EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 4, 2024, 2878-2897
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

A Novel Inspection of a Time-Delayed Rolling of a
Rigid Rod

Khalid Alluhydan1, Galal M. Moatimid2, T. S. Amer3, A. A. Galal4,∗

1 Department of Mechanical Engineering, College of Engineering, King Saud University,
P.O Box 800, Riyadh 11421, Saudi Arabia
2 Department of Mathematics, Faculty of Education, Ain Shams University, Egypt
3 Department of Mathematics, Faculty of Science, Tanta University, Tanta 31527, Egypt
4 Department of Engineering Physics and Mathematics, Faculty of Engineering,
Tanta University, Tanta 31734, Egypt

Abstract. The work examines the stability configuration of a rolling rigid rod in the presence
of a time-delayed (TD) in a square position as well as velocity. Examining time-delayed rolling
rigid rod bridges presents real engineering challenges and raises significant theoretical questions,
making it a desirable problem in applied and theoretical contexts. It is recommended to use the
non-perturbative approach (NPA) to find an equivalent linearized differential equation. Actually,
the NPA is based on the He’s frequency formula (HFF). The Mathematica Software (MS) is used
to compare and assess of this suitability. Using the appropriate numerical methodology (NM), a
matching between the strong nonlinear ordinary differential equation (ODE) and the equivalent
analytical linear one is obtained. This matching has revealed a very significant agreement for differ-
ent criteria. Stated differently, the new performance appears powerful, promising, and beneficial,
and it may be applied to different classes of nonlinear oscillators. It is well-precision, flexible, and
convenient. The new approach has many advantages in contrast to all other perturbed methods.
It avoids the usage of the Taylor expansion in expanding the restoring forces; particularly in the
topic of the dynamical systems. Therefore, the stability analysis is analyzed and the current work
no longer incorporates this shortcoming. Furthermore, the temporal histories of the obtained novel
outcomes and their various stable zones are accomplished. It is possible to examine the relevance
of the employed parameter and demonstrate the precision of the outcomes through an exploration
of the data. It is found that periodic solutions provide reliable and predictable behavior in a sys-
tem, whereas phase plane diagrams offer a visual and quantitative comprehension of dynamics and
stability. This appreciation is vital for ensuring safe operation under different conditions.

2020 Mathematics Subject Classifications: 70K20 , 70Q05, 70K05, 70K40

Key Words and Phrases: Rolling rigid rod, Nonlinear vibrations, Square position-velocity time
delay, He’s frequency formula, Non-perturbative approach, Stability configuration

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5398

Email address: abdallah.galal@f-eng.tanta.edu.eg (A. A. Galal),
kalluhydan@ksu.edu.sa (K. Alluhydan),
tarek.saleh@science.tanta.edu.eg (T. S. Amer), gal moa@edu.asu.edu.eg (G. M. Moatimid)

https://www.ejpam.com 2878 Copyright: © 2024 The Author(s). (CC BY-NC 4.0)



A. A. Galal et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2878-2897 2879

Nomenclature
L: Length of the rod.
θ: Angular displacement.
t: Time.
·: Time derivative.
τ : Time-decay.
M : Uniform rigid rod’s mass.
g: Gravity’s acceleration.
2r: Circular surface diameter.
C.M : Center of mass of the uniform rod.
a, b : Non-dimensional quantities.

1. Introduction

There are various amounts of nonlinear phenomena present in all practical structures.
The nonlinearity’s characteristics are usually understated enough to be not ignored. Con-
sequently, the linear theories cannot be used to study the dynamics and behavior of these
systems. On the other hand, many structures in the real-world have large nonlinearities
that make linear theories out-of-date. Furthermore, the system richer the dynamical be-
havior, such as period doubling, quasi-periodic behavior, and chaotic response, may be
disregarded if linear techniques are applied to highly nonlinear systems. Therefore, a full
understanding of nonlinear system dynamics is necessary to comprehend organizational
dynamics. Differential equation systems, ordinary or partial, are widely recognized for
their ability to represent a wide range of engineering systems. With a few exclusions,
their exact solutions seem too challenging and unachievable. In order to make handling
numerous nonlinear equations easier, many academics are focused their attention on the
approximation solutions for specific weak nonlinear issues, such as the mean approach
and the procedure of extremely tiny parameters [9, 10, 35, 42]. Analysis was done on
the magnetic spherical pendulum problem [16]. The issue was investigated using the ad-
vanced nonlinear frequency, additionally with the homotopy perturbation method (HPM),
and the Laplace transforms. It was studied how to move a simple pendulum with a light
spring and a rolling wheel attachment [14]. The basic motion equation experienced a
transformation into high nonlinear ODE under specific conditions. The complicated be-
havior of a perturbed Van der Pol-Duffing oscillator was effectively reduced by utilizing
TD position and velocity [15]. A time delay was implemented as an additional protection
to prevent the system under examination from undergoing the nonlinear vibrations. The
issue of this work was especially current since technologies with a TD are the subject of
many investigations recently. Over the recent years, the techniques involving TD have
increasingly drawn interest. Henceforth, research was conducted on a TD controller for a
damped, nonlinear, excited Duffing oscillator (DO) [19]. The modified periodic solution
that forms the basis of the current investigation is called the HFF. The examination of



A. A. Galal et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2878-2897 2880

the stability analysis concerning the disturbed pendulum movement was conducted using
the adapted HPM. The TD for position and velocity are employed in order to reduce the
complex oscillation’s behavior of the framework under consideration. Additional progress
in this field was performed [2, 24]. A precise solution for a nonlinear oscillator with a mass
that depends on the position of the coordinate was derived [1]. The findings collected
demonstrated that the proposed technique was a promising tool for solving the van der
Pol oscillator and providing information on the phase portrait, so proving the stability of
the system [3]. The study reported the use of first- and second-order difference techniques
to find numerical solutions for initial boundary value issues of both homogeneous and
nonhomogeneous Helmholtz equations [5]. The stability of these procedures is thoroughly
examined, guaranteeing their dependability and convergence over a broad spectrum of
problem situations. An approach that is dependable and utilizes an adapted version of
the conventional differential transform method was introduced [4]. The results collected
demonstrated that the proposed technique is a highly promising tool for solving the van
der Pol oscillator and providing accurate information on the phase portrait, so proving
the system’s stability.

In real-life, there have been situations where objects are just tangentially attached
to their framework, allowing rocks to fall on nearby materials. These include radioac-
tive fuel cells in reactors, air extraction columns, petroleum cracking towers, and liquid
petrol tanks. Undeniably, the most lucid illustration of structural responses during seis-
mic activities is their rocking motion. Despite the seemingly straightforward nature of
this phenomenon, the swaying and toppling of rigid structures in response to basic stimuli
present significant challenges. The primary inspiration for comprehending this rocking
issue stems from the potential to utilize it as a preventive measure. This understanding
could be instrumental in safeguarding structures, furnishings, and apparatus from collaps-
ing and posing threats to human safety during the tremors of an earthquake. Early and
good results were obtained from the shaking of the isolation system [13, 41]. The first-order
approximations of the iterative perturbation technique are explored to mimic pertains to
the oscillation patterns observed in a stiff bar that sways reciprocally on a cylindrical
surface without any slippage, along with the cubic-quintic DO [6]. An investigation of
the movement of a rigid rod swaying back and cubic-quintic DO employed He’s energy
balancing approach [11]. It is found that this approach does not require the linearization
process or small perturbations, and it is very practical and efficient. The problem of a stiff
rod rocking while slipping on a stiff circular surface was examined [17]. From the princi-
ples of the Euler-Lagrange theorem, one can potentially derive the overarching equation
governing motion. Analysis was done on the stability analysis of a rocking rigid rod using
TD square position as well as velocity [17]. The TD technologies have been the subject of
several researches lately; therefore this topic of investigation was especially relevant. The
TD served as a supplementary protection against the nonlinearly oscillating system that
was being examined. In all the previous efforts, the restorative powers were weakened
through the application of the Taylor expansion. On the other hand, the current represen-
tation is incomparable in that it maintains these forces. Recently, the HFF were adopted



A. A. Galal et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2878-2897 2881

in analyzing several works in the dynamical system as well as the Electrohydrodynamics
stability [8, 18, 20–23, 25–34].

Nonlinear oscillations are a part of ordinary life and comparable technical systems.
Nonlinear vibrations, as one of the most significant and often used prototypes in dynamic
structures, are vital in considering different nonlinear occurrences in electrical manufac-
turing and industry. The HFF is the most straightforward way to calculate the frequency-
amplitude relationship, while analyzing different classes of nonlinear dynamical systems.
The calculation of the remaining values is compared with the formulation of frequency-
amplitude and its diverse versions [40]. This study suggested introducing a free parameter
to offer an accurate estimation of the nonlinear oscillator frequency and to reduce the
number of techniques needed for the residual computation. Using residuals from two trial
solutions, the HFF is used to find the nonlinear oscillator frequency and amplitude re-
lationship [37]. This approach can still be improved, even in the unlikely event that an
extremely precise response is produced. For nonlinear oscillators, the HFF offered the
most straightforward and precise frequency formulation [36]. The un-damped DO and
its related family are successfully resolved with it [7]. The complexity characteristic in
evaluating the DO with advanced nonlinearity, as well as the quadratic damping equa-
tion, has marked this subject as a critical area necessitating comprehensive investigation
for more accurate solutions. Since linear equations are frequently perfect solutions, the
HFF for damped nonlinear oscillation is a hot topic. The extremely accurate solution,
also known as the linearized equation solution, illustrated how to approach the nonlinear
issue. An exact solution is obtained when equations having constant coefficients are made
linear through the application of the HPM. The existing study aims to utilize the HFF
in identifying the frequency of a nonlinear oscillation, which may be subject to linear or
nonlinear damping forces.

The TD rolling of a rigid rod is the occurrence in which the rolling motion of a rod is
affected by a delay in the application of forces or torques. This delay can lead to intricate
dynamics, such as oscillations, alterations in stability, or changing rolling trajectories.
The TD in mechanical systems frequently occurs as a result of factors such as control
system feedback, material deformation, or delayed actuation. Several distinguished uses
can be concisely summarized as follows: The TD rolling can have a considerable impact
on the movement control of robotic manipulators or mobile robots, especially when precise
control is required. Gaining comprehension of this TD can aid in formulating enhanced
control algorithms to alleviate unwanted oscillations or instability.

The TD rolling can be employed in engineering structures to serve as vibration damping
systems. Engineers can enhance the stability and durability of structures by implementing
controlled delays to mitigate or counter undesired vibrations. Understanding the delayed
rolling behavior of conveyor systems, particularly when handling elongated objects such as
rods, can result in improved design and control strategies to prevent jams or misalignment
during material transport.

At the micro and nano scales, the TD can be more pronounced because of the slower



A. A. Galal et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2878-2897 2882

response times in actuation and sensing. Precision is crucial when developing instruments,
as even slight delays can have a significant impact on the performance and accuracy of
the device. Understanding TD rolling can improve the dependability and predictability of
deployment mechanisms in military electronics, especially in situations when timing and
synchronization are critical. This understanding ensures that systems function appropri-
ately under time-sensitive conditions.

When designing tires or rolling elements for automotive and aerospace applications,
including the effects of time-delayed rolling can enhance the precision of simulations and
optimize the performance of these components in dynamic settings, especially in high-
speed or high-stress environments.

Comprehending and controlling these delays are essential for maximizing performance
and guaranteeing the stability and dependability of the system.

Owing to the aforementioned potentials, their exceptional sensitivity to dynamic load-
ing, geometric variations, and dissipation challenges, assessing the rocking and toppling
reactions of rigid blocks during motion events presents a substantial challenge. Therefore,
examining the dynamic behavior of a rigid rocking rod over a circular surface under the
assumption of pure rolling without slippage is the aim of this paper. According to classical
mechanics, the motion’s governing equation is controlled as a nonlinear ODE with highly
nonlinear orders. Regarding the innovative approach or noteworthy outcomes, it is vital
to emphasize the following:
The new technique yields highly nonlinear ODE that exactly has the same behavior as
the existing nonlinear one.
There must be a harmonious correlation between the mathematical expressions for the
new method to work.
All traditional methods utilize the Taylor series expansion to reduce the complexity of
the specified issue in the presence of restoring forces. The current paper has no longer
encompasses this shortcoming.
We can examine the problem’s stability analysis thanks to the current methodology, which
is not possible with other conventional techniques.
Lastly, the novel approach seems to be an easy-to-use, worthwhile, and interesting tool.
It is useful for the analysis of several types of nonlinear oscillators.
To cartelize the presentation of the paper, the remaining text is organized as follows: The
NPA yields an equivalent linear equation in § 2. This Section presents a confirmation using
the numerical solution via NM. The graphical plots, including time history and stability,
are presented in § 3 together with their interpretations based on the obtained data. Lately,
closing thoughts are offered in § 4.

2. Methodology Structure

The improvement of the existing equation is used to analyze the motion of a solid rod
that rocks. The suitable response to the periodic motions remains challenging due to the
structures of related essential equations. Therefore, the goal of this work is to find periodic



A. A. Galal et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2878-2897 2883

Figure 1: Sketches the model being examined.

analytical solutions using a more straightforward scheme for this type of dynamical system.
In this instance, any other conventional perturbation method in addition to the Taylor
expansion is not required. The subject under research is established by a consistently
inflexible rod actively rolling across a circular surface without sliding. The length and the
mass of the rod are both been retained. The rod is first resting horizontally at the tangency
point, which is situated at the center of mass (C.M.), as shown in Fig. (1). The arc rθ
with no-slip motion, which is defined as a small movement on the circle, must eventually
be used to characterize the distance between the center of mass and the tangency point.
Currently, going back to our earlier work [17], we can extract the uniform stiff swinging
rod’s regulating equation of motion, which took the following form [17]:

ML2

12
θ̈ +Mr2

(
θ̇2θ̈ + θ̈θ̇2

)
+Mg cos θ = 0. (1)

The foundation of this article provides a comprehensive inventory of all the variables
employed in Eq. (1). The anticipated model’s outline is illustrated in Fig. (1).

Simply, Eq. (1) will be converted to the subsequent simplified form:

θ̈ + a(θ2θ̈ + θ̇2) + b cos θ = 0, (2)

where a,and b are two non-dimensional physical quantities.
As was widely known, TD controllers were proposed to control the nonlinear distur-

bances. As was previously shown, the loop delay can significantly affect the stability of
the system’ or instability. However, when the ideal TD was considered, the subsequent
analytical and numerical evaluations demonstrated that the nonlinear position and non-
linear velocity are the best at damping the vibration. A simple procedure for determining
the ideal loop delay values in a way that enhances the system profile was suggested [38].
Moreover, the effectiveness of six iterations of linear and nonlinear feedback controllers
for position, acceleration, and velocity was studied [38]. It was discovered that the best
method for reducing vibration and suppressing bifurcations was the TD cubic acceleration
control. In view of the previous achievement, we consequently incorporate the square TD
in location and velocity into the existing model. Accordingly, the basic equation of motion
that controls all future motion is as follows:

θ̈ + a
[
θ2θ̈(t− τ) + θθ̇2(t− τ)

]
+ bθ cos θ = 0. (3)



A. A. Galal et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2878-2897 2884

One could ideally imagine the initial conditions (ICs), as in the next form:

θ(0) = A, θ̇(0) = 0. (4)

At this stage, returning again to the fundamental TD as given in Eq. (3), the NPA enables
us to transform the nonlinear ODE as shown in Eq. (3) into an equivalent linear one as
ü + Ω2u = 0 under the similar ICs that are given in Eq. (4), where Ω2 is known as the
total frequency that depends on all parameters of the original system.

u = A cos(Ωt), u̇ = −AΩcos(Ωt), and ü = −Ω2u, (5)

where A is the initial oscillation amplitude.
Appropriately, the shift of the independent time may be expressed as:

u(t− τ) = A cos(t− τ)

= A(cosΩt cosΩτ + sinΩt sinΩτ)

= u(t) cosΩt− 1

Ω
u̇(t) sinΩτ.

(6)

It follows that
ü(t− τ) = u̇(t) cosΩ(t) + Ωu(t) sinΩ(τ). (7)

Eq. (3) may be written as follows:

θ̈ + f(θ, θ̇, θ̈) + f(θ, θ̇, θ̈) = 0, (8)

where

f(θ, θ̇, θ̈) =α(θ̈2 + θ̇2) cos2Ωt+

(
Ω2θ̇2 +

1

Ω2
θ̈2
)
sin2Ωt+ bθ cos θ,

g(θ, θ̇, θ̈) =αΩ

(
θ̇θ̈ − 1

Ω2
θ̇3
)
sin 2Ωt.

(9)

An equivalent frequency ω2
eqv can by evaluated as shown previously by Moatimid et al.

[8, 18, 20–23, 25–34] in the following manner:

ω2
eqv =

∫ 2π/Ω
0 uf(u, u̇, ü)dt∫ 2π/Ω

0 u2dt
=

2b

A
(J1(A)−AJ2(A))− 1

2
aA2Ω2 cos 2Ωt, (10)

where J1(A), and J2(A)are the first kind of Bessel functions of the argument A of order
one and two, respectively. Additionally, following Moatimid et al. [8, 23, 27, 28], the
equivalent damping term can be evaluated as follows:

σeqv =

∫ 2π/Ω
0 u̇g(u, u̇, ü)dt∫ 2π/Ω

0 u̇2dt
= −1

2
aA2Ω2 sin 2Ωt. (11)



A. A. Galal et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2878-2897 2885

The equivalent linear ODE can now be constructed as follows:

ü+ σeqvu̇+ ω2
eqvu = 0. (12)

Furthermore, the standard normal form may be achieved concurrently with the trans-
formation u(t) = h(t)(−σeqvt/2) . Following the standard calculus, the unknown function
satisfies the following damped-simple harmonic differential equation:

ḧ+Ω2h = 0, (13)

where Ω2 = ω2
eqv − σ2

eqv/4.
In other words, the equivalence frequency can be obtained by combining the results in
Eqs. (10) and (11) with the previous relation to produce:

Ω2 =
2b

A
(J1(A)−AJ2(A))− 1

2
aA2Ω2 cos 2Ωτ − 1

4

(
1

2
aA2Ω2 sin 2Ωτ

)2

. (14)

Indeed, the governing equation of the total frequency as given in the previous Eq. (14)
is a transcendental one. Therefore, the numerical stability discussions are plotted via the
Command (PlotRegion) in the MS, for this purpose, see the subsequent Figs. (6), (7),(8),
and (9). Additionally, because the time-delay is infinitesimal, employing Taylor expansion,
then Eq. (14) will be polynomial in Ω2. In this case the stability standard requires:

Ω2 > 0, and σeqv > 0. (15)

To obtain the value of the equivalent frequency, consider the following data sample
and the MS:

a = 5.0, b = 3.0, τ = 0.01, and A = 0.1

Utilizing the assistance of the Mathematica Software through the command of Find-
Root, the value of the total frequency becomes 1.39425 . Once more, Eq. (14) gives the
characteristic equation in the total frequency equation. Actually, this equation is a tran-
scendental one; it has no exact solution. To analyze the stability configuration, we are
forced to adopt the MS via the command PlotRegion.

3. Graphical Representations and Discussions of the Results

The purpose of the current section is to highlight the graphical representations and
discuss the achieved results in light of the following data of the parameters:

A = (0.8, 0.9, 1), b = (0.1, 0.5, 1, 1.5), a = (0.1, 0.5, 1, 1.5), τ = (0.1, 0.5, 1).

It is easy to access to match the numerical solution (NS) of the previous Eq. (3)
via the knowledge of NDSolve of the MS with the solution of the equivalent linear ODE.
Therefore, Fig.(2) is graphed for A = 1, a = 0.1, and b = 1 when the TD parameter
τ has the value 0.1. As seen from this comparison, there is an excellent consistency



A. A. Galal et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2878-2897 2886

Figure 2: Explores the comparison connecting the AS and NS.

concerning the analytical solution and numerical one, which demonstrates good accuracy
of the NPA. Additionally, up to the time of 50 units, the MS showed that the absolute
error between the analytical and numerical solutions is 0.002025. To validate the absolute
error between them, Table (3) is presented. The inspection of the drawn curves of this
figure explores their periodic forms which give an impression of the stability and steady
behavior of mutual solutions.

Time Analytical Solution Approximate Solution Absolute Error

0 1 1 0.0000000000000000

5 -0.6367368984628483 -0.6840673392798011 0.04733044081695281

10 -0.1891322442718248 -0.165428295011344 0.02370394927069039

15 0.8775918556967678 0.8671456885746593 0.01044616712210844

20 -0.9284579883534055 -0.9463244488562964 0.017866460211890867

25 0.30437506401763743 0.37993757885446146 0.0755625143862673

30 0.540334930370593 0.521267116029197 0.01906781434157334

35 -0.9928774394082572 -0.9734314088762631 0.01944603040994164

40 0.7240684722745052 0.758824169662992 0.0347556793179396

45 0.0707952127866591 0.0599615698265702 0.01083361589400279

50 -0.8142243207060901 -0.8215978344057133 0.007373513699623246

The solutions of Eq.(3) at various values of A, a, and b are plotted, respectively, in
Figs. ((3), (4), and (5)). The curves in Fig. (3) are calculated when A = 0.8, 0.9, and 1
and with the similar amounts of other factors as shown in Fig. (2), while curves in Fig.
(4) are drawn at a = (0, 1, 0.5, 1), A = 1, b = 1, and τ = 0.1. In addition, the variation of



A. A. Galal et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2878-2897 2887

the value of b is examined in Fig. (5) at a = A = b = 1, and τ = 0.1.

The parts (a) of these figures show the temporal history of θ according to the varia-
tion of the above-mentioned parameters. These curves have a decaying form, where the
amplitudes of the waves are improved with the rise of A values, and the wave number
increases with time, as seen in Fig. (3a). The number of oscillations and their wave’s
number, are shown in Fig. 4(a), remain stationary with the change of a values, in which
their amplitudes decrease with the increase of the values of a. However, the change of the
values b of has a positive impact on the behavior of the waves describing θ, as seen in Fig.
(5a). Looking at the curves of this figure, in more detail, reveals that the fluctuation num-
bers increase with the increase of the values of b, i.e., as the wavelength decreases, even
though the amplitudes of these curves persist unaffected. When we going to compare the
curves of Fig. (3a), Fig. (4a), and Fig. (5a) with each other, one can say that the number
of wave fluctuations rises with the change of A than with the difference between both a
and b. Moreover, these waves have variable amplitudes, as seen in Figs. (4a) and (5a), but
in Fig. (5a), they remain constant. Figs. (3b), (4b), and (5b) reveal the corresponding
diagrams of phase planes which have the forms of closed curves to maintain the stability
of the solutions.

The drawn curves in Figs. (3b) and (4b) are close to each other and take the direction
inward, which indicates the stability and stationary behavior of the solution. The reason is
due to the change in the amplitudes of the waves. On the other hand, one finds that these
curves take symmetrical closed forms, as shown in Fig. (5b) because the amplitudes of the
drawn waves in this figure don not change to some extent. However, this interpretation
confirms the validity of what has been discussed in the previous discussion.

The symmetry in the phase plane curves for periodic waves arises due to the nature
of the underlying differential equations governing the system. These curves represent the
relationship between a system’s position and its velocity as it evolves over time. The
symmetry of these curves backs to the following reasons: In a periodic system, the motion
repeats itself after a certain time period. This implies that for every point in the phase
plane, the system will return to the same point after one period. This repetitive motion
is a key factor leading to the closed curves in the phase plane. In conservative systems,
the total mechanical energy is constant. This constancy is mirrored in the symmetry
of the phase plane. As the system oscillates, energy continuously shifts between kinetic
and potential forms, resulting in symmetric trajectories in the phase plane. If the system
is linear, the phase plane trajectories are elliptical and perfectly symmetric about the
origin. This symmetry is due to the linear relationship between the restoring force and
displacement in harmonic motion. Even in nonlinear systems, if the nonlinearities are
symmetric, the phase plane curves can remain symmetric. Although the type and degree
of nonlinearity may change the shape of the curves, symmetry in the potential function
frequently leads to symmetric trajectories in the phase plane.

For more convenience, since the TD must be of a positive real value, the inequality
Eq. (15) has been drawn to reflect the influences of the physical parameters A.a, and
b, as explored in Figs (6,7,8), and (9). These figures are graphed to show the drawn



A. A. Galal et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2878-2897 2888

Figure 3: (a) Explores curves of at different amounts of θ(t), and (b) Reveals the phase plane trajectories in
(a).

Figure 4: (a) Demonstrates curves of θ(t) at different amounts of a, and (b) Shows the phase plane diagrams
in (a).

stable/unstable zones, in which the green areas denote the stabile zones, while the white
ones express the unstable areas. These zones are plotted in view the intersection of the
instability and stability regions of the inequalities Eq. (15).

Fig. (6) show the variation of these zones according to the various values of A = (0.4, 0.7, 1)
at a = b = 0.01, while parts of Fig. (7) are drawn when A = (0.4, 0.7) and a = b = 1.
Looking at the parts (a) and (b) of these figures, besides the part (c) of Fig. (7), we con-
clude that the stability regions decrease with the increase of the values of A. This means
that the stable areas shrink to account for the unstable areas. The comparison of Fig. (6)
with Fig. (7), shows that more deformation is observed with the increase of a and b values.

The graphed areas in the potions of Fig. (8) are calculated at A = b = 1 when a =
(0.1, 0.5). The increase of the values produces a decrease in the stability areas, which is



A. A. Galal et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2878-2897 2889

Figure 5: (a) Shows curves of θ(t) at different amounts of b, and (b) Presents the phase plane paths in (a).

Figure 6: Show the stability/instability areas in the plane (τ − Ω) at a = b = 0.01 when (a) A = 0.4, (b)
A = 0.7, and (c) A = 1.

the reason backs to the mathematical form of the first inequality in Eq. (15), where the
terms conclude the parameter a in Eq. (15) have negative signs. The inspection in the
drawn areas in Fig. (9) shows that they are graphed at A = a = 1 when b = (0.01, 0.1). A
closer look at the Fig. (9a) and Fig. (9b), shows the positive influence of the b values, i.e.
when the values of b grows, the stability areas increases. We find that the reason behind
this increase is that the term containing the parameter b in the first inequalities Eq. (15)
has a positive sign.

Referring to the previous analysis, one can say that periodic solutions provide insights
into the regular and predictable behavior of systems. The diagrams of phase plane Offer
a visual and qualitative understanding of system dynamics and stability, while stability
areas are crucial for ensuring systems remain stable and operate safely under different
conditions.



A. A. Galal et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2878-2897 2890

Figure 7: Presents the stability/instability zones in the plane (τ − Ω) at a = b = 1 when (a) A = 0.4, and (b)
A = 0.7.

4. Evaluation of the Impact of Outcomes

Assessing the impact of solving dynamical systems with periodic solutions requires a
nuanced understanding of both the theoretical aspects and practical implications. The
periodic nature of the solution implies that the system exhibits repeating behavior over
time, which can have distinct short-term and long-term effects [12, 39]. Below is a com-
prehensive analysis of the impact assessment.
(i) Short-Term Effects:
Solving a dynamical system with periodic solutions allows for an immediate assessment of
system stability. In the short term, it becomes possible to determine whether the system
will remain in a stable periodic orbit or if there is a risk of divergence. This is crucial
for predicting short-term behavior, particularly in systems where stability is paramount.
In systems where periodic motion is a factor, short-term solutions can lead to immediate
gains in energy efficiency by optimizing the periodic cycles. For instance, in resonance-
based systems, ensuring that the system operates at or near resonance can minimize energy
losses.

(ii) Long-Term Effects: Periodic solutions allow for better prediction of long-term be-
havior, including the identification of potential failure points due to fatigue in mechanical
systems or wear and tear in repetitive processes. Once a periodic solution is established,
long-term optimization strategies can be employed to fine-tune system performance. This
could involve adjusting system parameters to improve overall system performance. Over
time, systems may need to adapt to changing conditions, such as varying external forces.
The periodic nature of the solution can guide the adaptation process, ensuring that the
system remains in a desirable state despite these changes.



A. A. Galal et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2878-2897 2891

Figure 8: Presents the time variation ofΩ at A = b = 1 when: (a)a = 0.1, and (b) a = 0.5.

5. Concluding Remarks

The work investigates the stability of a rolling rigid rod with square TD position as
well as velocity. The TD and other influential aspects are used to evaluate the approx-
imate analytical outcome technique. The investigation’s motivation stems from the fact
that TD technology has recently been the subject of numerous technological applications.
When attempting to solve an equivalent linearized ODE, it is advised to utilize the NPA.
This satisfactoriness is compared and evaluated using the MS. The stability criterion is
illustrated physically and explored conceptually. A matching between the strong nonlinear
ODE and the corresponding analytical linear ODE is established by using the suitable NM.
For several criteria, this matching has shown a highly significant agreement. Put another
way, the new performance is robust, promising, and practical, and it could be used with
other types of oscillators with high nonlinearity. It is accurate, adaptable, and practical.
As is well-known, all traditional methods use the Taylor expansion to reduce their com-
plexity when the restoring forces are present. This flaw is no longer included in the current
development. Moreover, different stable and unstable zones are graphed according to the
various values of the used parameters. These charts confirm the stability performance of
the stability criteria as given in Eq(15). By analyzing the data, it is feasible to assess the
applicability of the used parameter and show how accurate the results are. This inquiry
scrutinizes the problem of an inflexible rotating rod without slipping on a rigid circular
surface. The Euler-Lagrange theorem worked as the foundation for the development of the
governing equation of motion in our earlier work [17]. The obtained Gaylord’s oscillator
has a semi-precise periodic analytic solution because the expected non-linear frequency
was anticipated based on the NPA. The below mentioned specifics ought to be highlighted
pertaining to the innovative method or results that are important:



A. A. Galal et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2878-2897 2892

Figure 9: Describes the time variation of Ω at A = a = 1 when: (a) b = 0.01, and (b) b = 0.1.

• In essence, the novel approach creates a parallel linear ODE that corresponds to the
existing nonlinear one.

• A good match between these two equations is achieved to ensure the effectiveness of
the novel method.

• Every traditional approach employed the Taylor expansion to simplify the given
problem in the case of existence of the restoring forces. The current strategy has
eliminated this shortcoming.

• Away from the traditional perturbations, the current methodology enables us to
investigate the stability analysis of the problem.

• Finally, NPA appears to be a straightforward, effective, and promising tool. This
can be utilized to examine different nonlinear oscillator classes.

The implications of the original nonlinear Gaylord oscillator led to the establishing of
the equivalent frequency. Consequently, this strategy enables us to tackle a wide range
of issues related to the use of various oscillators in mechanical systems. To analyze the
stability analysis in the current work, a TD square position and velocity are used. To
achieve an analogous frequency and subsequently an equivalent linear ODE, the new NPA
has been used. It becomes clear that the total frequency includes every physical vari-
able in the fundamental governing equation of motion. The equivalent frequency of the
alternative linear ODE that includes trigonometric and Bessel functions is expressed as
a transcendental relationship. A comparison between the numerical and the theoretical
outcomes reveals a precise accuracy. The study purposes a careful analysis of the con-
clusions reached utilizing the analytical strategy of practical approximation. A secondary
argument together with the system potential for nonlinear vibration is provided by the



REFERENCES 2893

TD. Additionally, the time history of the achieved results and their stabilities are plotted
to show the positive effect of the used parameter.
Coupled dynamical systems constitute interrelated systems that mutually affect one other’s
behavior, offering valuable insights into intricate phenomena like as biological rhythms,
disease transmission, brain activity, and social dynamics. These systems have the poten-
tial to exhibit emergent behavior, which can offer valuable insights into their functioning,
enable predictions of future behavior, and provide suggestions for controlling or optimizing
them in practical applications. Therefore, in subsequent papers, the NPA will be devel-
oped to analyze such systems.

Authors Statements

Khalid Alluhydan: Resources, Methodology, Formal analysis, Validation, Visualiza-
tion and Reviewing.
Galal M. Moatimid: Conceptualization, Resources, Methodology, Formal analysis, Val-
idation, Writing- Original draft preparation, Visualization and Reviewing.
T. S. Amer: Investigation, Methodology, Data duration, Conceptualization, Validation,
Reviewing and Editing.
A. A. Galal: Examination, Organization, Data duration, Validation, Corroboration,
Rereading and Editing.

Conflict of Interest
There are no conflicts of interest declared by the authors.

Data Availability
All data generated or analysed during this study are included in this published article.

References

[1] Ata Abu-As’ ad and Jihad Asad. Exact solution for nonlinear oscillators with
coordinate-dependent mass. European Journal of Pure and Applied Mathematics,
15(2):496–510, 2022.

[2] TS Amer, Galal M Moatimid, and WS Amer. Dynamical stability of a 3-dof auto-
parametric vibrating system. Journal of Vibration Engineering & Technologies,
11(8):4151–4186, 2023.

[3] VS Erturk, B Rath, Taqwa M Al-Khader, Noorhan Alshaikh, P Mallick, and Jihad
Asad. Two-dimensional coupled asymmetric van der pol oscillator. European Journal
of Pure and Applied Mathematics, 17(2):1254–1264, 2024.

[4] Bawar Mohammed Faraj, Shnyar Karim Rahman, Deni Adnan Mohammed,
Hozan Dlshad Hilmi, and Ali Akgul. Efficient finite difference approaches for solving



REFERENCES 2894

initial boundary value problems in helmholtz partial differential equations. Contem-
porary Mathematics, pages 569–580, 2023.

[5] Bawar Mohammed Faraj, Dana Taha Mohammed Salih, Bnar Hama Zaman Hama
Ali, Bahadin Muhammad Hussien, Sarkhel Akbar Mahmood, and Shewa Abid Hama.
On the numerical solution for two dimensional laplace equation with initial boundary
conditions by using finite difference methods. Journal of Studies in Science and
Engineering, 2(4):50–59, 2022.

[6] Seyed S Ganji, Amin Barari, S Karimpour, and G Domairry. Motion of a rigid rod
rocking back and forth and cubic-quintic duffing oscillators. Journal of theoretical
and applied mechanics, 50(1):215–229, 2012.

[7] Ji-Huan He. Amplitude-frequency relationship for conservative nonlinear oscillators
with odd nonlinearities. International Journal of Applied and Computational Math-
ematics, 3:1557–1560, 2017.

[8] Gamal Ismail, Galal Moatimid, and Mohammed Yamani. Periodic solutions of
strongly nonlinear oscillators using he’s frequency formulation. European Journal
of Pure and Applied Mathematics, 17(3):2155–2172, 2024.

[9] Goran Janevski, Predrag Kozić, Ratko Pavlović, and Strain Posavljak. Moment
lyapunov exponents and stochastic stability of a thin-walled beam subjected to ax-
ial loads and end moments. Facta Universitatis, Series: Mechanical Engineering,
19(2):209–228, 2021.

[10] Weiming Ji, Hui Wang, and Meng Liu. Dynamics analysis of an impulsive stochas-
tic model for spruce budworm growth. Applied and Computational Mathematics,
19(3):336–359, 2020.

[11] H Ebrahimi Khah and DD Ganji. A study on the motion of a rigid rod rocking back
and cubic-quintic duffing oscillators by using he’s energy balance method. Interna-
tional Journal of Nonlinear Science, 10(4):447–451, 2010.

[12] Craig A Kluever. Dynamic systems: modeling, simulation, and control. John Wiley
& Sons, 2020.

[13] JW Meek. Dynamic response of tipping core buildings. Earthquake Engineering &
Structural Dynamics, 6(5):437–454, 1978.

[14] Galal M Moatimid and TS Amer. Analytical solution for the motion of a pendulum
with rolling wheel: stability analysis. Scientific Reports, 12(1):12628, 2022.

[15] Galal M Moatimid and TS Amer. Nonlinear suppression using time-delayed controller
to excited van der pol–duffing oscillator: analytical solution techniques. Archive of
Applied Mechanics, 92(12):3515–3531, 2022.



REFERENCES 2895

[16] Galal M Moatimid and TS Amer. Analytical approximate solutions of a magnetic
spherical pendulum: Stability analysis. Journal of Vibration Engineering & Tech-
nologies, 11(5):2155–2165, 2023.

[17] Galal M Moatimid and TS Amer. Dynamical system of a time-delayed of rigid rocking
rod: analytical approximate solution. Scientific Reports, 13(1):5570, 2023.

[18] Galal M Moatimid and TS Amer. Dynamical system of a time-delayed ϕ 6-van der
pol oscillator: a non-perturbative approach. Scientific Reports, 13(1):11942, 2023.

[19] Galal M Moatimid, TS Amer, and WS Amer. Dynamical analysis of a damped
harmonic forced duffing oscillator with time delay. Scientific Reports, 13(1):6507,
2023.

[20] Galal M Moatimid, TS Amer, and YY Ellabban. A novel methodology for a time-
delayed controller to prevent nonlinear system oscillations. Journal of Low Frequency
Noise, Vibration and Active Control, 43(1):525–542, 2024.

[21] Galal M Moatimid, TS Amer, and AA Galal. Studying highly nonlinear oscillators
using the non-perturbative methodology. Scientific Reports, 13(1):20288, 2023.

[22] Galal M Moatimid, TS Amer, and AA Galal. Studying highly nonlinear oscillators
using the non-perturbative methodology. Scientific Reports, 13(1):20288, 2023.

[23] Galal M Moatimid, TS Amer, and AA Galal. Inspection of some extremely non-
linear oscillators using an inventive approach. Journal of Vibration Engineering &
Technologies, pages 1–11, 2024.

[24] Galal M Moatimid, TS Amer, and Marwa H Zekry. Analytical and numerical study of
a vibrating magnetic inverted pendulum. Archive of Applied Mechanics, 93(6):2533–
2547, 2023.

[25] Galal M Moatimid, AT El-Sayed, and Hala F Salman. Different controllers for sup-
pressing oscillations of a hybrid oscillator via non-perturbative analysis. Scientific
Reports, 14(1):307, 2024.

[26] Galal M Moatimid, Mona AA Mohamed, and Khaled Elagamy. Nonlinear kelvin-
helmholtz instability of a horizontal interface separating two electrified walters’ b
liquids: A new approach. Chinese Journal of Physics, 85:629–648, 2023.

[27] Galal M Moatimid, Mona AA Mohamed, and Khaled Elagamy. An innovative ap-
proach in inspecting a damped mathieu cubic–quintic duffing oscillator. Journal of
Vibration Engineering & Technologies, pages 1–18, 2024.

[28] Galal M Moatimid, Mona AA Mohamed, and Khaled Elagamy. Insightful inspec-
tion of the nonlinear instability of an azimuthal disturbance separating two rotating
magnetic liquid columns. The European Physical Journal Plus, 139(7):1–23, 2024.



REFERENCES 2896

[29] Galal M Moatimid, Mona AA Mohamed, and Khaled Elagamy. Inspection of the
nonlinear instability of electrified casson fluids: a novel approach. Waves in Random
and Complex Media, pages 1–31, 2024.

[30] Galal M Moatimid and Yasmeen M Mohamed. Nonlinear electro-rheological insta-
bility of two moving cylindrical fluids: An innovative approach. Physics of Fluids,
36(2), 2024.

[31] Galal M Moatimid and Yasmeen M Mohamed. A novel methodology in analyzing
nonlinear stability of two electrified viscoelastic liquids. Chinese Journal of Physics,
89:679–706, 2024.

[32] Galal M Moatimid and DM Mostafa. Nonlinear stability of two superimposed elec-
trified dusty fluids of type rivlin-ericksen: Non-perturbative approach. Partial Dif-
ferential Equations in Applied Mathematics, page 100745, 2024.

[33] Galal M Moatimid, DM Mostafa, and Marwa H Zekry. A new methodology in eval-
uating nonlinear electrohydrodynamic azimuthal stability between two dusty viscous
fluids. Chinese Journal of Physics, 90:134–154, 2024.

[34] Galal M Moatimid and Aya Sayed. Nonlinear ehd stability of a cylindrical interface
separating two rivlin-ericksen fluids: A novel analysis. Chinese Journal of Physics,
87:379–397, 2024.

[35] Ivan R Pavlović, Ratko Pavlović, Goran Janevski, Nikola Despenić, and Vladimir
Pajković. Dynamic behavior of two elastically connected nanobeams under a white
noise process. Facta Universitatis, Series: Mechanical Engineering, 18(2):219–227,
2020.

[36] Na Qie, Wei-Fan Houa, and Ji-Huan He. The fastest insight into the large amplitude
vibration of a string. Reports in Mechanical Engineering, 2(1):1–5, 2021.

[37] Zhong-Fu Ren and Gui-Fang Hu. He’s frequency–amplitude formulation with average
residuals for nonlinear oscillators. Journal of Low Frequency Noise, Vibration and
Active Control, 38(3-4):1050–1059, 2019.

[38] Nasser A Saeed, Galal M Moatimid, Fawzy M Elsabaa, Yomna Y Ellabban, Mo-
hammed A El-Meligy, and Mohamed Sharaf. Time-delayed nonlinear feedback con-
trollers to suppress the principal parameter excitation. IEEE Access, 8:226151–
226166, 2020.

[39] Steven H Strogatz. Nonlinear dynamics and chaos: with applications to physics,
biology, chemistry, and engineering. CRC press, 2018.

[40] Yue Wu and Yan-Ping Liu. Residual calculation in he’s frequency–amplitude formula-
tion. Journal of Low Frequency Noise, Vibration and Active Control, 40(2):1040–1047,
2021.



REFERENCES 2897

[41] Chik-Sing Yim and Anil K Chopra. Earthquake response of structures with par-
tial uplift on winkler foundation. Earthquake engineering & structural dynamics,
12(2):263–281, 1984.

[42] Yuting Zuo. A gecko-like fractal receptor of a three-dimensional printing technology:
a fractal oscillator. Journal of Mathematical Chemistry, 59(3):735–744, 2021.