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Abstract

This study introduces a novel approach to analyzing a four-degree-of-freedom (DoF) nonlinear system by leveraging advanced numerical and analytical techniques to comprehensively examine its dynamic behavior. The system’s nonlinear differential equations (DEs) are obtained through the application of Lagrange’s equations (LE). The solutions are obtained using the fourth-order Runge–Kutta method (4-RKM). The investigation involves analyzing the relationships between the angular solutions and their corresponding first-order derivatives, commonly referred to as phase plane analysis. The study aims to examine bifurcation diagrams and Lyapunov exponent spectra to reveal the various modes of motion within the system and visualize Poincaré maps. These tools are used to analyze a unique system configuration. Lastly, the conditions for solvability and the characteristic exponents are identified by examining resonance scenarios. The examination of resonance scenarios through characteristic exponents and solvability conditions, coupled with the application of Routh-Hurwitz criteria (RHC) for stability evaluation, provides an innovative framework for understanding frequency response and nonlinear stability across stable and unstable ranges. By exploring both theoretical and practical aspects of vibrational dynamics in applications like aviation, robotics, and underwater exploration, this work offers a significant advancement in analyzing complex systems, with wide-ranging implications for various engineering fields, including aerospace, structural mechanics, and energy harvesting.

Keywords

vibrating motions nonlinear dynamics phase plane numerical solution bifurcation analyses Poincare map stability

Introduction

Vibration analysis is a fundamental aspect of engineering and applied sciences, playing a crucial role in understanding the dynamic behavior of mechanical and structural systems subjected to external forces or excitations. From bridges and buildings to aircraft and automotive components, the study of vibrations is indispensable for ensuring structural integrity, optimizing performance, and enhancing safety across various engineering disciplines. The solution of vibration systems typically involves determining dynamic responses, mode shapes, and natural frequencies under different loading circumstances. Analytical methods, such as modal analysis and the method of undetermined coefficients, offer valuable insights into the fundamental behavior of vibrating systems and are widely used for preliminary design and analysis. However, for systems with complex geometries, material properties, or loading conditions, analytical solutions may be challenging or impractical to obtain. Numerical techniques offer a versatile and efficient alternative for vibrating systems, allowing for the accurate simulation and analysis of complex systems with arbitrary geometries and boundary conditions. Techniques such as the finite element method (FEM), finite difference method (FDM), and boundary element method (BEM) enable engineers and researchers to discretize and solve the governing equations of motion, providing detailed insights into the vibrational behavior of structures and components. An introduction to the numerical techniques used for solving ordinary DEs is provided in Ref. 1, including detailed discussions on the Euler method, 4-RKM, and multistep method, along with their theoretical underpinnings and practical applications. In Ref. 2, a wide range of topics in scientific computing, including numerical solutions of ordinary DEs and partial DEs, are introduced. It emphasizes practical implementation, providing a balanced approach to theory and application with numerous examples and exercises. A practical guide to numerical techniques for scientists and engineers, with an emphasis on using MATLAB for implementation, is presented in Ref. 3. It covers ordinary DEs, partial DEs, systems of equations, numerical optimization, and eigenvalue problems, focusing on algorithm development and practical applications.

The study of complex dynamical systems has been greatly influenced by the pioneering work of mathematicians Aleksandr Mikhailovich Lyapunov and Henri Poincaré, whose insights laid the foundation for modern nonlinear dynamics. Furthermore, the concept of bifurcation has emerged as a central theme in understanding the qualitative changes in system behavior. Lyapunov made significant contributions to stability theory through his seminal work on Lyapunov functions and stability analysis. His groundbreaking monograph established the framework for assessing the stability of equilibrium points in dynamical systems. Lyapunov’s stability theory, based on the concept of Lyapunov exponents and functions, has found widespread applications in fields such as control theory, celestial mechanics, and population dynamics.⁴ Poincaré made fundamental contributions to the understanding of chaotic behavior in dynamical systems. His investigations into the three-body problem in celestial mechanics revealed the existence of chaotic orbits and sensitive dependence on initial conditions, anticipating the modern concept of chaos theory.⁵

Bifurcation theory, originating from the works of mathematicians such as Andronov, Pontryagin, and Poincaré, focuses on the qualitative changes in the behavior of dynamical systems as control parameters vary. Bifurcations represent critical points where the system undergoes a qualitative transition, giving rise to new dynamical regimes such as periodic orbits, chaos, or multiple stable states. Understanding bifurcation phenomena is essential for predicting and controlling the behavior of complex systems in diverse scientific domains, including physics, engineering, ecology, and economics.⁶ The authors introduce the concept of strange attractors and analyze the behavior of nonlinear dynamical systems, particularly in the context of turbulence. They discuss the role of bifurcations in determining the transitions between different dynamical regimes and propose methods for characterizing chaotic behavior.⁷ The discovery of deterministic chaos in a simple model of atmospheric convection is presented in Ref. 8. He discusses the emergence of chaotic behavior through bifurcations and highlights the sensitivity to initial conditions, which has profound implications for weather forecasting and climate science. May explores the dynamics of simple ecological models and demonstrates how they can exhibit complex behavior, including bifurcations, oscillations, and chaos. He discusses the implications of these findings for understanding population dynamics, biodiversity, and ecological stability.⁹ The stability of predator-prey systems using graphical methods and Lyapunov stability theory is analyzed in Ref. 10. The authors provide graphical representations of stability regions and discuss the conditions under which predator-prey interactions lead to stable or oscillatory dynamics. The application of Lyapunov distribution patterns is crucial for investigating stability and chaos in dynamic systems, especially those integrating energy harvesting (EH) devices. EH systems encounter the challenge of enhancing device performance and efficiency in the face of fluctuating and unpredictable environmental conditions. Through the utilization of Lyapunov distribution patterns, one can identify areas of stability, periodic behavior, and chaos within these systems.¹¹ An analytical scheme designed to analyze the stability of a 4DoF mechanical system under friction-induced vibrations is introduced in Ref. 12. The study investigates the dynamic behavior of such systems by considering the impact of friction on stability. Utilizing analytical methods such as (LE) and stability analysis techniques, the research explores how friction affects the system’s stability boundaries. Through extensive numerical simulations and analytical derivations, the study identifies critical stability thresholds and regions susceptible to instability. The findings offer valuable insights into understanding the stability characteristics of 4DoF mechanical systems subjected to friction-induced vibrations, with implications for diverse engineering applications, including machinery, robotics, and automotive systems. The bifurcations and stability of high-frequency periodic motions with limited amplitude, along with the associated stability criteria, are also addressed. Recent studies in Refs. 13 and 14 investigate the stability of unstretched double and triple pendulums subjected to external harmonic moments and forces. The stability and instability regions for various frequency response parameters are determined.

The bifurcation and chaotic behaviors in a Duffing oscillator are examined in Ref. 15 using Poincaré maps and bifurcation diagrams. It also explores methods for controlling chaotic dynamics. Whereas the nonlinear dynamics in rotor systems, focusing on bifurcation behavior and chaos, is investigated in Ref. 16. The Poincaré maps are used to analyze transitions between periodic and chaotic regimes. In Ref. 17, the vibrational dynamics and stability of a planar double pendulum, particularly near resonance conditions is explored. Furthermore, the authors investigated the nonlinear behaviors, stability regimes, and resonance phenomena, making it relevant for mechanical systems with complex dynamic interactions. A nonlinear dynamical system incorporating a piezoelectric transducer is analyzed in Ref. 18. The coupling effects of damping and piezoelectric interactions, offering insights into energy harvesting and system optimization, are emphasized. In Ref. 19, the authors investigated a 2DOF system with piezoelectric devices and feedback control. Stability, bifurcation behaviors, and feedback mechanisms are provided to improve control and performance. The chaotic and stable dynamics of the Duffing oscillator under sinusoidal excitation are delved into Ref. 20. The bifurcation, stability limits, and chaotic transitions in nonlinear oscillators are explored. The presented work in Ref. 21 investigates coupled damped oscillators integrated with a piezoelectric energy harvester. The study examines bifurcation, stability, and dynamics, aiming to optimize energy harvesting and dynamic performance. These works contribute significantly to understanding nonlinear dynamics, stability, and energy systems across diverse engineering fields, such as robotics, vibration control, and energy harvesting.

The research outlined in this paper delves into the dynamics of a 4DoF system. The nonlinear DEs governing this system are deduced through the application of LE. Numerical solutions are then obtained utilizing the 4-RKM. The inquiry entails scrutinizing the connections between angular solutions and their respective first-order derivatives, a process commonly known as phase plane analysis. The objective is to analysis the bifurcation diagrams and Lyapunov exponent spectra to unveil the diverse modes of motion within the system and depict Poincaré maps. The RHC are applied to evaluate the stability and instability domains, followed by an analysis of the steady-state solutions. The outcomes of this research are applicable to disciplines like construction and structural design, vehicle dynamics, industrial machinery, seismic impact mitigation, and renewable energy harvesting.

Overview of the problem

We have a hollow block with a mass $m_{1}$ connected to three damped springs as shown in Figure 1. Two of these springs, located on the right and left sides, are connected to smooth vertical planes that slide along red-colored vertical walls. The third damped spring has its other end connected to a horizontal plane that slides along the red-colored horizontal floor. Inside the hollow block, another mass $m_{2}$ oscillates. The inner mass is connected to four damped springs, with their other ends connected to planes that slide along the inner surfaces of the hollow block. The external block is subjected to horizontal and vertical forces $F_{1}$ and $F_{2}$ , respectively, while the internal mass is subjected to horizontal and vertical forces $F_{3}$ and $F_{4}$ , respectively. The spring constants and damping coefficients are $k_{s}$ and $b_{s}$ , respectively, where $s = 1, . . ., 7 .$ as indicated by the numbering on the damped springs. The horizontal and vertical displacements of the external block are $x_{1}$ and $x_{2}$ , respectively, while the horizontal and vertical displacements of the internal mass are $x_{3}$ and $x_{4}$ , respectively. Note that friction between the contact surfaces is neglected and the displacements are real, not relative. This system could be used as a model for vibration isolation platforms or systems that protect sensitive equipment (in buildings, laboratories, or manufacturing plants). The hollow block represents the external structure, while the inner oscillating mass represents a dynamic isolator tuned to reduce transmitted vibrations from external forces (Figure 1).

Figure 1.

Illustrates the vibrating system.

We have defined the kinetic and potential energies in the following manner:

T = \frac{1}{2} m_{1} (x_{1}^{' 2} + x_{2}^{' 2}) + \frac{1}{2} m_{2} (x_{3}^{' 2} + x_{4}^{' 2}), V = \frac{1}{2} (k_{1} + k_{2}) x_{1}^{2} + \frac{1}{2} (k_{4} + k_{5}) {(x_{1} - x_{3})}^{2} + \frac{1}{2} k_{3} x_{2}^{2} + \frac{1}{2} (k_{6} + k_{7}) {(x_{2} - x_{4})}^{2} .

(1)

The derivation of the equations of motion (EOM) for the described system utilizing the Lagrangian $L = T - V$ employed a specific variant of LE termed the second type.

\frac{d}{d t} (\frac{\partial L}{\partial {\dot{x}}_{i}}) - \frac{\partial L}{\partial x_{i}} = Q_{i}, i = 1, 2, . . ., 4,

(2)

where the generalized velocities

{\dot{x}}_{i}

are linked with distinct generalized coordinates

x_{i}

within the system, whereas the generalized forces

Q_{j} (j = 1, 2, . . ., 4)

can be represented using specific mathematical expressions

\begin{array}{l} Q_{1} = F_{1} - (b_{1} + b_{2}) {\dot{x}}_{1} - (b_{4} + b_{5}) ({\dot{x}}_{1} - {\dot{x}}_{3}), \\ Q_{2} = F_{2} - (b_{3} + b_{6} + b_{7}) {\dot{x}}_{2} + (b_{6} + b_{7}) {\dot{x}}_{4}, \\ Q_{3} = F_{3} - (b_{4} + b_{5}) ({\dot{x}}_{1} - {\dot{x}}_{3}), Q_{4} = F_{4} - (b_{6} + b_{7}) ({\dot{x}}_{2} - {\dot{x}}_{4}), \end{array}

(3)

where

F_{1} = F_{1}^{*} \cos δ_{1} t, F_{2} = F_{2}^{*} \cos δ_{2} t, F_{3} = F_{3}^{*} \cos δ_{3} t, F_{4} = F_{4}^{*} \cos δ_{4} t .

Here,

F_{j}^{*},

and

δ_{j} (j = 1, 2, . . ., 4)

are the amplitudes and frequencies of

F_{j}

In order to streamline the system analysis, we introduce dimensionless variations of the system parameters.

\begin{array}{l} ω_{1}^{2} = \frac{k_{1} + k_{2} + k_{4} + k_{5}}{m_{1}}, ω_{2}^{2} = \frac{k_{4} + k_{5}}{m_{4}}, ω_{3}^{2} = \frac{k_{3} + k_{6} + k_{7}}{m_{1}}, ω_{4}^{2} = \frac{k_{6} + k_{7}}{m_{1}}, \\ ω_{5}^{2} = \frac{k_{4} + k_{5}}{m_{2}}, ω_{6}^{2} = \frac{k_{6} + k_{7}}{m_{2}}, ω^{2} = \frac{ω_{2}^{2}}{ω_{1}^{2}}, ϖ^{2} = \frac{ω_{3}^{2}}{ω_{1}^{2}}, Ω_{1}^{2} = \frac{ω_{4}^{2}}{ω_{1}^{2}}, Ω_{2}^{2} = \frac{ω_{5}^{2}}{ω_{1}^{2}}, \\ Ω_{3}^{2} = \frac{ω_{6}^{2}}{ω_{1}^{2}}, B_{1} = \frac{b_{1} + b_{2}}{m_{1} ω_{1}}, B_{2} = \frac{b_{3} + b_{5}}{m_{1} ω_{1}}, B_{3} = \frac{b_{6} + b_{7}}{m_{1} ω_{1}}, B_{4} = \frac{b_{3} + b_{6} + b_{7}}{m_{1} ω_{1}}, \\ B_{5} = \frac{b_{4} + b_{5}}{m_{2} ω_{1}}, B_{6} = \frac{b_{6} + b_{7}}{m_{2} ω_{1}}, f_{1} = \frac{F_{1}^{*}}{m_{1} ω_{1}}, f_{2} = \frac{F_{2}^{*}}{m_{1} ω_{1}}, f_{3} = \frac{F_{3}^{*}}{m_{2} ω_{1}}, f_{4} = \frac{F_{4}^{*}}{m_{2} ω_{1}}, p_{1} = \frac{δ_{1}}{ω_{1}}, p_{2} = \frac{δ_{2}}{ω_{1}}, p_{3} = \frac{δ_{3}}{ω_{1}}, p_{4} = \frac{δ_{4}}{ω_{1}}, τ = ω_{1} t . \end{array}

(4)

Inserting formula (4) in the equations (1)–(3) to derive the dimensionless forms of the equations of motion (EOM).

{\ddot{x}}_{1} + x_{1} = ω^{2} x_{3} - (B_{1} + B_{2}) {\dot{x}}_{1} + B_{2} {\dot{x}}_{3} + f_{1} \cos (p_{1} τ),

(5)

{\ddot{x}}_{2} + ϖ^{2} x_{2} = - Ω_{1}^{2} x_{4} + B_{3} {\dot{x}}_{4} - B_{4} {\dot{x}}_{2} + f_{2} \cos (p_{2} τ),

(6)

{\ddot{x}}_{3} + Ω_{2}^{2} x_{3} = Ω_{2}^{2} x_{1} - B_{5} ({\dot{x}}_{1} - {\dot{x}}_{3}) + f_{3} \cos (p_{3} τ),

(7)

{\ddot{x}}_{4} + Ω_{3}^{2} x_{4} = Ω_{3}^{2} x_{2} - B_{6} ({\dot{x}}_{2} - {\dot{x}}_{4}) + f_{4} \cos (p_{4} τ) .

(8)

Numerical analysis

The primary goal of the 4-RKM is to approximate the solution of an initial value problem for an ordinary differential equation, where an equation describes how a function changes with respect to an independent variable. These problems commonly arise in various scientific and engineering fields, including physics, biology, chemistry, and engineering. By analyzing the shape and direction of trajectories, we can infer important properties such as stability, and oscillatory behavior. Now, we have to transform equations (5)–(8) into a system of first-order equations. Introduce a new variable $u_{s} (s = 1, . . ., 8)$ such that $x_{1} = u_{1}, {\dot{x}}_{1} = u_{2}, x_{2} = u_{3}, {\dot{x}}_{2} = u_{4}, x_{3} = u_{5}, {\dot{x}}_{3} = u_{6}, x_{4} = u_{7}, {\dot{x}}_{4} = u_{8} .$ This transforms the second-order DE (5)–(8) into eight first-order DEs²²

\begin{array}{l} {\dot{x}}_{1} = u_{2}, {\dot{u}}_{2} + u_{1} = ω^{2} u_{5} - (B_{1} + B_{2}) u_{2} + B_{2} u_{6} + f_{1} \cos (p_{1} τ), \\ {\dot{x}}_{2} = u_{4}, {\dot{u}}_{4} + ϖ^{2} u_{3} = - Ω_{1}^{2} u_{7} + B_{3} u_{8} - B_{4} u_{4} + f_{2} \cos (p_{2} τ), \\ {\dot{x}}_{3} = u_{6}, {\dot{u}}_{6} + Ω_{2}^{2} u_{5} = Ω_{2}^{2} u_{1} - B_{5} (u_{2} - u_{6}) + f_{3} \cos (p_{3} τ), \\ {\dot{x}}_{4} = u_{8}, {\dot{u}}_{8} + Ω_{3}^{2} u_{7} = Ω_{3}^{2} u_{3} - B_{6} (u_{4} - u_{8}) + f_{4} \cos (p_{4} τ) . \end{array}

(9)

The numerical solution of the system presented in equation (9) has been computed and visually represented in Figures 2 and 3, utilizing the provided parameter values.

b_{1} = 0.001, b_{2} = 0.001, b_{3} = 0.002, b_{4} = 0.004, m_{1} = 8, m_{2} = 4, b_{5} = 0.004, b_{6} = 0.003, b_{7} = 0.005, k_{1} = 20, k_{2} = 18, k_{3} = 20, k_{4} = 10, k_{5} = 15, k_{6} = 12, k_{7} = 7 .

Figure 2.

Demonstrates how changes in $ω (= 0.5, 0.54, 0.58),$ values affect the time evolution of the solutions: (a) $x_{1},$ (b) $x_{2},$ (c) $x_{3},$ and (d) $x_{4}$ .

Figure 3.

Illustrates the impact of alterations in $B_{5} (= 0.00026, 0.00029, 0.00032)$ data on the temporal evolution of the curves: (a) $x_{1},$ (b) $x_{2},$ (c) $x_{3},$ and (d) $x_{4}$ .

These data are selected values for the used parameters, where the stability of the proposed solver is indeed influenced by the specific parameter values of the system. In general, the 4-RKM is applied for a wide range of parameter values, provided that the time step is chosen appropriately. However, there are certain conditions that must be taken into account to ensure stability in the context of the system being analyzed.

Upon detailed analysis of Figures 2–5, it becomes evident that these plots are generated under the circumstance where $ω (= 0.5, 0.54, 0.58),$ and $B_{5} (= 0.00026, 0.00029, 0.00032) .$ As the parameter $ω$ increases, we observe the emergence of periodic waves, leading to higher amplitudes for the waves representing $x_{1}, x_{2}, x_{3},$ and $x_{4}$ , accompanied by a modest increase in the frequency of oscillations, as illustrated in Figure 2. It should be noted that with the change of the frequency value $ω$ , all values of $B_{1}$ through $B_{7}$ change. The change of the $B_{5}$ values on the behavior of $x_{1}, x_{2}, x_{3},$ and $x_{4}$ has a positive effect resulting in the generation of periodic waves, as illustrated in Figure 3. The amplitudes of the waves describing $x_{1},$ and $x_{3}$ decrease with a rise in the value of $B_{5}$ as seen in Figure 3(a) and (c), conversely, the amplitudes of $x_{2},$ and $x_{4}$ increase as shown in Figure 3(b) and (d).

Figure 4.

Unveils the phase plane diagrams under the condition that $ω (= 0.5, 0.54, 0.58),$ in the planes: (a) $x_{1} {\dot{x}}_{1},$ (b) $x_{2} {\dot{x}}_{2},$ (c) $x_{3} {\dot{x}}_{3},$ and (d) $x_{4} {\dot{x}}_{4}$ .

Figure 5.

Phase plane diagrams under the constraint that $B_{5} (= 0.00026, 0.00029, 0.00032)$ across the planes: (a) $x_{1} - {\dot{x}}_{1},$ (b) $x_{2} - {\dot{x}}_{2},$ (c) $x_{3} - {\dot{x}}_{3},$ and (d) $x_{4} - {\dot{x}}_{4}$ .

Phase plane analysis also allows us to identify and classify different types of trajectories exhibited by the system. These trajectories may include periodic orbits, limit cycles, spirals, and other complex patterns, each offering unique glimpses into the system’s dynamics. Moreover, phase plane analysis enables the study of bifurcations, which are qualitative changes in the system’s behavior as control parameters are varied. Bifurcation points mark critical transitions where the system undergoes a qualitative shift in its dynamics, leading to the emergence of new phenomena such as chaos, multiple stable states, or periodic oscillations. The connections between the angular solutions and their respective first-order derivatives are depicted in Figures 4 and 5 to construct phase plane diagrams for various values of $ω$ and $B_{5}$ . It’s worth mentioning that we observe partially closed symmetric trajectories, suggesting the stable nature of the angular solutions over the examined time span, aligning with previous predictions. Due to the complexity of the model, we set the amplitude of all forces to zero to simplify the study in Sections 3 and 4.

Chaotic motion

Chaotic motion is a fascinating phenomenon observed in dynamic systems across various scientific disciplines, ranging from physics and engineering to biology and economics. Unlike regular or periodic motion, chaotic motion is characterized by its unpredictability, sensitivity to initial conditions, and non-repeating trajectories. This unpredictability arises from the inherent complexity and nonlinear dynamics of chaotic systems, making them challenging yet intriguing subjects of study. It also emerges as a form of irregular and unpredictable movement observed in systems that exhibit extreme sensitivity to their starting conditions. Unlike truly random motion, chaotic behavior is governed by deterministic rules, where future states are entirely determined by initial conditions. However, despite its deterministic nature, practical prediction of chaotic motion proves challenging. Conversely, quasi-periodic motion lacks the regularity of periodic motion and the randomness of chaotic motion, yet maintains a semblance of order. This phenomenon arises when a system simultaneously embodies two or more frequencies that are not related by simple numerical ratios, thereby preventing repetitive patterns from emerging.

Here, we will demonstrate the two types of motion mentioned earlier, as observed in our dynamical system. Specifically, we will showcase Lyapunov spectra for $λ_{j} (j = 1, 2, . . ., 8)$ , present bifurcation diagrams for $x_{i}$ , and analyze the spectrum under the influence of $B_{4}$ . The Lyapunov spectrum showcased a variety of motion patterns, featuring two distinct intervals of $B_{4}$ , each associated with a unique dynamic behavior in Figures 6 and 7. Within the initial range of $B_{4} \in [0, 4.6]$ , multiple Lyapunov spectra are observed, divided into two clusters with positive and negative values as seen in Figure 7(b)). Consequently, the system’s behavior can be characterized as chaotic motion. In the subsequent interval of $B_{4} \in 4.6, 10$ , we noted that the lines of the Lyapunov spectrum tend to cluster closely, resembling a straight line originating near zero (see Figure 7(c)). This suggests adherence to a quasi-periodic pattern of motion, a characteristic also evident in the bifurcation diagrams corresponding to the respective range of $x_{1}$ and $x_{3}$ as shown in parts (a), and (b) of Figure 6.

Figure 6.

Diagrams of bifurcation in (a) and (b) for $x_{1}$ and $x_{3}$ .

Figure 7.

Diagrams of the spectrum of Lyapunov for $x_{1}$ and $x_{3}$ .

Poincaré maps present dots and phase portraits in Figures 8–10, with red dots representing points and blue curves depicting trajectories. For Figures 8 and 9, we selected two values of $B_{4} = 6.5, 5$ falling within the second interval. These figures show the blue closed curves that eventually converge into a single red point, indicating periodic motion as observed in Figure 8(a) and 8(b). In Figure 9(a) and 9(b), the arrangement of red dots in the Poincaré map illustrates nearly closed trajectories for both variables $x_{1}$ and $x_{3}$ , confirming our earlier conclusion regarding the quasi-periodic nature of the system’s movement. Conversely, in Figure 10(a) and 10(b), due to chaotic behavior, the red dots at $B_{4} = 0.5$ begin to exhibit random divergence.

Figure 8.

Phase portraits and maps of Poincaré for $x_{1}$ and $x_{3}$ at $B_{4} = 6.5$ .

Figure 9.

Phase portraits and maps of Poincaré for $x_{1}$ and $x_{3}$ at $B_{4} = 5$ .

Figure 10.

Phase portraits and maps of Poincaré for $x_{1}$ and $x_{3}$ at $B_{4} = 0.5$ .

In Figure 11, the Lyapunov spectrum displayed diverse patterns of motion, delineated into two distinct intervals of $B_{4}$ . Each interval corresponds to a distinct dynamical behavior. Within the initial range of $B_{4} \in [0, 4.6]$ , the Lyapunov spectrum exhibits multiple spectra divided into two clusters, one featuring positive values and the other negative. Consequently, the system’s behavior may be characterized as chaotic movement. In the subsequent interval of $B_{4} \in 4.6, 7$ , we observed that the lines of the Lyapunov spectrum tend to align closely, resembling a straight line originating near zero. This suggests adherence to a quasi-periodic pattern of motion, a phenomenon also evident in the bifurcation diagrams corresponding to the respective range of $x_{2}$ and $x_{4}$ .

Figure 11.

Diagrams of bifurcation in (a) and (b) for $x_{2}$ and $x_{4}$ , and the corresponding Lyapunov spectrum in (c).

Poincaré maps depict dots and phase diagrams, represented by red dots and blue curves in Figures 12–14. For Figures 12 and 13, we selected two values of $B_{4} = 4.8, 6.8$ falling within the second interval. These figures show closed blue curves that gradually converge into a single red point, indicating periodic motion as observed in Figure 12(a) and 12(b). In Figure 13(a) and 13(b), the arrangement of red dots in the Poincaré map illustrates nearly closed trajectories for both $x_{2}$ and $x_{4}$ , confirming our earlier conclusion regarding the quasi-periodic nature of the system’s movement. Conversely, in Figure 14(a) and 14(b), due to chaotic behavior, the red dots at $B_{4} = 0.2$ begin to exhibit random divergence. Table 1 shows the values of $B_{4}$ that generate periodic, quasi-periodic and chaotic behavior.

Figure 12.

Phase portraits and maps of Poincaré for $x_{2}$ and $x_{4}$ at $B_{4} = 6.8$ .

Figure 13.

Poincaré maps and phase portraits for the variables $x_{2}$ and $x_{4}$ at $B_{4} = 4.8$ .

Figure 14.

Phase portraits and maps of Poincaré for $x_{2}$ and $x_{4}$ at $B_{4} = 0.2$ .

Table 1.

Shows periodic, quasi-periodic, and chaotic regions at different values of $B_{4}$ .

Figure	Type of motion	$B_{4}$
Figures 6 and 7	Chaotic	$[0, 4.6]$
Figures 6 and 7	Quasi-periodic, periodic	$4.6, 10$
Figure 11	Chaotic	$[0, 4.6]$
Figure 11	Quasi-periodic, periodic	$4.6, 7$

Analysis of system stability

It is possible to easily derive the controlling equations after obtaining the solvability conditions by studying four cases of external resonance simultaneously in the following form

\begin{array}{l} a_{1} (σ_{1} - \frac{d θ_{1}}{d τ} + B_{1} + B_{2} + \frac{B_{3} Ω_{2}^{2}}{Ω_{2}^{2} - 1}) - \frac{f_{1} \cos θ_{1}}{2} = 0, \frac{d a_{1}}{d τ} + \frac{B_{3} a_{1}}{2 {(Ω_{2}^{2} - 1)}^{2}} (B_{5} - \frac{2 Ω_{2}^{2}}{Ω_{2}^{2} - 1} (B_{1} + B_{2} + \frac{B_{5}}{2} + \frac{B_{3} Ω_{2}^{2}}{Ω_{2}^{2} - 1}) + \frac{f_{1} \sin θ_{1}}{2} = 0, \\ a_{2} (σ_{2} - \frac{d θ_{2}}{d τ} + \frac{B_{3} Ω_{3}^{2}}{2 (Ω_{3}^{2} - ϖ^{2})} - \frac{B_{4}}{2}) - \frac{f_{2} \cos θ_{2}}{2} = 0, ϖ \frac{d a_{2}}{d τ} - \frac{B_{3} ϖ^{3}}{2 (Ω_{3}^{2} - ϖ^{2})} (B_{6} + \frac{Ω_{3}^{2}}{{(Ω_{3}^{2} - ϖ^{2})}^{2}} (B_{4} + B_{6} - \frac{B_{3} Ω_{3}^{2}}{Ω_{3}^{2} - ϖ^{2}})) a_{2} + \frac{f_{2} \sin θ_{2}}{2} = 0, \\ a_{3} (σ_{3} - \frac{d θ_{3}}{d τ} + \frac{1}{2} (B_{5} + \frac{1}{1 - Ω_{2}^{2}} (B_{3} Ω_{2}^{2} - \frac{Ω_{2}^{2} (B_{1} + B_{2})}{2 (1 - Ω_{2}^{2})}) + (\frac{3 B_{3} Ω_{2}^{2}}{2 (1 - Ω_{2}^{2})} (1 - \frac{4 Ω_{2}^{2}}{3 (1 - Ω_{2}^{2})}) - \frac{B_{5}}{2}) (\frac{B_{3} Ω_{2}^{2}}{1 - Ω_{2}^{2}} + B_{5})) - \frac{f_{3} \cos θ_{3}}{2} = 0, Ω_{2} (\frac{d a_{3}}{d τ} - \frac{a_{3}}{2} (\frac{B_{3} Ω_{2}^{2}}{1 - Ω_{2}^{2}} + B_{5}) B_{3}) + \frac{f_{3} \sin θ_{3}}{2} = 0, \\ a_{4} (σ_{4} - \frac{d θ_{4}}{d τ} + \frac{1}{4} (\frac{B_{3} Ω_{3}^{2}}{ϖ^{2} - Ω_{3}^{2}} + B_{6}) (1 + 2 (\frac{B_{3} Ω_{3}^{2}}{ϖ^{2} - Ω_{3}^{2}} + B_{6}) + 2 (\frac{B_{3} Ω_{3}^{2}}{ϖ^{2} - Ω_{3}^{2}} (1 + \frac{2 Ω_{3}^{2}}{ϖ^{2} - Ω_{3}^{2}} + B_{6})) - \frac{f_{4} \cos θ_{4}}{2} = 0, Ω_{3} (\frac{d a_{4}}{d τ} + (\frac{B_{3} Ω_{3}^{2}}{ϖ^{2} - Ω_{3}^{2}} + B_{6}) \frac{a_{4} B_{3} B_{6}}{4 (ϖ^{2} - Ω_{3}^{2})}) + \frac{f_{4} \sin θ_{4}}{2} = 0, \end{array}

(10)

where

θ_{j} (j = 1, 2, . . ., 4)

represent phases, and

a_{j}

stand for the amplitudes.

It is important to note that resonance happens when an external force aligns with the natural frequency of a system, causing substantial oscillations. As a result, it is demonstrated in $p_{1} \approx 1, p_{2} \approx ω, p_{3} \approx Ω_{2},$ and $p_{4} \approx Ω_{3}$ .

By approximating the proximity of $p_{1}, p_{2}, p_{3}, p_{4}$ to $1, ω, Ω_{2}, Ω_{3},$ the detuning parameters $σ_{1}, σ_{2}, σ_{3},$ and $σ_{4}$ can be utilized to represent the deviation between the vibrations and exact resonance, as shown:

p_{1} = 1 + σ_{1}, p_{2} = ω + σ_{2}, p_{3} = Ω_{2} + σ_{3}, p_{4} = Ω_{3} + σ_{4} .

(11)

These concepts can be expressed through the variable $ε$ by

σ_{j} = ε {\tilde{σ}}_{j} (j = 1, 2, . . ., 4) .

(12)

Now, we will examine the steady-state solution by setting the initial derivatives of variables $θ_{j}$ and $a_{j}$ to zero.²³ Consequently, the following set of algebraic equations involving the variables $θ_{j}$ and $a_{j}$ can be written as follows:

\begin{array}{l} \frac{f_{1} \cos θ_{1}}{2} = a_{1} (σ_{1} + B_{1} + B_{2} + \frac{B_{3} Ω_{2}^{2}}{Ω_{2}^{2} - 1}), \\ \frac{f_{1} \sin θ_{1}}{2} = \frac{B_{3} a_{1}}{2 {(Ω_{2}^{2} - 1)}^{2}} (\frac{2 Ω_{2}^{2}}{Ω_{2}^{2} - 1} (B_{1} + B_{2} + \frac{B_{5}}{2} + \frac{B_{3} Ω_{2}^{2}}{Ω_{2}^{2} - 1} - B_{5}), \\ \frac{f_{2} \cos θ_{2}}{2} = a_{2} (σ_{2} + \frac{B_{3} Ω_{3}^{2}}{2 (Ω_{3}^{2} - ϖ^{2})} - \frac{B_{4}}{2}), \\ \frac{f_{2} \sin θ_{2}}{2} = \frac{B_{3} ϖ^{3}}{2 (Ω_{3}^{2} - ϖ^{2})} (B_{6} + \frac{Ω_{3}^{2}}{{(Ω_{3}^{2} - ϖ^{2})}^{2}} (B_{4} + B_{6} - \frac{B_{3} Ω_{3}^{2}}{Ω_{3}^{2} - ϖ^{2}})) a_{2}, \\ \frac{f_{3} \cos θ_{3}}{2} = a_{3} (σ_{3} + \frac{1}{2} (B_{5} + \frac{1}{1 - Ω_{2}^{2}} (B_{3} Ω_{2}^{2} - \frac{Ω_{2}^{2} (B_{1} + B_{2})}{2 (1 - Ω_{2}^{2})}) + (\frac{3 B_{3} Ω_{2}^{2}}{2 (1 - Ω_{2}^{2})} (1 - \frac{4 Ω_{2}^{2}}{3 (1 - Ω_{2}^{2})}) - \frac{B_{5}}{2}) (\frac{B_{3} Ω_{2}^{2}}{1 - Ω_{2}^{2}} + B_{5})), \frac{f_{3} \sin θ_{3}}{2} = \frac{a_{3} Ω_{2}}{2} (\frac{B_{3} Ω_{2}^{2}}{1 - Ω_{2}^{2}} + B_{5}) B_{3}), \\ \frac{f_{4} \cos θ_{4}}{2} = a_{4} (σ_{4} + \frac{1}{4} (\frac{B_{3} Ω_{3}^{2}}{ϖ^{2} - Ω_{3}^{2}} + B_{6}) (1 + 2 (\frac{B_{3} Ω_{3}^{2}}{ϖ^{2} - Ω_{3}^{2}} + B_{6}) + 2 (\frac{B_{3} Ω_{3}^{2}}{ϖ^{2} - Ω_{3}^{2}} (1 + \frac{2 Ω_{3}^{2}}{ϖ^{2} - Ω_{3}^{2}} + B_{6})), \\ \frac{f_{4} \sin θ_{4}}{2} = - \frac{Ω_{3} a_{4} B_{3} B_{6}}{4 (ϖ^{2} - Ω_{3}^{2})} (\frac{B_{3} Ω_{3}^{2}}{ϖ^{2} - Ω_{3}^{2}} + B_{6}) . \end{array}

(13)

The process of reducing this system to a more suitable form necessitates the elimination of the modified phases $θ_{j}$ . As a result, the desired new system, characterized by four nonlinear algebraic equations involving amplitudes $a_{j}$ and frequency responses $σ_{j}$ , may be expressed as below:

\begin{array}{l} \frac{f_{1}^{2}}{4} = a_{1}^{2} {{(σ_{1} + B_{1} + B_{2} + \frac{B_{3} Ω_{2}^{2}}{Ω_{2}^{2} - 1})}^{2} + ({\frac{B_{3}}{2 {(Ω_{2}^{2} - 1)}^{2}} (\frac{2 Ω_{2}^{2}}{Ω_{2}^{2} - 1} (B_{1} + B_{2} + \frac{B_{5}}{2} + \frac{B_{3} Ω_{2}^{2}}{Ω_{2}^{2} - 1} - B_{5}))}^{2}}, \\ \frac{f_{2}^{2}}{4} = a_{2}^{2} {{(σ_{2} + \frac{B_{3} Ω_{3}^{2}}{2 (Ω_{3}^{2} - ϖ^{2})} - \frac{B_{4}}{2})}^{2} + {(\frac{B_{3} ϖ^{3}}{2 (Ω_{3}^{2} - ϖ^{2})} (B_{6} + \frac{Ω_{3}^{2}}{{(Ω_{3}^{2} - ϖ^{2})}^{2}} (B_{4} + B_{6} - \frac{B_{3} Ω_{3}^{2}}{Ω_{3}^{2} - ϖ^{2}})))}^{2}}, \\ \frac{f_{3}^{2}}{4} = a_{3}^{2} {({σ_{3} + \frac{1}{2} (B_{5} + \frac{1}{1 - Ω_{2}^{2}} (B_{3} Ω_{2}^{2} - \frac{Ω_{2}^{2} (B_{1} + B_{2})}{2 (1 - Ω_{2}^{2})}) + (\frac{3 B_{3} Ω_{2}^{2}}{2 (1 - Ω_{2}^{2})} (1 - \frac{4 Ω_{2}^{2}}{3 (1 - Ω_{2}^{2})}) - \frac{B_{5}}{2}) (\frac{B_{3} Ω_{2}^{2}}{1 - Ω_{2}^{2}} + B_{5}))}^{2} + {(\frac{Ω_{2} B_{3}}{2} (\frac{B_{3} Ω_{2}^{2}}{1 - Ω_{2}^{2}} + B_{5}))}^{2}}, \\ \frac{f_{4}^{2}}{4} = a_{4}^{2} {(σ_{4} + \frac{1}{4} (\frac{B_{3} Ω_{3}^{2}}{ϖ^{2} - Ω_{3}^{2}} + B_{6}) (1 + 2 (\frac{B_{3} Ω_{3}^{2}}{ϖ^{2} - Ω_{3}^{2}} + B_{6}) + 2 {(\frac{B_{3} Ω_{3}^{2}}{ϖ^{2} - Ω_{3}^{2}} (1 + \frac{2 Ω_{3}^{2}}{ϖ^{2} - Ω_{3}^{2}} + B_{6}))}^{2} + {(\frac{Ω_{3} B_{3} B_{6}}{4 (ϖ^{2} - Ω_{3}^{2})} (\frac{B_{3} Ω_{3}^{2}}{ϖ^{2} - Ω_{3}^{2}} + B_{6}))}^{2}} . \end{array}

(14)

It’s crucial to highlight that a fundamental aspect of stability assessment entails investigating the situation of steady-state solutions.^24,25 To delve into the behavior near fixed point neighborhoods, we’ll insert the subsequent expressions into the previously mentioned system (10)

a_{j} = a_{j 0} + a_{j 1}, θ_{j} = θ_{j 0} + θ_{j 1} (j = 1, 2, . . ., 4),

where

a_{j 1}

represents the steady-state solutions of equation (13) and denotes the corresponding small perturbations. Hence, the linearized form of equation (10) takes the shape of

\begin{array}{l} a_{10} \frac{d θ_{11}}{d τ} = a_{11} (σ_{1} + B_{1} + B_{2} + \frac{B_{3} Ω_{2}^{2}}{Ω_{2}^{2} - 1}) + \frac{f_{1}}{2} θ_{11} \sin θ_{10}, \frac{d a_{11}}{d τ} = \frac{B_{3} a_{11}}{2 {(Ω_{2}^{2} - 1)}^{2}} (\frac{2 Ω_{2}^{2}}{Ω_{2}^{2} - 1} (B_{1} + B_{2} + \frac{B_{5}}{2} + \frac{B_{3} Ω_{2}^{2}}{Ω_{2}^{2} - 1}) - B_{5}) - \frac{f_{1}}{2} θ_{11} \cos θ_{10}, \\ a_{20} \frac{d θ_{2}}{d τ} = a_{21} (σ_{2} + \frac{B_{3} Ω_{3}^{2}}{2 (Ω_{3}^{2} - ϖ^{2})} - \frac{B_{4}}{2}) + \frac{f_{2}}{2} θ_{21} \sin θ_{20}, ϖ \frac{d a_{21}}{d τ} = \frac{B_{3} ϖ^{3}}{2 (Ω_{3}^{2} - ϖ^{2})} (B_{6} + \frac{Ω_{3}^{2}}{{(Ω_{3}^{2} - ϖ^{2})}^{2}} (B_{4} + B_{6} - \frac{B_{3} Ω_{3}^{2}}{Ω_{3}^{2} - ϖ^{2}})) a_{21} - \frac{f_{2}}{2} θ_{21} \cos θ_{20}, \\ a_{30} \frac{d θ_{3}}{d τ} = a_{31} (σ_{3} + \frac{1}{2} (B_{5} + \frac{1}{1 - Ω_{2}^{2}} (B_{3} Ω_{2}^{2} - \frac{Ω_{2}^{2} (B_{1} + B_{2})}{2 (1 - Ω_{2}^{2})}) + (\frac{3 B_{3} Ω_{2}^{2}}{2 (1 - Ω_{2}^{2})} (1 - \frac{4 Ω_{2}^{2}}{3 (1 - Ω_{2}^{2})}) - \frac{B_{5}}{2}) (\frac{B_{3} Ω_{2}^{2}}{1 - Ω_{2}^{2}} + B_{5})) + \frac{f_{3}}{2} θ_{31} \sin θ_{30}, Ω_{2} \frac{d a_{31}}{d τ} = \frac{a_{31} Ω_{2}}{2} (\frac{B_{3} Ω_{2}^{2}}{1 - Ω_{2}^{2}} + B_{5}) B_{3} + \frac{f_{3}}{2} θ_{31} \cos θ_{30}, \\ a_{40} \frac{d θ_{4}}{d τ} = a_{41} (σ_{4} + \frac{1}{4} (\frac{B_{3} Ω_{3}^{2}}{ϖ^{2} - Ω_{3}^{2}} + B_{6}) (1 + 2 (\frac{B_{3} Ω_{3}^{2}}{ϖ^{2} - Ω_{3}^{2}} + B_{6}) + 2 (\frac{B_{3} Ω_{3}^{2}}{ϖ^{2} - Ω_{3}^{2}} (1 + \frac{2 Ω_{3}^{2}}{ϖ^{2} - Ω_{3}^{2}} + B_{6})) + \frac{f_{4}}{2} θ_{41} \sin θ_{40} = 0, Ω_{3} \frac{d a_{4}}{d τ} = - a_{41} (\frac{B_{3} Ω_{3}^{2}}{ϖ^{2} - Ω_{3}^{2}} + B_{6}) \frac{Ω_{3} B_{3} B_{6}}{4 (ϖ^{2} - Ω_{3}^{2})} - \frac{f_{4}}{2} θ_{41} \cos θ_{40} . \end{array}

(15)

The solutions of system (15) can be attained by expressing $a_{j 1}$ and $θ_{j 1}$ exponentially, as in the forms $q_{s} e^{λ T}$ where $q_{s} (s = 1, 2, . . ., 8)$ , are invariants and $λ$ represents the unknown perturbations associated with their eigenvalues. The real parts of the roots of the characteristic equations should not be positive values if the steady-state solutions $a_{j 0}$ and $θ_{j 0}$ are stable asymptotically²⁶

λ^{8} + Γ_{1} λ^{7} + Γ_{2} λ^{6} + Γ_{3} λ^{5} + Γ_{4} λ^{4} + Γ_{5} λ^{3} + Γ_{6} λ^{2} + Γ_{7} λ + Γ_{8} = 0 .

(16)

Here,

Γ_{j}

denote functions dependent on the unperturbed parameters

a_{j 0}

and

θ_{j 0}

, that can be obtained easily.

The RHC offers the essential conditions required for steady-state solutions²⁷ (see Appendix 1). These criteria were utilized in this study as a well-established method for assessing the stability of linearized systems by analyzing the coefficients of the characteristic polynomial. It is particularly suitable for the examined system, as it identifies stability conditions efficiently without explicitly calculating eigenvalues.

Now, we explore the dynamical behavior using linear stability analysis of this system. In addition to deriving equations that capture the system’s nonlinear aspects, we investigate the necessary conditions for stability. Key factors, like frequencies $1, ω, Ω_{2}, Ω_{3},$ and detuning parameters $σ_{j}$ , are found to be essential in satisfying stability criteria.

A tailored approach with diverse the parameters has been employed to generate the stability plots of the system (10). The fluctuation amplitudes $a_{j}$ over time are graphed for various parametric regions in Figures 15–18, illustrating the effect of these $f_{1}, f_{2}, f_{3}, f_{4}, ω$ values on the potential fixed points. The regions of stable and unstable fixed points, given the current constraints of the chosen impacted parameter values, fall within the range $σ_{1} \leq 0$ and $0 < σ_{1}$ . Solid curves indicate the region of stable fixed points, while dashed lines represent the unstable ranges.

Figure 15.

Illustrates resonance curves corresponding to the planes $σ_{1} a_{j} (j = 1, 2, \dots, 4)$ at $f_{1} (= 0.0001, 0.0002, 0.0003), σ_{2} = 0.2, σ_{3} = 0.1$ and $σ_{4} = 0.3$ .

Figure 16.

Expresses resonance curves corresponding to the planes $σ_{1} a_{j} (j = 1, 2, \dots, 4)$ at $f_{2} (= 0.0002, 0.0004, 0.0006), σ_{2} = 0.2, σ_{3} = 0.1$ , and $σ_{4} = 0.3$ .

Figure 17.

Shows resonance curves corresponding to the planes $σ_{1} a_{j} (j = 1, 2, \dots, 4)$ at $f_{3} (= 0.006, 0.007, 0.008), σ_{2} = 0.2, σ_{3} = 0.1$ , and $σ_{4} = 0.3$ .

Figure 18.

Reveals resonance curves corresponding to the planes $σ_{1} a_{j} (j = 1, 2, \dots, 4)$ at $f_{4} (= 0.01, 0.0012, 0.013), σ_{2} = 0.2, σ_{3} = 0.1$ , and $σ_{4} = 0.3$ .

In Figures 15(a), 15(d), and 15(c), the three and two peaks fixed points are directed downwards, respectively. In contrast, in Figure 15(b), the three peaks are directed upwards. A closer examination of Figure 16 reveals that it is graphed when

(f_{2} = 0.0002, 0.0004, 0.0006),

and

(σ_{2} = 0.2, σ_{3} = 0.1, σ_{4} = 0.3)

. Multi peaks have been noted in the planes

σ_{1} a_{1}, σ_{1} a_{2}, σ_{1} a_{3},

and

σ_{1} a_{4}

. In Figure 17, there are three downward-directed fixed points in the

a_{1} σ_{1}

and

a_{4} σ_{1}

planes. Conversely, part (b) shows three upward-directed fixed points at

f_{3} (= 0.006, 0.007, 0.008)

. Two points are upward-directed at

f_{3} = 0.007, 0.008

, and the third is downward-directed at

f_{3} = 0.006

with three peaks, as illustrated in part (c) of Figure 17. Examining the plotted curves in Figure 18 reveals that they are depicted in the

a_{j} σ_{1}

planes. These curves represent the impact of the force’s amplitude

f_{4}

which has the values

(0.01, 0.012, 0.013)

. Table 2 shows the relation between the critical and peaks fixed points at different values of

f_{j}

Table 2.

Shows the critical and peaks fixed points when external forces $f_{j}$ have different values.

Figure	Peaks points	Critical peak points	$f_{j} (j = 1, 2, 3)$
Figure 15(a)	(±0.00939742, 0)	(0, −1.04815)	$f_{1} (= 0.0001, 0.0002, 0.0003)$
	(±0.00965491, 0)	(0, −0.492404)
	(±0.00939742, 0.003520)	(0, −0.120031)
Figure 15(b)	(±0.0104049, 0.0018214)	(0, 0.00191013)
	(±0.0100676, 0.0018215)	(0, 0.00184549)
	(±0.0104049, 0.0018212)	(0, 0.00191045)
Figure 15(c)	(±0.0099619, 0.0035024)	(0, −0.001016551)
Figure 15(c)	(±0.0102613, 0.0035)	(0, 0.000400826)
Figure 15(d)	(±0.0106089, 606886 × 10⁻⁶)	(0, −1.89233 × 10⁻⁶)
	(±0.00995585, 6.79446 × 10⁻⁶)	(0, 1.5204 × 10⁻⁶)
	(±0.0107738, 6.73899 × 10⁻⁶)	(0, 3.9401 × 10⁻⁶)
Figure 16(a)	(±0.0101081, 0)	(0, −3.11299)	$f_{2} (= 0.0002, 0.0004, 0.0006)$
	(±0.0101083, 0)	(0, −3.13948)
	(±0.00965491, 0)	(0, −3.14342)
Figure 16(b)	(±0.0105593, 0.001821)	(0, 0.0182996)
	(±0.0105593, 0.0018206)	(0, 0.00183901)
	(±0.0100676,0.0018215)	(0, 0.00184549)
Figure 16(c)	(±0.0102613, 0.0035)	(0, 0.000518295)
	(±0.0091662, 0.0034735)	(0, −0.00369101)
	(±0.0099661, 0.0035024)	(0, −0.00477795)
Figure 16(d)	(±0.00924156, 0.66489 × 10⁻⁶)	(0, −2.00672 × 10⁻⁶)
	(±0.0100759, 6.72128 × 10⁻⁶)	(0, −2.4649 × 10⁻⁶)
	(±0.00995585, 6.79446 × 10⁻⁶)	(0, −1.89233 × 10⁻⁶)
Figure 17(a)	(±0.00965491, 0)	(0, −3.13948)	$f_{3} (= 0.006, 0.007, 0.008)$
	(±0.0102665, 0)	(0, −3.53897)
	(±0.0101081, 0)	(0, −4.17398)
Figure 17(b)	(±0.0100676, 0.0018211)	(0, 0.00184549)
	(±0.0097591, 0.0018995)	(0, 0.0019127)
	(±0.00988671, 0.001829)	(0, 0.00182644)
Figure 17(c)	(±0.0099619, 0.0030246)	(0, 0.00477795)
	(±0.00961662, −0.17)	(0, −0.012406)
	(±0.0103782, 0)	(0, −0.00521694)
Figure 17(d)	(±0.00995585, 6.79446 × 10⁻⁶)	(0, −1.89233 × 10⁻⁶)
	(±0.0106689, 7.8935 × 10⁻⁶)	(0, −1.93448 × 10⁻⁶)
	(±0.00936017, 0)	(0, −1.90575 × 10⁻⁶)
Figure 18(a)	(±0.00965491, 0)	(0, −3.13948)	$f_{4} (= 0.01, 0.012, 0.013)$
	(±0.00958971, 0)	(0, −3.58176)
	(±0.0101081, 0)	(0, −3.73194)
Figure 18(b)	(±0.0100676, 0.0018215)	(0, 0.00184549)
	(±0.0105593, 0.0019347)	(0, 0.00192595)
	(±0.0105593, 0.0018950)	(0, 0.00192301)
Figure 18(c)	(±0.0099619, 0.0035026)	(0, 0.00477795)
	(±0.0099612, −0.008967)	(0, −0.015458)
	(±0.0103782, −0.004719)	(0, −0.014659)
Figure 18(d)	(±0.00995585, 6.79446 × 10⁻⁶)	(0, −1.89233 × 10⁻⁶)
	(±0.00924156, −8.77667 × 10⁻⁶)	(0, −2.40032 × 10⁻⁶)
	(±0.0108, −0.00001595)	(0, 5.13588 × 10⁻⁶)

Applications of the examined system

The examined system depicts a mechanical model with two masses, interconnected via springs and dampers, with external several forces acting on them. Therefore, it represents a coupled multi MDOF system. The significant applications of such a model include the following:

• The design could resemble components of advanced suspension systems used in vehicles to absorb shocks and isolate vibrations from road irregularities.

• The system might model vibration isolation in structures subjected to environmental forces like earthquakes or wind.

• Buildings, particularly skyscrapers, often use similar tuned mass damper systems to mitigate oscillations.

• The system may provide insights into seismic base isolators used in critical infrastructure to reduce earthquake-induced forces transmitted to a structure.

• The multi-degree-of-freedom design (with internal and external components) could model vibration control systems in spacecraft or aircraft, where managing forces from engines or external disturbances is critical.

• Using the system to model and mitigate vibrations via active or passive damping mechanisms.

• Modeling base isolators for buildings or equipment to protect against seismic shocks.

• Investigating coupled dynamic modes in spacecraft or aircraft structures.

• Studying systems that convert mechanical energy from vibrations into electrical energy.

Conclusion

The dynamics of a multi-DoF system are examined in this paper. The nonlinear DEs governing the system were derived using LE and solved numerically with the 4-RKM. The phase plane analysis, exploring the connections between angular solutions and their corresponding first-order derivatives, has been examined. The analysis revealed three motion types in the system: periodic, quasi-periodic, and chaotic, which were explored using bifurcation diagrams, Poincaré maps, and Lyapunov spectra. The stability and instability regions of the fixed points in steady-state conditions were evaluated using the RHC. A nonlinear stability analysis was conducted to examine the steady-state behavior of the system under investigation. Addressing the practical implications of vibrational control across multiple engineering disciplines, this research holds significant scientific and application-orientated importance. The described system offers a practical model for designing vibration isolation platforms. The outer hollow block symbolizes the supporting structure, while the oscillating mass simulates an isolator precisely tuned to absorb and reduce external vibrations impacting sensitive equipment.

Footnotes

Author contributions

T. S. Amer: Resources, Formal analysis, Conceptualization, Supervision, Reviewing, Visualization, Reviewing.

Galal M. Moatimid: Methodology, Conceptualization, Methodology, Data curation, Supervision, Validation.

S. K. Zakria: Visualization, Conceptualization, Investigation, Writing-Original draft preparation.

A. A. Galal: Supervision, Resources, Formal analysis, Conceptualization, Methodology, Visualization, Reviewing and Editing.

Declaration of conflicting interests

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

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

TS Amer

AA Galal

Data availability statement

Data sharing is not applicable to this research as no datasets were created or analyzed.*

Appendix

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Dynamical analysis of a four-degree-of-freedom vibratory structure: Bifurcation,stability,and resonance exploration

Abstract

Keywords

Introduction

Overview of the problem

Numerical analysis

Chaotic motion

Analysis of system stability

Applications of the examined system

Conclusion

Footnotes

Author contributions

Declaration of conflicting interests

Funding

ORCID iDs

Data availability statement

Appendix

References