A new framework in analysing an auto-parametric pendulum vibration damper

Abstract

The analysis offers a thorough analytical examination of the auto-parametric pendulum vibration damper, focused on enhancing vibration mitigation in engineering structures via accurate modelling of nonlinear systems and optimization of energy transfer processes. The work aims to enhance the practical application of auto-parametric pendulum vibration dampers in engineering systems. The non-perturbative approach offers a distinctive framework that does not depend on Taylor series expansions. The primary objective of the non-perturbative approach is to transform nonlinear ordinary differential equations of time varying control systems into analogous linear ones. Approximate solutions are obtained without employing series expansions, unlike conventional perturbation methods. The study seeks to advance beyond traditional perturbation methods to evaluate the behaviour of systems exhibiting small-amplitude parametric fluctuations. To generate successive approximations that define the system’s nonlinear parametric behaviour, it is crucial to precisely delineate the relationship between frequency and amplitude. The numerical implementations are employed to validate the derived parametric formulation, which demonstrates significant agreement with the original governing ordinary differential equation. The influence of several factors on the stability of equilibrium states is systematically analysed. The findings indicate that the proposed strategy is very successful, thoroughly validated, and intuitive. Moreover, it could be extended too many applications in dynamical systems and fluid dynamics. The multiple-time scales method is employed to assess the stability structure for a more comprehensive analysis of the system’s behaviour. The morphology of the bifurcation curves alters significantly with variations in the bifurcation parameters. Phase portraits, Poincaré maps, and bifurcation diagrams are employed to do a comprehensive bifurcation analysis. As the excitation amplitude rises, it detects changes from periodic to chaotic motion. The method facilitates the differentiation and classification of the many dynamic reactions exhibited by the system.

Keywords

nonlinear systems non-perturbative approach Auto-parametric pendulum time varying control systems poincaré map & bifurcation

1. Introduction

As some of the first models to demonstrate nonlinear behaviour, pendulum systems are crucial in expanding our knowledge of nonlinear physics. Among them, the cart-pendulum system is a traditional mechanical arrangement that is the subject of much control theory research in recent decades. Despite the apparently straightforward dynamics of the cart-pendulum system, numerous traditional nonlinear control techniques are useless. Stabilizing this system is a crucial aspect of its study. The cart-pendulum system is a notable mechanical system of 2DOF. The system has emerged as a prominent physical prototype in nonlinear regulator. To comprehend their dynamic reactions, a broad range of pendulum-based dynamic models were thoroughly examined, including basic pendulums,¹ double pendulums,² inverted pendulums,³ parametrically stimulated pendulums,⁴ and spring-coupled pendulums.⁵ Various pendulum system configurations are the subject of several studies. Specifically, the employment of a moveable dynamic absorber placed in the longitudinal or transverse direction of an excited pendulum was improved the efficacy of weakly damped systems.⁶ Additionally, the application of oscillatory control to a 3DOF pendulum mounted on a cart was thoroughly studied.⁷ The persistence of bounded periodic oscillations, stability criteria, and bifurcation behaviour was well studied. The stability properties of idealized multi-pendulum systems under external harmonic stimulation were examined.⁸ Furthermore, they examined the dynamics of a spherical pendulum under horizontal stress.⁹ The angular displacement in the horizontal plane and the inclination angle with respect to the vertical axis are 2DOF. To facilitate NI, a mathematical formulation was created, and colour-coded distributions of the maximal Lyapunov exponent were used to show the findings. Additionally, attractor structure visualisations utilizing Poincaré sections and trajectory density maps on a chosen control plane were shown. The dynamics of a 2DOF auto-parametric pendulum connected to a damped subsystem were studied.¹⁰ Using the proper generalized coordinates, Lagrange’s formalism was used to obtain the governing equations of motion. Furthermore, the complicated vibrational behaviour of a recently suggested 3DOF of auto-parametric system was analysed.¹¹ A damped DO served as the primary subsystem, and it was connected to a secondary subsystem that was represented by a damped spring-pendulum arrangement. The mathematical modelling of a dynamical system connected to a piezoelectric device was investigated.¹² The NPA in contrast to other methodologies of traditional perturbation provides a novel conceptual base that enhances flexibility and expands applicability. NPA distinguishes itself from prior models by integrating and rectifying specific shortcomings identified in earlier research. This divergence is not only incremental but signifies a significant methodological progress.

A variety of engineering systems, including rotating machinery, aeroplanes, and civil infrastructure (such as buildings, bridges, and dams), are widely designed and used in practical applications in recent years. A thorough analysis of both traditional and modern advancements in the identification of nonlinear dynamical systems was provided.¹³ The purpose of this study was to provide an overview of the main approaches that were documented in the literature, validate those using NI and experimental research, evaluate their advantages and disadvantages, and suggest potential directions further research. Engineering constructions are susceptible to varying degrees of vibration from a variety of factors, including seismic activity, wind forces, vehicle loads, and imbalanced rotational components, regardless of how complex or simple their geometric arrangements or material behaviours.¹⁴ By examining several approaches from other fields, the function of damping in identifying structural degradation was investigated.¹⁵ It evaluated the use of damping in fibre-reinforced composite and reinforced concrete structures, identified the critical factors influencing damage identification through damping characteristics, and suggested future lines of inquiry to enhance the efficacy of damping-based detection methods. Furthermore, high vibrations raise the danger of instability and possible failure by seriously impairing structural performance, decreasing durability, and increasing maintenance requirements. A useful dynamic feature that lessens system vibrations to a manageable level is structural damping. The on-going loss of energy caused damping and its fluctuations.¹⁶ A technique in determining Coulomb and viscous friction coefficients from the responses of a harmonically excited dual-damped oscillator with linear stiffness was provided. Established analytical solutions for non-sticking reactions activated close to resonance were used in the identification process. Both internally and externally connected to elements like boundary conditions, geometric changes, material deterioration, were examined.¹⁷ Based on complicated mechanics, structural damping is frequently divided into three groups. The hydrodynamic or aerodynamic forces applied to the structures were the source of fluid damping.¹⁸ Second, intricate atomic and molecular interactions within materials were the source of material damping.¹⁹ Third, it was recognised that Coulomb friction between parts of a structural system caused structural damping.²⁰ For the purpose of characterising damping, several simpler models were proposed, such as Coulomb frictional damping models, hysteretic damping models, and viscous damping models.²¹ One basic model of a parametrically driven system with cubic nonlinearities is the Mathieu–DO. Parametric resonances and related limit cycles appear at both fundamental and combination resonance frequencies in multi-DOF systems were studied.²² The subject of piezoelectric active thin-walled structures has garnered significant interest over the past two decades. An isogeometric finite element formulation for shell structures composed of composite laminates, incorporating piezoelectric layers defined by electro-mechanical coupling, was investigated.²³ A selection of examples from the existing literature was examined to illustrate the applicability of the proposed numerical tool and evaluate its performance. Semi-nonclassical controller effects, including strain gradient, nonlocal, and Gurtin–Murdoch surface/interface theories, were introduced for the analysis of nonlinear vibrations in piezoelectric nanoresonators, in contrast to classical theory.²⁴ The attributes of nonlinear oscillators featuring quasi-linear terms in mechanical engineering were analysed.²⁵ A revised frequency formulation for this category of nonlinear oscillator was also introduced. Although the mathematical proof was absent, illustrative examples were included to showcase the notable simplicity and dependability of the method. Comprehending the dynamics of nonlinear vibrations in stringer-stiffened shell structures is essential for improving the stability and efficacy of advanced aerospace and marine systems. These systems frequently demonstrate intricate responses owing to geometric and material complexities, necessitating analytical approaches adept in accurately capturing essential dynamics. Three effective analytical methods for deriving closed-form expressions for the nonlinear frequency of such systems were examined: He’s adaptive location point-based formulation, the square error minimization frequency formulation, and the Hamiltonian-based frequency-amplitude formulation.²⁶ These methods offer rapid and straightforward solutions for analyzing nonlinear oscillators without requiring complex iterative procedures. The nonlinear frequency of the stringer-stiffened shell is determined by each method and validated against both exact analytical solutions and numerical results. Novel coupled vibration energy harvesting systems were intentionally constructed and examined to exploit nonlinearity and structural coupling effects, utilizing a nonlinear X-shaped structure linked to piezoelectric harvesters via two distinct mounting configurations (horizontal and vertical cases).²⁷ Deliberate nonlinearities were incorporated to enhance energy harvesting efficiency. Nevertheless, limited findings are documented regarding the combined exploitation of bistable and quasi-zero stiffness structures or mechanisms to improve ocean energy conversion in the literature. A unique bistable electromagnetic wave energy converter was proposed, featuring a bio-inspired X-shaped supporting structure and a mechanical motion rectifier.²⁸

Chinese mathematician Professor Ji-Huan He created an effective technique known as He’s frequency formula to deal with nonlinear oscillations. The amplitude-frequency equations of nonlinear oscillators were summarised and an improvement was suggested. The chosen location affects both the original and modified versions, and there are currently no set standards for site selection.²⁹ To solve the problem, the weighted residuals approach was used. It was crucial to quickly evaluate the amplitude-frequency correlation of a nonlinear oscillator exhibiting discontinuity.³⁰ This approach produces an approximation result with adequate precision and little processing. A location point selection criterion was introduced.³¹ Many authors use He’s frequency formula with great success. In order to solve a nonlinear oscillator with a given frequency, a different trial function was presented. The technique’s effectiveness was demonstrated using the DO.³² He’s frequency formula shows an effective approach to dealing with nonlinear oscillations and is derived from an old Chinese mathematical technique. A modified version of the formulation was suggested after a different theoretical analysis of it showed its dependability and practicality in real-world situations.³³ By contrasting a nonlinear oscillator with its linear equivalent in a domain where oscillations were studied, a formula connecting the amplitude and period of an ODE was obtained.³⁴ Algebraic techniques using an inequality that restricts the divergence between linear/nonlinear functions are used to evaluate the nonlinear period. A number of examples were given that contrast the approximation with He’s frequency formula.³⁵ The amplitude-period formula depends on an integer that is ideally chosen for certain cases. A thorough solution defining a conservative nonlinear oscillator’s ideal period is given. A conclusive conclusion that was shown to be a good estimate of the exact time in many common circumstances is made possible by the inclusion of an adequate weight function in the objective functional.³⁶ By analysing the residuals of two trial solutions, He’s frequency formula determines the connection between frequency and amplitude in a nonlinear oscillator. Even though this approach can yield extremely precise results, there is room for improvement. It is possible to improve the residuals computation without compromising accuracy. The extremely nonlinear DO was recorded as a model to verify the accuracy and approach of the solution.³⁷ There are examples in the literature where the NPA was applied to study non-conservative oscillation, focusing on specific scenarios and using advanced techniques to study a particular kind of nonlinear vibration. The NPA is an independent approach that doesn’t depend on perturbation expansions to function. The NPA finds it difficult to comprehend the fundamental elements of a problem, in contrast to perturbation theory, which effectively produces starting series terms and yields polynomial approximations. From a polynomial approximation to the NPA conclusion, this approach used an approximate analytic continuation on occasion ^38–48. This method makes use of cutting-edge methods and strategies that go beyond conventional constraints, allowing for a more thorough understanding of complex challenges.

Investigating a unique framework of the analysis of an APVD holds extensive experimental and useful consequence in varied engineering and industrial fields. It enables researchers to formulate sophisticated mathematical models and NI to comprehend the intricate nonlinear dynamics of the system, encompassing stability analysis, resonance occurrences, and energy transfer mechanisms. These investigations enhance experimental validation techniques by utilizing high-precision sensors, accelerometers, and motion capture devices to assess the damper’s response under various operating situations. The framework facilitates the design of controlled laboratory experiments that investigate the impacts of modifying system characteristics, including mass, length, damping coefficients, and excitation frequencies, yielding useful insights in optimizing damper efficiency. This research has extensive applications in structural engineering, automotive industries, aerospace systems, and mechanical vibration control. In civil engineering, APVD can be incorporated into tall structures and bridges to alleviate vibrations induced by wind, seismic activity, or automobile traffic. In automotive engineering, these dampers can improve ride comfort and vehicle stability by efficiently dispersing vibrational energy inside suspension systems. Aerospace applications encompass the mitigation of undesirable oscillations in aircraft structures and spacecraft components, hence enhancing safety and performance. Furthermore, in mechanical and industrial gear, such dampers can be utilized to extend the lifespan of spinning equipment by reducing vibrations that contribute to wear and failure. The innovative framework enables the creation of intelligent, adaptive vibration control strategies through sophisticated control algorithms, resulting in improved efficiency and real-time adaptability in dynamic settings. The merger of experimental and practical methodologies in the examination of APVD enhances theoretical understanding and fosters innovations across many engineering applications. To be easily followed: The remainder of the present work is structured as follows: Section 2 specifies the issue formulation; Section 3 describes the NPA’s methodology and approach validation. Section 4 provides the stability analysis. This Section encompasses time history procedure, and phase plane structure. The chaotic motion is depicted in Section 5. Section 6 presents the main findings from the primary results.

2. Formulation of issue

Consider a damped main oscillator coupled to an APVD. A harmonic stimulation $q \cos σ t$ with a specified amplitude $q$ and frequency $σ$ is applied to the system, which is operating close to the primary parametric resonance state. A linear spring with a normalized dimensionless stiffness of unity and a viscous damping term $α_{1} x^{'}$ sustain the primary oscillator. A linear dissipation function $α_{2} φ^{'}$ represents the pendulum’s own dampening in the meanwhile. The second variant of Lagrange’s equations, cited in 38, is used to obtain the governing equations of motion for this 2DOF of auto-parametric system as shown in Figure 1. Lagrange’s equations of the second type are used to derive the following Eqs (1) and (2) of a 2DOF auto parametric system that consists of a pendulum vibration absorber and a main oscillator. The following equation describes how an excited system, or main oscillator, moves when it is exposed to a harmonic force:

x^{''} + α_{1} x^{'} + x + μ λ (φ^{''} \sin φ + φ^{' 2} \cos φ) = q \cos σ t

(1)

where

x

represents the shift of the first structure.

φ

is the angle of pendulum,

μ

and

λ

are coefficients that responsible of coupling between the oscillator and the pendulum. Finally,

q \cos σ t

gives the external excitation force.

Figure 1.

Shows the model of APVD.

It should be noted that Eq. (1) includes coupling terms between terms between the oscillator and the pendulum.

The second equation may be formulated as follows:

φ^{″} + α_{2} φ^{'} + λ (x^{″} + 1) \sin φ = 0

(2)

where

λ \sin φ

accounts the restoring force due to gravity, and

φ^{″} \sin φ

represents the coupling with the oscillator’s acceleration.

Eqs. (1) and (2) were derived using Lagrangian mechanics and describe the interaction between the primary system and the pendulum, where the energy transfer can lead to effective absorption.³⁹

3. Methodology of solution

Following our previous work⁴⁰ in analysing the NPA of 2DOF, therefore, Eqs. (1) and (2) may be formulated as follows:

To apply the NPA, we firstly assume two guessing (trial) solutions of each equation. For Eq. (1), assuming the trial solutions as follows:

\tilde{x} (t) = A \cos Ω_{1} t, and \tilde{φ} (t) = B \cos Ω_{1} t

(3)

where

A

and

B

are initial amplitudes,

Ω_{1}

reprersents the total frequency to be identified subsequently.

Trial solutions are thought of as informed suggestions that might not immediately result in a precise linear solution. The phrase acknowledges that the initial assumption may need to be improved or modified, reflecting their temporary nature. Iterative techniques use a trial solution as a starting point and apply consecutive improvements to get closer to the right answer over time. It is frequently used in analytical methods to provide a broad form of the solution, which is then modified by adding the system’s particular boundary conditions or constraints. Additionally, the ICs are addressed as⁴⁰:

\tilde{x} (0) = A, {\tilde{x}}^{'} (0) = 0, \tilde{φ} (0) = B, and {\tilde{φ}}^{'} (0) = 0

(4)

As previously shown,⁴⁰ the correspondent frequency is shown as:

ω_{e q v 1}^{2} = \int_{0}^{2 π / Ω_{1}} \tilde{x} (\tilde{x} + \frac{1}{2} μ λ {\tilde{φ}}^{″ 2} \sin φ) d t / \int_{0}^{2 π / Ω_{1}} {\tilde{x}}^{2} d t = \frac{1}{2 π A} [2 π A + μ λ Ω_{1}^{4} (6 π J_{2} (B) - 2 π J_{3} (B) B) B]

(5)

Moreover, the equivalent coefficient of damping may be determined as⁴⁰:

μ_{e q v 1} = \int_{0}^{2 π / Ω_{1}} {\tilde{x}}^{'} (α_{1} {\tilde{x}}^{'} + \frac{1}{3} μ λ {\tilde{φ}}^{' 3} \cos φ) d t / \int_{0}^{2 π / Ω_{1}} {\tilde{x}}^{' 2} d t = \frac{1}{3 π A} [3 α_{1} π A + 6 π λ μ Ω_{1}^{2} B J_{2} (| B |)]

(6)

The equivalent linear ODE is then specified as:

u^{''} + μ_{e q v 1} u^{'} + ω_{e q v 1}^{2} u = q \cos σ t

(7)

To could be complete the process without missing the external force in the stability discussion, the right hand side must be integrated w.r.t. $u$ and Eq. (7) converts to be:

u^{''} + μ_{e q v 1} u^{'} + (ω_{e q v 1}^{2} - q \cos σ t) u = 0

(8)

which is a Mathieu equation form.

It should be noted that the NPA primarily addresses secular terms, which pertain exclusively to the odd functions. Since the function $q \cos σ t$ can be expressed as a constant function with a time-dependent parameter $u$ . This action may be referred to as the “masking technique”. In accordance with El-Dib,⁴¹ the utilization of the masking technique in nonlinear dynamics, particularly in scenarios involving periodic forces, constitutes a strategic and distinctive scientific methodology. This method primarily focuses on altering the governing ODE of the system to conceal the explicit influence of the periodic force. This reformulation optimizes the analytical process while efficiently maintaining the fundamental attributes of the system’s reaction to periodic input. The “masking technique” circumvents the direct complexities of periodic forces while still providing a comprehensive understanding of their impact on the system’s behaviour. This method is especially beneficial in clarifying the complex dynamics of nonlinear systems influenced by periodic forces, hence offering a more efficient analytical framework. The application of the masking technique commences with the initial formulation, as illustrated by Eq. (7) in our study, and progresses to efficiently eliminate the explicit representation of the periodic force. This crucial transformation involves devising an alternative equation that obscures the influence of the periodic force on the system. The objective is to develop a comparable system that emulates the behaviour of the original system under periodic forcing, while omitting the force from the equation itself. Eq. (8) is derived by a sequence of mathematical transformations or approximations to accomplish this. These steps aim to subtly integrate the impact of the periodic force into the system’s dynamics. In few words, the NPA process pertains exclusively to odd functions of the time-dependent context $u$ . Since, the time-dependent function $q \cos σ t$ is independent of $u$ , consequently, the application of “masking technology” enables us to convert this function into $u q \cos σ t$ . Finally, it should be noted that the inclusion of the non-homogeneous part in the actual ODE as presented in Eq. (10) has a certain advantage. This advantage enables us to incorporate the external force throughout the stability configuration.

The normal standard formula of Eq. (8) assumes that

u (t) = \tilde{x} (t) Exp (- μ_{eqv 1} t / 2)

(9)

It follows that $\tilde{x} (t)$ is a periodic function of time that satisfies,

{\tilde{x}}^{″} + (Ω_{1}^{2} - q \cos σ t) \tilde{x} = 0

(10)

which is the form of the standard Mathieu equation with a new frequency

Ω_{1}

which is stated by:

Ω_{1}^{2} = ω_{e q v 1}^{2} - \frac{1}{4} μ_{e q v 1}^{2}

(11)

Now, the objective is to transform Eq. (10) into the following comparable equation with constant coefficients:

g^{″} + Σ^{2} (σ) g = 0

(12)

Utilizing parallel procedures to those given by^41–51 regarding to Eq. (10), the total frequency may be attained as:

Σ^{2} (σ) = \int_{0}^{2 π / Σ} g (t) h (g, t) d t / \int_{0}^{2 π / Σ} g^{2} (t) d t

(13)

given,

h (g, t) = (Ω_{1}^{2} - q \cos σ t) g

(14)

The form of $Σ$ is evaluated and gotten as:

Σ^{2} (σ) = \frac{1}{8} (- 4 q + σ^{2} + 4 Ω_{1}^{2} + \sqrt{16 q^{2} + 24 q σ^{2} + σ^{4} - 32 q Ω_{1}^{2} - 8 σ^{2} Ω_{1}^{2} + 16 Ω_{1}^{4}})

(15)

Again, the trial solutions of Eq. (2) could be assumed as follows:

\tilde{x} (t) = A \cos Ω_{2} t, and \tilde{φ} (t) = B \cos Ω_{2} t

Once more, applying the previous procedure for Eq. (2) and parallel to Eq. (7), one gets the following characteristics.

μ_{e q v 2} = α_{2}

(16)

Furthermore, the equivalent frequency is given as follows:

ω_{e q v 2}^{2} = \int_{0}^{2 π / Ω_{2}} \tilde{φ} (λ \sin φ + \frac{1}{2} λ {\tilde{x}}^{″ 2} \sin φ) d t / \int_{0}^{2 π / Ω_{2}} {\tilde{φ}}^{2} d t = \frac{λ}{π B^{2}} [π B (2 + Ω_{2}^{4} A^{2}) J_{1} (B) - π Ω_{2}^{4} A^{2} J_{2} (B)]

(17)

The equivalent linear ODE of Eq. (2) is defined as:

v^{″} + α_{2} v^{'} + ω_{e q v 2}^{2} v = 0

(18)

where, the total frequency

Ω_{2}

is given as:

Ω_{2}^{2} = ω_{e q v 2}^{2} - \frac{1}{4} α_{2}^{2}

(19)

Which satisfies the simple harmonic equation:

{\tilde{φ}}^{″} (t) + Ω_{2}^{2} \tilde{φ} (t) = 0

(20)

The simplification of Eqs (11). and (19) yields:

Ω_{1}^{2} = \frac{1}{2 a_{1}} [- 1 - \frac{α_{1} B λ μ J_{2} (| B |)}{A} - \frac{1}{4} \sqrt{{(\frac{4 α_{1} B λ μ J_{2} (| B |)}{A} + 4)}^{2} - 16 a_{1} (α_{1}^{2} - 4)}]

(21)

and,

Ω_{2}^{2} = \frac{1}{2 a_{2}} [B^{2} + \frac{1}{4} \sqrt{16 (B^{4} - a_{2} (8 B λ J_{1} (B) - α_{2}^{2} B^{2}))}]

(22)

where,

a_{1} = \frac{B^{2} λ^{2} μ^{2} {(J_{2} (| B |))}^{2}}{A^{2}} + \frac{B^{2} λ μ J_{3} (B)}{A} - \frac{3 B λ μ J_{2} (B)}{A}

(23)

and

a_{2} = A^{2} B λ J_{1} (B) - A^{2} λ J_{2} (B)

(24)

where,

J_{1} (r)

J_{2} (r)

, and

J_{3} (r)

are the Bessel’s functions of first, second, and third orders, respectively.

3.1. Approach validation

Typically, an auto-parametric pendulum damper is made composed of a pendulum that serves as a nonlinear energy sink connected to a main configuration (like a plate or a beam) that is sensitive to base or external stimulation. The system becomes auto-parametric due to a parametric resonance of the pendulum caused by the main system’s excitation. The objective of NPA is to convert the APVD as a nonlinear ODE into a linear one without requiring minor changes or approximations. The NPA in the current oscillator has several characteristics, including:

Where linear or weakly nonlinear approximations fail, NPA manages large-amplitude movements. Large pendulum swings, non-sinusoidal responses, and nonlinear coupling effects are all examples of APVD. NPA is not dependent on tiny parameters (e.g., small damping, small excitation, weak coupling) like perturbation techniques (e.g. multiple scales). This makes it possible to investigate global dynamics, such as amplitude-dependent frequency shifts and hardening/softening stiffness effects. APVD functions close to internal resonance conditions, which are usually 2:1 between the pendulum and the structure. NPA reveals the nonlinear energy exchange between modes, which perturbation approaches could overlook or distort. It draws attention to frequency-energy dependency, which is a crucial aspect of nonlinear systems. The facilitation of the prediction of energy distribution on the pendulum and the visualization of energy transfer routes.

As previously shown, it is beneficial to match the nonlinear ODEs and their corresponding linear ones in order to support the NPA. A data of the pertinent constants is chosen to achieve this goal as:

μ = 0.05, α_{1} = 1.0, α_{2} = 0.2, A = 0.1, B = 0.1, λ = 1.5, q = 0.1 and σ = 0.5

To validate our outcomes, it is appropriate to use the MS to draw the numerical solutions of the nonlinear ODEs with their linear ones. For this purpose, using the above data to draw and compare the functions $x (t)$ and $u (t)$ in one graph, on gets the following Figure 2 as well as Table 1.

Figure 2.

Shows a very good agreement between the nonlinear and linear ODEs as given in Eqs. (1) and (7), respectively.

Table 1.

Authorises the correspondence concerning values of $x (t)$ and its equivalent NPA one $u (t)$ .

t	Nonlinear	Linear	Absolute error	Relative error
0	0.1	0.1	0.	0.
5	0.0180769	0.017835	0.000241816	0.0133771
10	-0.052741	-0.0527668	0.0000258367	0.00048988
15	-0.0341328	-0.0337517	0.00038111	0.0111655
20	0.0169841	0.0173948	0.000410755	0.0241847
25	0.0283663	0.0279627	0.00040354	0.0142261
30	0.00220054	0.00158341	0.000617137	0.280447
35	-0.0165727	-0.0162559	0.000316842	0.0191183
40	-0.00873568	-0.00801108	0.000724606	0.0829478
45	0.00659377	0.00639163	0.000202146	0.0306571
50	0.00838402	0.00761266	0.000771355	0.092003

The persistent concordance between nonlinear and linear ODEs through the NPA stems from techniques that directly examine the complete nonlinear framework without depending on minor perturbations. Methods such as the HPM, Adomian decomposition, and variational techniques provide accurate or systematically convergent approximate solutions that maintain the intrinsic characteristics of nonlinear systems. These methods demonstrate that given particular transformations, constraints, or asymptotic conditions, nonlinear ODEs can display solution structures that substantially resemble their linear equivalents, hence maintaining consistency in their dynamical behaviour. This agreement is especially important in situations where NPA expansions are inadequate, providing a more resilient foundation for comprehending nonlinear processes. It should be indicated that the absolute error in the previous figure is estimated and found to equal 0.00048.

Once more, the second comparable equations as given in Eqs. (2) and (18) gets the following Figure 3 and Table 2.

Figure 3.

Displays an excellent agreement concerning nonlinear/linear ODEs as specified in Eqs. (2) and (18).

Table 2.

Approves the correspondence between the NS of $φ (t)$ and its equivalent NPA one $v (t)$ .

t	Nonlinear	Linear	Absolute error	Relative error
0	0.1	0.1	0.	0.
5	0.0579794	0.0591204	0.00114098	0.019679
10	0.0319604	0.0340369	0.00207643	0.0649688
15	0.0168953	0.0190262	0.0021309	0.126124
20	0.00874897	0.0102714	0.00152247	0.174018
25	0.00437168	0.00530559	0.000933913	0.213628
30	0.00191609	0.00257729	0.0006612	0.345078
35	0.000620818	0.00113565	0.000514835	0.829286
40	0.0000404247	0.000412319	0.000371895	9.19969
45	-0.000166818	0.0000761496	0.000242968	1.45648
50	-0.000211259	-0.000060462	0.000150797	0.713801

The enduring agreement between nonlinear and linear ODEs via the NPA arises from methods that thoroughly investigate the entire nonlinear structure without relying on negligible perturbations. Techniques such as the HPM, Adomian decomposition, and variational methods yield precise or systematically convergent approximate solutions that preserve the fundamental properties of nonlinear systems. These methods illustrate that under specific transformations, constraints, or asymptotic conditions, nonlinear ODEs can exhibit solution structures that closely resemble their linear counterparts, hence preserving consistency in their dynamical behaviour. This agreement is particularly significant in scenarios where NPA expansions are insufficient, offering a more robust basis for understanding nonlinear processes. The absolute error in the preceding figure is calculated to be 0.00096.

3.2. Error estimations

i. Absolute Error:

The absolute error displays the numerical discrepancy between the estimated and actual values. A more accurate estimate is shown by a lower absolute error, which shows that the computation closely resembles the actual data.

ii. Relative Absolute Error:

The relative absolute error is quantified by the ratio between the error and the original value. This concept is essential to the experimental sciences, engineering, and physics, where precise quantities and approximations are required for precision, strategy, and analysis. A large relative error denotes a substantial deviation from the actual number, endangering the precision and reliability of calculations, whereas a small relative error shows a fairly similar approximation with less volatility. It may be expressed numerically as: $Relativeabsoluteerror = \frac{| Real ‐ Approximate |}{| Real |}$ .

4. Stability outline

As an extension of the NPA, the present goal is to examine the stability of the structured APVD framework. The stability diagrams of the analogous coupled linear system specified in Eqs. (9) and (15) are shown to guarantee clarity. Tables 1 and 2, together with Figures 2 and 3, show how reliable this strategy is. It is more practical to study the appropriate replacement ODEs proposed in Eqs. (9) and (15) under the examined conditions rather than directly analyzing the nonlinear system supplied in Eqs. (1) and (2). Considering the current example that deals with nonlinear stability, this strategy turns out to be a novel, simple, and effective substitute that performs better than conventional approaches. Furthermore, a variety of nonlinear systems may be analyzed using this method. Therefore, the two oscillators’ stability requirements may be stated as follows:

Σ^{2} > 0, and Ω_{2}^{2} > 0

(25)

Figures 4–10 are designed to examine the stability diagram with respect to the two restrictions (20). As shown in Eqs (16). and (17), inequalities of the parameters $α_{1}, α_{2}, λ, μ, A$ , and $B$ exemplify these requirements. The bright sections beneath the curves in those graphs specify the unstable regions, while the colourful portions over these arcs specify the stable zones of the scenarios shown. Plotting $Σ^{2} and Ω_{2}^{2}$ against the original amplitudes $A$ and $B$ , respectively, shows these stability profiles in relation to the data:

α_{1} = 0.5, α_{2} = 0.2, λ = 5, μ = 0.1, q = 0.1, σ = 0.5, A = 2, and B = 2

which differ from figure to one more owing to the discussed factor.

Figure 4.

Specifies the stabilizing effect of the displacement damping coefficient $α_{1}$ on the $x$ oscillator.

Figure 5.

Specifies the destabilizing effect of the angular damping coefficient $α_{2}$ on $φ$ oscillator.

Figure 6.

Indicates the destabilizing effect of the initial amplitude $B$ on $x$ oscillator.

Figure 7.

Displays the stabilizing effect of the initial amplitude $A$ on $φ$ oscillator.

Figure 8.

Shows the stabilizing effect of the coupling coefficient $λ$ on $x$ oscillator.

Figure 9.

Shows the stabilizing effect of the coupling coefficient $λ$ on $φ$ oscillator.

Figure 10.

Shows the dual effect of the excitation amplitude $q$ on $x$ oscillator.

In the NPA, the stability configuration of an APVD refers to the collection of stable/unstable response regimes that the coupled oscillator may display under various parameter and excitation conditions. The stability configuration in the APVD has physical relevance since it indicates whether or not the pendulum functions as a dependable vibration absorber. The stability configuration study of the APVD is significant physically. In a stable design, energy is gradually transferred from the main structure into the pendulum and dissipated by damping. This indicates the direction of energy flow. When this energy transmission is disrupted or fails in an unstable configuration, the fundamental structure vibrates excessively (poor damping performance). The pendulum’s capacity to absorb vibrations is directly related to its stability. Stability maps are used by engineers to avoid operating frequencies that cause the damper to destabilize. Through stability analysis, we can determine the ideal pendulum length, mass ratio, and damping to optimize energy pumping, reduce structural vibration amplitude, and ensure robust stability across a wide frequency range. The pendulum’s stability configuration indicates when it will remain in small-angle oscillations (the safe area), move to quasi-periodic or chaotic states (uncontrolled energy transfer), or enter big oscillations or rotations (unstable, dangerous).

Figures 4 and 5 show the stability area with the variation of the oscillators’ damping coefficients $α_{1}$ and $α_{2}$ of the the main structure’s displacement $x$ and the pendulum’s angular displacement $φ$ , respectively. It is seen from Figure 4 that the growth in $α_{1}$ leads to an increase of the stable area of the $x$ oscillator, meanwhile Figure 5 shows an opposite impact for the stable areas of $φ$ under the growth of $α_{2}$ . The term $α_{1} x$ represents the viscous damping in the primary oscillator. Energy dissipation inside the structure is immediately increased by increasing $α_{1}$ . This causes the oscillations of the structure $x (t)$ to diminish more quickly in the direction of equilibrium. The stabilizing impact of $α_{1}$ on $x$ is to reduce big oscillations and resonant responses, promote energy dissipation in the main structure, and improve the system’s overall stability. An ideal $α_{1}$ must be selected for balance, though, as too much damping may lessen the pendulum’s ability to absorb energy. Physically, the dampening in pendulum should assist in reducing the pendulum’s huge swings. However, excessive damping on $φ$ can decrease its capacity to absorb energy from $x$ oscillator in the auto-parametric damper mechanism causing less stable in the $φ$ oscillator, as seen in Figure 5.

Figures 6 and 7 are schemed to inspect the influence of the amplitudes $B$ and $A$ on the main structure’s displacement $x$ and the pendulum’s angular displacement $φ$ stability areas. It is found that the influences of $B$ and $A$ are dissimilar, where the increase of $B$ , from 0.2 to 0.4, decreases the stability areas of the $x$ oscillator, as seen in Figure 6. Whereas, the rise of $A$ in a very small range from 0.5 to 0.53, enlarges the stability regions of the $φ$ oscillator, as obtained by Figure 7. Physically, the amount of energy that is instantly accessible to excite the pendulum is determined by the starting amplitude of $x$ . By inducing dependable energy transfer into a bounded periodic pendulum motion (a desirable captured state), a modest starting $x$ can stabilize $φ$ . However, the impact is non-monotonic: a beginning $x$ that is too small does not engage the pendulum; meanwhile an initial $x$ that is too big might cause the pendulum to rotate in huge circles, move in quasi-periodic motion, or even become chaotic (which disrupts the absorber). Furthermore, the auto-parametric interaction relies on the way $x (t)$ modifies the torque or effective length of the pendulum. Proper phasing enhances the likelihood of steady capture, therefore timing (phase and velocity) of the initial $x$ motion is important. Therefore, stability is affected by starting amplitude plus beginning velocity/phase. High starting pendulum amplitude allows the pendulum to accumulate mechanical energy. It is possible to pump that energy back into the primary oscillator $x$ in an auto-parametric system through the nonlinear coupling. If that transferred energy is significant enough or comes with the “wrong” phase, it destabilizes $x$ by increasing its amplitude, as shown by Figure 7, causing bifurcations, excitation of resonances, or even temporary growth and chaotic motion. Moreover, an excessively energetic pendulum may not function as a sink but rather as a generator of $x$ , reversing the desired energy transfer and intensifying the vibration of the main structure.

Figures 8 and 9 indicate the influence of $λ$ which represents the coupling coefficient between the pendulum’s angular motion $φ$ and the main structure’s displacement $x$ on the stability regions of the two oscillators. The stable parts are noted extensively increase with the growth of $λ$ for both $x$ and $φ$ . Physically, the connection between the pendulum and the main structure is strengthened by larger $λ$ , allowing the pendulum to absorb more vibrational energy from $x$ . Higher connection enables energy to go to $φ$ more quickly when external stimulus excites $x$ , reducing the steady-state amplitude of $x$ . The structure becomes less responsive to excitation frequency as $λ$ increases because the frequency-response curve of $x$ flattens and resonance peaks diminish. When there is enough coupling, the pendulum is not inactive. Rather, it is “captured” into a steady periodic oscillation, avoiding erratic or disorderly fluctuations. Energy is pushed into $φ$ , therefore damping in the pendulum can dissipate energy more efficiently and prevent $φ$ from becoming out of control. A small $λ$ can cause the pendulum to go into chaotic or rotating motion, whereas a greater $λ$ tends to stabilize the dynamics by locking $φ$ into limited oscillations.

Furthermore, Figures 10 and 11 are conspired to discuss the effects of the excitation force factors $q$ and $σ$ , respectively. Figure 10 indicates that as $q$ rises, the system is transferring from unstable to stable manner at some region. Internal resonance between the pendulum $φ$ and the main mass x cannot be activated by the excitation at low forcing $q$ (destabilizing area). The pendulum either oscillates very little or stays almost at rest. Almost like an uncontrollable nonlinear oscillator, the primary mass reacts when the absorber effect is ineffectual. A little change in forcing might cause the system to move in an irregular or weakly chaotic manner, which makes $q$ destabilizing overall. The internal resonance between the main oscillator $x$ and the pendulum angle of oscillation $φ$ is activated when $q$ is sufficiently large to excite the pendulum. Targeted energy transfer from $x$ into $φ$ is made possible by this resonance, which captures the pendulum. This energy is released by pendulum damping, which lowers the vibration of the main structure and produces a stabilizing effect. Moreover, Figure 11 displays that the increase of $σ$ has a destabilizing consequence on the current system, where the stable zones shrink with the rise of. The APVD depends on the internal resonance (often a 2:1 resonance) between the pendulum angle $φ$ and the main mass displacement x. Energy effectively transfers to the pendulum leading to stabilizing when $σ$ (the main structure’s excitation frequency) is near resonance. However, energy transmission is weakened and the pendulum is unable to efficiently absorb vibrations as $σ$ moves away from this tuned resonance. Consequently, fluctuations in lead cause the absorber to lose synchronization, so destabilize the system.

Figure 11.

Shows the destabilizing effect of the excitation frequency $σ$ on $x$ oscillator.

4.1. Time history examination

By discussing the APVD time history, we may determine the operating regime (trivial, captured periodic, quasi-periodic, or chaotic), quantify targeted-energy-transfer events, and resolve the whole transient–to-steady development of the coupled $x - φ$ dynamics. Envelopes and instantaneous frequencies are extracted from $x (t)$ and $φ (t)$ to identify internal resonance and phase-locking, calculate decay rates and log decrements to evaluate the effectiveness of damping, and measure capture time, settling time, overshoot, and peak/crest factors to assess absorber performance. When the time histories and their energy partitions are examined together, net energy flow and dissipation are shown, and spectrograms indicate frequency sweeps and intermittency during transients.

Consequently, the solutions of Eqs. (7) and (18), utilizing NPA, are graphically illustrated with respect to time through Figures 12–16, using the following data:

μ = 0.5,, λ = 1.5, q = 0.001, α_{1} = 0.1, α_{2} = 0.5, σ = π / 2, A = 0.1, and B = 0.1

Figure 12.

Shows the influence of the primary amplitudes $A$ and $B$ on $u (t)$ .

Figure 13.

Shows the impact of the primary amplitudes $A$ and $B$ on $v (t)$ .

Figure 14.

Shows the influence of the coupling coefficient $λ$ on the functions $u (t)$ and $v (t)$ .

Figure 15.

Displays the impact of the damping factors $α_{1}$ and $α_{2}$ on $u (t)$ and $v (t)$ .

Figure 16.

Displays the impact of the excitation coefficients $q$ and $σ$ on the linear solution $u (t)$ .

Based on the mentioned factor, this data differs from graph to graph. The purpose of Figures 12–16 is to examine how the postulated parameters affect the performance of and over time. It is clear from the time-history waves in the figures that these parameters have an impact on the waveform shape, number of oscillations, and amplitudes of both solutions.

Figures 12 and 13 show the effects of the oscillator’s starting amplitudes $A$ and $B$ . These graphs show that the amplitudes of both $u (t)$ and $v (t)$ waves are increasing as $A$ and $B$ expand, respectively. Meanwhile, the amplitude of $u (t)$ decays with the increase of $B$ and the amplitude of $v (t)$ enhances to some extent with the growth of $A$ . The number of vibrations of $u (t)$ and $v (t)$ stays constant, while the wavelengths shrink with the rise of $B$ and $A$ , respectively. The waves are accelerating as noticed also due to the development of both $A$ and $B$ . The quantity that shows the departure of the system from balance at $t = 0$ is known as the initial amplitude in time history analysis. This impact manifests itself in a variety of ways, including changes in energy transfer, phase connections, and frequency dynamics between the coupled oscillators. The impact on the oscillators’ time-varying behavior depends on the strength and strength of the link. Increasing amplitudes have a major effect on the time-varying behavior of oscillations in coupled dynamical systems. These effects result in beats, synchronization, or chaotic oscillations and involve modifications to frequency shifts, phase correlations, and waveforms. For stable and expected applicable wave dynamics extending from biology to engineering, appropriate amplitude management is required.

Figure 14 specify the influence of the coupling coefficient $λ$ on the temporary oscillators $u (t)$ and $v (t)$ . It is noted from these figures that the amplitudes of the two waves are decaying for $u (t)$ and growing for $v (t)$ with the rise of $λ$ . The waves are accelerating and the wavelengths are diminishing with the growth of $λ$ . As said above, the joining between the main structure and the pendulum is reinforced by greater $λ$ , permitting the pendulum to absorb more oscillating energy from $x$ . In the APPVD system, the temporary responses of $u (t)$ and $v (t)$ are clearly influenced by the coupling coefficient $λ$ . The pendulum does not interact significantly with the fundamental structure when $λ$ is small, which results in massive oscillations in $u (t)$ and weak or irregular $v (t)$ , which indicates inadequate energy transfer. While $v (t)$ exhibits confined periodic oscillations, indicating efficient capture into internal resonance and focused energy transfer, the time history of $u (t)$ shows a discernible decrease in amplitude and a quicker decline toward steady state as $λ$ grows. The two signals become phase-locked at big enough $λ$ , and their envelopes show smooth energy exchange cycles.

Figure 15 are plotted to examine the influence of the damping coefficients $α_{1}$ and $α_{2}$ on both $u (t)$ and $v (t)$ , separately. It is observed that the effects of $α_{1}$ and $α_{2}$ are almost similar. The amplitudes of the waves weaken significantly with the growth of $α_{1}$ and $α_{2}$ , which is reliable with the normal effect of these two factors. The number of oscillations and the wavelengths stay constant for both $u (t)$ and $v (t)$ with the growth of $α_{1}$ and $α_{2}$ . The time history responses of $u (t)$ and $v (t)$ in the APPVD are significantly shaped by the damping coefficients $α_{1}$ (main structure) and $α_{2}$ (pendulum). Increasing $α_{1}$ causes the envelope of $u (t)$ to decay more quickly, which decreases overshoot and settling time and results in smoother oscillations. On the other hand, too much $α_{1}$ lowers the energy that can be transferred to the pendulum, which weakens absorber action. While high $α_{2}$ over damps $v (t)$ , causing its oscillations to decay quickly and the main mass $u (t)$ to retain higher amplitudes, low or moderate values allow sustained bounded swings in $v (t)$ , enabling efficient targeted energy transfer and stabilizing $u (t)$ . In contrast, $α_{2}$ controls how the pendulum manages the energy it absorbs.

The forcing amplitude $q$ and excitation frequency $σ$ strongly influence the temporary response of the main structure movement in the APVD system. So, Figure 16 are plotted to scrutinize their effects on the behavior of $u (t)$ , which is the equivalent function to the structure displacement $x (t)$ . It is found that the amplitude of $u (t)$ enlarges with the growth of both $q$ and $σ$ . Additionally, the waves become not uniform with the rise of $q$ , which is in covenant with the excitation nature of $q$ .When $q$ is small, the pendulum remains weakly engaged, and $u (t)$ shows nearly un-damped oscillations with slow decay, indicating poor energy transfer. As $q$ increases to moderate values, the pendulum is activated, and the time history of $u (t)$ exhibits reduced amplitudes and faster envelope decay as energy is efficiently directed to the pendulum. However, at excessively high $q$ the time history may show large bursts, irregular oscillations, or even beating phenomena, reflecting destabilization of the coupled dynamics. The excitation frequency $σ$ also governs stability: when $σ$ is close to the internal resonance condition (e.g., 2:1), $u (t)$ rapidly decays to a bounded periodic response with clear envelope reduction, while off-resonant values of $σ$ produce weaker damping, allowing higher oscillation amplitudes to persist. At certain detuned or critical frequencies, the time history of $u (t)$ can display quasi-periodic or chaotic patterns, showing that $q$ have a dual role moderate values promote stabilization, while excessive or mismatched values of $σ$ destabilize the system.

4.2. Phase plane structure

Deep insight into the coupled dynamics of the main structure and the pendulum is provided by the analysis of phase plane configurations for the APVD system. State trajectories, such as uu’ and vv’, can be plotted to identify stable, periodic, and unstable regimes that are difficult to discern from time histories alone. Phase pictures in the stable case show restricted oscillations and efficient energy transfer in the form of converging spirals or closed limit cycles. Smooth circular or near-periodic loops in the phase plane indicate synchronized energy exchange between the main structure and the pendulum when forcing and damping are adjusted to their ideal values. The trajectories, on the other hand, spread outward, cross over themselves, or display dispersed patterns under inadequate parameter tuning or severe forcing clear signs of instability, quasi-periodicity, or chaos. Consequently, the phase plane serves as a diagnostic tool, showing the basins of attraction, illustrating how beginning circumstances change toward various attractors, and indicating if the absorber stabilizes or destabilizes the coupled motion. The phase plane studying significance performs in many applications, as:

Analyze how oscillatory stability is affected by amplitudes and predict when a system will become chaotic. Assess how well energy is distributed and transmitted in interconnected systems. Optimize stable and predictable motion characteristics in engineering designs. Examine nonlinear occurrences in mechanics and physics applications that are caused by amplitude.

Because of the aforementioned features, the connections between the periodic corresponding NPA functions $u (t)$ , and $v (t)$ and their time-derivatives $u^{'} (t)$ , $v^{'} (t)$ are exposed by Figures 17–19. These graphs illustrate the stability and steady-state properties of solutions using symmetric circular loops with the vertical or horizontal axes at their centers. The resulting elliptical, symmetric, and periodic trajectories are expected since the phase plane diagram of the linear system solutions as shown in Eqs (12). and (20). Elevating elliptical trajectories on a phase plane graph indicate a rise in the system’s energy level. This behavior is often induced by several circumstances, such as increased coupling in the system, external energy input, or increasing amplitude.

Figure 17.

Shows the $u u^{'}$ - phase plane with the variation of A.

Figure 17. is schemed to inspect the phase-plane trajectories of $u (t)$ under the variation of the preliminary amplitude $A$ . These graphics show that the enclosed curves are expanding around the function $u (t)$ ’s centre. Weak oscillations and quick decay under damping are reflected by the trajectories at small $A$ , which are limited close to the origin and form tight spirals or small closed loops. The systems stored energy increases as $A$ rises, and the trajectories widen into loops around the centre, exhibiting longer transient responses and more intense oscillations. This outward expansion shows that bigger reaction amplitudes are produced by larger initial displacements before energy is progressively transferred to the pendulum and dissipated. Thus, the expansion of system energy and its effect on stability are immediately shown by the growing dispersion of trajectories around the centre with greater $A$ . It is found that the second initial amplitude $B$ doesn’t affect the phase plane paths of $u (t)$ .

Similarly, the same effects are found on the trajectories of $v (t)$ under the rise of $B$ as well as $A$ , where it is noted that these bounded curves are growing about their midpoint for this function, as seen in Figure 18(a) and (b). For the pendulum phase plane, the expansion and dispersion of trajectories around the equilibrium center clearly show the effect of raising the two starting amplitudes. Weak excitation and restricted energy exchange are indicated when both $A$ and $B$ are modest because the pendulum trajectories stay around the origin and form tight elliptical loops or tiny spirals that fade rapidly. As either $A$ or $B$ rises, the pendulum receives more energy input: higher $A$ drives greater coupling from the main system into the pendulum, while larger $B$ directly increases the pendulum’s stored potential energy. Larger oscillations and longer-lived transient motion are thus reflected by the trajectories expanding into bigger closed loops around the center. The pendulum phase plane exhibits distorted loops, crossings, or dispersed patterns with high enough $A$ and $B$ , indicating nonlinear effects, multi-stability, or shifts toward quasi-periodic or chaotic responses. The direct relationship between the initial energy content in both subsystems and the complexity of pendulum dynamics is therefore seen by the outward development of trajectories with increasing starting amplitudes.

Figure 18.

Displays the $v v^{'}$ - phase plane with the variation of (a) $A$ and (b) $B$ .

Figure 19(a) and (b) illustrate the impacts of the damping factors $α_{1}$ and $α_{2}$ on $u (t)$ , and $v (t)$ . It is evident that as $α_{1}$ increases, the phase plane trajectories of $u (t)$ expand, but those of $v (t)$ decrease as $α_{2}$ increases. Increasing the damping coefficient $α_{1}$ has a visible influence on the trajectories in the primary structure phase plane. The trajectories create broad loops around the origin for tiny $α_{1}$ , indicating that oscillations continue with huge amplitudes and decay slowly since only a little amount of energy is lost each cycle. Faster attenuation of oscillations and a stabilizing effect on the system are reflected in the trajectories’ increased inward spiral and the loops’ gradual shrinkage toward the centre as $α_{1}$ rises. The absorber performs better in this regime as the main system successfully transfers energy to the pendulum, and damping in the main structure aids in the suppression of resonance peaks. Nevertheless, if $α_{1}$ grows too much, the trajectories stop expanding considerably and instead collapse too soon close to the centre, which indicates that the motion of the main oscillator is over-suppressed and that energy transmission to the pendulum is diminished. Thus, excessive damping can change the coupled response of the system and diminish absorber efficiency, whereas a moderate increase in $α_{1}$ stabilizes the dynamics and reduces the spread of $u (t)$ - phase trajectories. Furthermore, increasing the damping coefficient $α_{2}$ in the pendulum phase plane has the opposite impact from the main structure. Because the pendulum can effectively absorb and release energy from the basic structure, the $v (t)$ trajectories are relatively constrained for low $α_{2}$ , with modest loops around the center. The pendulum’s capacity to waste energy, however, decreases as $α_{2}$ increases. Rather of stabilizing, the trajectories of $v (t)$ extend outward, suggesting that the pendulum oscillates with greater amplitudes and continues to move for longer. This happens because increased pendulum damping disrupts the auto-parametric vibration absorber’s precise energy-exchange process, decreasing synchronization with the main mass and permitting unstable increases in pendulum motion. The pendulum may even become ineffective as an absorber at very high $α_{2}$ , resulting in distorted, wider, or irregular phase trajectories that indicate decreased stability.

Figure 19.

Displays the phase planes $u u^{'}$ and $v v^{'}$ with the variation of (a) $α_{1}$ and (b) $α_{2}$ .

5. Chaotic behavior

The BDs of the dynamical systems can be examined to find the critical moments, where the system’s personality fundamentally changes. Then, you zoom in on one of those moments and use the PM to understand the intricate structure of its new, complex behaviour.⁴⁰ One shows you when the drama happens; the other shows you the face of the drama itself. Bifurcation curves graphically depict how a system behaves when a parameter changes.^38–40 The formation of new equilibrium points or periodic cycles is an instance of a transitional phase in the system’s behaviour that they identify.^38–40 The bifurcation charts of $x (τ)$ with $q$ are shown in Figure 20. The corresponding with PMs at various values of $q$ is shown in Figures 21 and 22 to reveal the characteristics of different motion kinds.

Figure 20.

Shows the BD of $x (τ)$ and $θ (τ)$ with $q$ .

Figure 21.

Shows PMs, and phase portraits of $x (τ)$ and $θ (τ)$ at $q = 0.5$ .

Figure 22.

Shows PMs, and phase portraits of $x (τ)$ and $θ (τ)$ at $q = 5$ .

A dynamical system’s behaviour as a parameter changes is represented graphically using bifurcation curves. The bifurcation diagrams of $x (τ)$ and $θ (τ)$ with $q$ are presented in Figure 20. They show when the behaviour of the system undergoes qualitative changes, such as the formation of new equilibrium points or periodic orbits. The phase portraits and Poincare’ maps are shown in Figures 21 and 22; the Poincare’ maps are shown by red dots, while the phase portraits are shown by blue curves. The resultant figures were displayed for a range of $q$ values in order to demonstrate the various motions of the system. Through the period $0 \leq q \leq 1.8$ the behaviour of the system is a periodic motion, but in $1.8 < q \leq 5$ the motion is chaotic. In Figure 21, we examine a value $q = 0.5$ . The red-dot pattern exhibited by PM closely resembles a point of $x (τ)$ and a four points of $θ (τ)$ , contributing to the earlier conclusion regarding the periodic motion of the system. But in Figure 22, we investigate a value $q = 5$ . The red-dot pattern exhibited by PM are randomly distribution so the motion is chaotic.

6. Conclusions

The current work offered a more thorough comprehension of how vibration suppression is controlled by auto-parametric coupling between a main structure and a pendulum absorber. The NPA captures the system dynamics without linearization, revealing the true nonlinear properties, including multi-frequency interactions, stability transitions, and potential chaotic reactions. The findings demonstrated the opposing stabilizing/destabilizing impacts of damping coefficients, starting amplitudes, excitation levels, and coupling parameters. This opposition emphasized the fine balance needed for efficient energy transfer: initial energy in the main system encourages suppression, meanwhile initial pendulum energy destabilizes the host structure; primary damping improves stability, although pendulum damping reduces absorber efficiency. Furthermore, factors such as excitation, forcing amplitude, and the coupling coefficients affect whether the system diverges into unstable motion or achieves regulated vibration mitigation. Physically speaking, this means that the APVD was a nonlinear energy redistributor rather than just a passive absorber, and its efficacy was dependent on precisely adjusting its parameters. Engineering applications that needed targeted vibration management under nonlinear and coupled oscillatory situations, such as tall buildings, bridges, ship rolling, and aerospace structures, depend on such understanding. The NPA limitations could be summarized as, the methodology applies to any weakly nonlinear oscillator described by an ODE, the ICs remain unaltered, the initial amplitude must be below one. The results of the current study could be summarizing in the following points:

(1) The damping coefficient of the main structure stabilized the main system; contracts $x$ -trajectories and accelerates decay.

(2) The damping coefficient of the pendulum destabilizes pendulum motion; expands $φ$ -trajectories and reduced absorber effectiveness.

(3) The Initial amplitudes stabilized the pendulum by injecting energy into pendulum motion and destabilized the main displacement by feeding back oscillatory energy.

(4) The coupling coefficient stabilized both the main displacement and the pendulum by strengthening coupled energy exchange.

(5) The excitation force played a dual role, where moderate levels activate absorption, but excessive forcing destabilized both motions.

(6) The excitation amplitude destabilized the system by amplifying nonlinear oscillations.

Recent developments include semi-active and active control systems that improve resilience to resonance by adapting to excitation frequencies. Methods used encompass NI, analytical modelling, and experimental validation of applications in skyscrapers and bridges. Future research may focus on smart materials, real-time sensors, and machine learning to optimize energy transfer and vibration mitigation in complex systems.

Footnotes

Acknowledgement

The researchers would like to thank the Deanship of Graduate Studies and Scientific Research at Qassim University () for its financial support (QU-APC-2026).

ORCID iDs

T. S. Amer

Mona A.A. Mohamed

Author Contributions

Ahmad Almutlg: Resources; Data duration. Galal M. Moatimid: Data duration; Visualization, Writing, review and editing. T. S. Amer: Methodology; Investigation; Formal analysis; Data analysis and validation; Software implementation; Writing – original draft; Writing – review and editing. Mona A. A. Mohamed: Software implementation; Writing – original draft; Writing – review and editing. All authors have read and agreed to the published version of the manuscript.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The researchers would like to thank the Deanship of Graduate Studies and Scientific Research at Qassim University () for its financial support (QU-APC-2026).

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Appendix

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