiency of energy harvester 
devices or to facilitate the conservation of mechanical vibrations, thereby reducing the emission 
of greenhouse gases. Furthermore, the frequency formulation can assist in the identification and 
search for optimal parameters for the devices currently under development. 

To date, a number of analytical methods have been proposed for this purpose, with the 
homotopy perturbation method and the variational iteration method representing the most 
commonly used mathematical tools. The frequency formulation represents a particularly 
simple method.  

This paper identifies the most fascinating aspect of the frequency formulation: its ability 
to link two primary factors that control periodic motion in a complex oscillator. The 
aforementioned examples illustrate the efficacy of the straightforward yet impactful frequency 
formulation in the context of scientific and technological advancement. The new frequency 
formulation presented in this paper represents a continuous process of improvement and 
expansion of its applications.  

The discovery constituted a profound breakthrough that astonished the scientific community. 
While the effectiveness is illustrated by examples, it remains in its nascent stages, and there are 
numerous challenges and prospects for future research.  

A rigorous mathematical proof of the frequency formulation is required to confirm its 
effectiveness. The formulation may encounter counter-examples in practical applications, 
which could lead to further technological developments. The innovation represents a 
significant advancement in technology and has the potential to transform existing practices.  



 A Modified Frequency Formulation for Nonlinear Mechanical Vibrations 209 

 

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

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

REFERENCES  

1. Ipek, C., 2023, Vibration analysis of shear deformable cylindrical shells made of heterogeneous anisotropic 

material with clamped edges, Journal of Applied and Computational Mechanics, 9(3), pp. 861-869. 
2. Tekin, G., Ecer, S., Kadioglu, F., 2023, An alternative procedure for longitudinal vibration analysis of bars 

with arbitrary boundary conditions, Journal of Applied and Computational Mechanics, 9(1), pp. 294-301. 

3. Barbosa, A., Kacem, N., Bouhaddi, N., 2024, A design methodology for nonlinear oscillator chains enabling 

energy localization tuning and soliton stability enhancement with optimal damping, Mechanical Systems and 

Signal Processing, 213, 111358. 

4. He, J.-H., Bai, Q., Luo, Y.-C., Kuangaliyeva, D., Ellis, G., Yessetov, Y., Skrzypacz, P., 2025, Modeling and 
numerical analysis for an MEMS graphene resonator, Front. Phys. 13, 1551969. 

5. Campante, T., 2024, Smallest known starquakes are detected with a subtle shift of colour, Nature, 628, pp. 

239-239. 
6. Maiuri, M., 2024, Vibration isolation could boost performance of near-infrared organic LEDs, Nature, 629, 

pp. 289-290. 

7. Elías-Zúñiga, A., 2014, Solution of the damped cubic-quintic Duffing oscillator by using Jacobi elliptic 
functions, Applied Mathematics and Computation, 246, pp. 474-481. 

8. Elías-Zúñiga, A., 2013, Exact solution of the cubic-quintic Duffing oscillator, Applied Mathematical 

Modelling, 37(4), pp. 2574-2579. 
9. Beléndez, A., Alvarez, M. L., Francés, J., Bleda, S., Beléndez, T., Nájera, A., Arribas, E.,2012, Analytical 

approximate solutions for the Cubic-Quintic Duffing oscillator in terms of elementary functions, Journal of 

Applied Mathematics, 2012, 286290. 
10. Guo, X.T., Zhang, W., Zhang, Y.F., Experimental and numerical investigations on nonlinear snap-through 

vibrations of an asymmetrically composite laminated bistable thin plate simple supported at four corners, 

Engineering Structures, 296, 116926. 
11. He, J.H., 2024, Frequency formulation for nonlinear oscillators (part 1), Sound & Vibration, 59(1), 1687. 

12. Remmi, S.K., Latha, M.M., 2018, Cubic Quintic-Septic Duffing oscillator: An analytical study, Chinese 

Journal of Physics, 56(5), pp. 2085-2094. 
13. Lev, B.I., Tymchyshyn, V.B., Zagorodny, A.G., 2017, On certain properties of nonlinear oscillator with 

coordinate-dependent mass, Physics Letters A, 381, pp. 3417-3423. 

14. Ren, Z.Y., 2018, The frequency-amplitude formulation with ω4 for fast insight into a nonlinear oscillator, 
Results in Physics, 11, pp. 1052-1053. 

15. He, C.H., Liu, C., 2022, A modified frequency-amplitude formulation for fractal vibration systems, Fractals, 

30(3), 2250046. 
16. Ismail, G.M., Moatimid, G.M., Yamani, M.I., 2024, Periodic solutions of strongly nonlinear oscillators using 

He's frequency formulation, European Journal of Pure and Applied Mathematics, 17(3), pp. 2155-2172. 
17. Hashemi, G., 2023, A novel analytical approximation approach for strongly nonlinear oscillation systems 

based on the energy balance method and He's frequency-amplitude formulation, Computational Methods for 

Differential Equations, 11(3), pp. 464-477. 
18. Kawser, M.A., Alim, M.A., Sharif, N., 2024, Analyzing nonlinear oscillations with He's frequency-amplitude 

method and numerical comparison in jet engine vibration system, Heliyon, 10(2), e24261. 

19. Tsaltas, K., 2023, An improved one-step amplitude-frequency relation for nonlinear oscillators, Results in 

Physics, 54, 107090. 

20. Tian, Y., Shao, Y.B., 2024, Mini-review on periodic properties of MEMS oscillators, Frontiers in Physics, 12, 

1498185. 
21. He, J.H., 2024, An old Babylonian algorithm and its modern applications, Symmetry, 16, 1467. 

22. Zhao, L., 2009, He's frequency-amplitude formulation for nonlinear oscillators with an irrational force, 

Computers and Mathematics with Applications, 58(11-12), pp. 2477-2479. 
23. Ebaid, A.E., 2009, Oscillations in an x(2n+2)/(2n+1) potential via He's frequency-amplitude formulation, 

Zeitschrift fur Naturforschung A, 64 (12), pp. 877-878. 

24. Elías-Zúñiga, A., Palacios-Pineda, L.M., Jiménez-Cedeño, I.H., Martínez-Romero, O., Olvera-Trejo, D., 
2020, Enhanced He's frequency-amplitude formulation for nonlinear oscillators, Results in Physics, 19, 

103626. 



210 C.H. HE, J.H. HE, J. MA, A.A. ALSOLAM, X.J. YANG 

 

25. Mohammadian, M., 2024, Application of He's new frequency-amplitude formulation for the nonlinear 
oscillators by introducing a new trend for determining the location points, Chinese Journal of Physics, 89, pp. 

1024-1040. 

26. Li, Y.D., Yang, Y.R., 2017, Vibration analysis of conveying fluid pipe via He’s variational iteration method, 
Applied Mathematical Modelling, 43, pp. 409-420. 

27. Brodsky, Y., 2024, Analyzing and simulating the oscillation in multidimensional odd competitive systems, 

Advances in Differential Equations and Control Processes, 31 (3), 2441. 
28. Alshomrani, N., G., Alharbi, W., Alanazi, I., Alyasi, L., Alrefaei, G., A.Al’amri, S.,  Alanzi, A., 2024,  

Homotopy perturbation method for solving a nonlinear system for an epidemic, Advances in Differential 

Equations and Control Processes, 31(3), 2442. 
29. Öziş, T., Yıldırım, A., 2007,  A study of nonlinear oscillators with u1/3 force by He's variational iteration 

method, Journal of Sound and Vibration, 306(1–2), pp. 372-376. 

30. Belendez, A., Pascual, C., Gallego, S., Ortuño, M., Neipp, C., 2007, Application of a modified He's homotopy 
perturbation method to obtain higher-order approximations of an x1/3 force nonlinear oscillator, Physics 

Letters A, 371(5–6), pp. 421-426. 

31. Beléndez, A., Hernández, A., Beléndez, T., Neipp, C., Márquez, A., 2008, Higher accuracy analytical 
approximations to a nonlinear oscillator with discontinuity by He's homotopy perturbation method, Physics 

Letters A, 372(12), pp. 2010-2016. 

32. Bayat, M., Bayat, M., Pakar, I., 2014, Accurate analytical solutions for nonlinear oscillators with 
discontinuous, Structural Engineering and Mechanics, 51(2), pp. 349-360.  

33. Liu, H.M., 2005, Approximate period of nonlinear oscillators with discontinuities by modified Lindstedt–

Poincare method, Chaos, Solitons & Fractals, 23(2), pp. 577-579. 
34. Ji, Q.P., Wang, J., Lu, L.X., Ge, C.F., 2021, Li-He's modified homotopy perturbation method coupled with the 

energy method for the dropping shock response of a tangent nonlinear packaging system, Journal of Low 

Frequency Noise, Vibration & Active Control, 40(2), pp. 675-682. 
35. Zhang, H.L., 2008, Application of He’s amplitude–frequency formulation to a nonlinear oscillator with 

discontinuity, Computers & Mathematics with Applications, 58(11–12), pp. 2197-2198. 

36. Wang, S.Q., He, J.H., 2008, Nonlinear oscillator with discontinuity by parameter-expansion method, Chaos 
Solitons and Fractals, 35(2008), pp. 688-691. 

37. Shou, D.H., 2009, Variational approach for nonlinear oscillators with discontinuities, Computers & 

Mathematics with Applications, 58(11–12), pp. 2416-2419. 
38. He, J.-H., 2025, Frequency-Amplitude Relationship in Nonlinear Oscillators with Irrational Nonlinearities, 

Spectrum of Mechanical Engineering and Operational Research, 2(1), pp. 121-129.