Inspection of a mix control technique for vibration control in a spring-coupled double pendulum

Abstract

This paper evaluates the perspective of a spring-coupled double pendulum with strict vibration control applied using a positive position negative derivative feedback (PPNDF) controller. The approach used in this case involved applying one of the most well-known perturbation techniques to discover the approximate mathematical solution of the equations of motion of the spring-coupled double pendulum with PPNDF damper. Our approach to obtaining the analytical results was to use the multiple-scale perturbation technique (MSPT) of the first-order approximation. Through the use of frequency response equations close to the simultaneous primary and internal resonance scenarios, the stability of the system has been studied and monitored. The MATLAB and Maple software were used to complete and clarify all the numerical discussions. It was very interesting and important to compare the effects of negative derivative feedback (NDF) only, positive position feedback (PPF) only, and PPNDF simply on the framework using the phase plane and time history. Moreover, a comparison was conducted between the numerical and analytical solutions. Ultimately, the study of the frequency response curves in three dimensions revealed the stable and unstable regions in each picture, illustrating the impact of altering the parameter values on the amplitude.

Keywords

PPNDF control nonlinear systems dynamical vibration spring-coupled double pendulum stability multiple scale perturbation technique

Introduction

The Gamma Function Method (GFM) is effectively used to solve three real-world nonlinear differential equation problems analytically. These examples show that, in comparison to other known analytical approaches, the analytical approximation answers agree better with the associated numerical solutions. Based on the results obtained, the authors decided that the GF method is, in principle, very simple, applicable, and efficient in finding analytical approximate solutions to different types of nonlinear differential equations, and it produces highly accurate numerical solutions to nonlinear problems in comparison to other methods in the field of analysis.¹ These same applications are studied using an innovative technique, which just converts the nonlinear ordinary differential equation (ODE) into a linear one, known as the non-perturbative methodology (NPM). The method yields a new frequency that is equivalent to the linear ODE, which may be used to generate a new damping term. For the benefit of the reader, a comprehensive explanation of the NPM is provided.² A description of the positive position feedback PPF control of the nonlinear giant magnetostrictive actuator GMA framework may be found in Ref. 3. By adjusting the PPF restrictions, it is possible to confine the fundamental resonance and framework amplitude stability successfully. Just a handful of experiments have been conducted to confirm the accuracy of the results. The analysis showed how the structure was developed and compared its utility to other studies. The 3D plot enhances and demonstrates the accuracy of the job. To control the nonlinear vibration of the primary resonance vibration of a spinning shaft, Ref. 4 recommends the use of nonlinear integral positive position feedback (NIPPF) controllers. A comparison of the controllers for the two frequency response curves (FRCs) under investigation, NIPPF and adaptive nonlinear integral positive position (ANIPPF), is one of these. EL-Sayed⁵ introduced the delay positive position feedback control (DPPF) to lessen vibrations in the Van der Pol oscillator system with external forces. A few factors are examined in order to assess the effectiveness of vibration suppression using DPPF and establish the stability condition of the model. A negative derivative feedback NDF controller was just developed by Cazzulani et al.⁶ and was constructed on a resonant control strategy. A decentralized application of an ideal NDF controller has been devised.⁷ Overall, these findings show how effective and high-power the NDF controller is at reducing high modal density vibration in bladed structures at higher frequencies. They additionally gave⁸ a simple process for the H2-optimization approach and maximum damping in the optimal design of a negative derivative feedback NDF controller. By stopping control activity far from the natural frequencies connected to the controlled modes, NDF functions as a controller that acts as a band-pass filter, minimizing the effect of spillover. Because it is a bandpass filter, disturbances at lower or higher frequencies can be successfully controlled. Furthermore, it can be applied to high-performing vibration mitigation applications. To determine which controller works best in the vibration system (quarter-vehicle car), a comparison between PPF and NDF controllers has been made in Ref. 9, and it was found that NDF is better than PPF.

A piezoelectric functionally graded laminated composite cantilever rectangular plate supported with graphene experiences nonlinear vibrations with four degrees of freedom when controlled by a PPF control scheme. Furthermore, the outcomes disclose the energy transformation that occurs between the PPF controller and the host system. It is anticipated that this Ref. 10 would offer theoretical direction for nonlinear large amplitude vibration reduction of structures reinforced with graphene. Moreover, in Ref. 11, the Multiple-Input Multiple-Output (MIMO) Positive Position Feedback PPF controller is tuned using the Particle Swarm Optimization (PSO) algorithm to lessen vibrations on a composite sandwich plate with free boundary conditions, which has been outfitted with piezoelectric actuators and sensors. A newly developed piezoelectric vibration control method for wind turbines is put forth. The tower vibrations are sensed, and then a control signal is applied to the piezoelectric transducers, which operate as actuators, via an internally established Positive Position Feedback PPF controller. This creates a closed-loop control system.¹² In Ref. 13, they used the PPF to reduce the MEMS’s vibration. They examined the vibrating system’s worst resonance condition and noted how various factors affected both the primary system and the PPF. The macro-fiber composite’s vibration was managed by Bauomy and El-Sayed¹⁴ by connecting a nonlinear proportional-derivative NDF. When using this damper, they were very successful. Additionally, they¹⁵ took into account the nonlinear dynamic vibrations of a composite plate that was subjected to parametric and external excitations and had square and cubic nonlinear components. Three PPF controllers are used to control this system, which results in a new six-degree-of-freedom model. Some of the most important crack indicators remain the same, but when nonlinear elements are present, the vibration response changes in the form of a changed frequency composition, a rising trend of higher circular harmonics, or a rising level of subharmonic resonances. These vibration-based techniques were used to a variety of engineering projects, such as block assemblies, trusses, beams, and rotors.^16–22

The nonlinear cantilever beam’s vibrations were lessened by nonlinear saturation controller (NSC) simulation.²³ To obtain the mathematical solution of the equations for dynamical modeling using NSC, a perturbation method treatment is established. The dynamical system’s approximate solution is found using the perturbation strategy. To demonstrate the degree of relationship between the perturbation and numerical solutions, validation curves are included. In Ref. 24, a single-link flexible manipulator with a piezoelectric actuator was utilized to apply the PPF and NDF controller approaches. The NDF controller is generally more effective than the PPF controller at suppressing vibrations, according to the comparison of the two controllers based on the system under study. Ref. 25 suggests that the Nonlinear Saturation Controller (NSC) be used to lessen the spring pendulum’s oscillations. Using the frequency response equations, they examine the system’s stability near the resonance state. The impacts of differentiable controller parameters on the fundamental behavior of the system are investigated numerically. The authors of²⁶ carried out stability analysis and provided an explanation of the closed-loop system’s responses for the nonlinear integral positive position feedback NIPPF, integral resonant controllers (IRC), and positive position feedback PPF controllers. However, utilizing the PPF controller, the authors of Ref. 27 investigated how to reduce the mechanical oscillations of the galloping system. The findings demonstrate the effectiveness of the PPF controller in reducing the D-shaped prism’s galloping amplitude. A PD controller and a time-delayed PD controller were proposed by Eissa et al.^28,29 to lessen vibrations in magnetic systems with cubic and quadratic nonlinear coefficients under parametric principal forces.

The primary objective of the current work is to suggest using the PPNDF controller as an acceptable control mechanism for removing growing oscillation amplitudes of the nonlinear dynamical system that was picked from applications in Ref. 1, which is known as a spring-coupled double pendulum. When we applied the multiple scale perturbation technique MSPT, Nayfeh^30–33 introduced perturbation strategies to handle the nonlinear differential equations, which were very valuable to us. In order to provide precise mathematical solutions, frequency response equations (FREs), and stability tests using PPNDF controllers, the MSPT was put into use. The results of the conducted analytical and numerical simulations indicate that, regardless of the excitation frequency and amount, the PPNDF controller can almost completely suppress the amplitude of the nonlinear vibration system. A comparison is made between the perturbation analysis and numerical results through the time history graph. Additionally, this comparison yields a very good agreement result. The system was studied both with PPF, NDF, and PPNDF controllers and without any of them being delivered numerically and graphically in the primary and 1:1 internal resonance case. A MATLAB program was used to simulate how various settings and the controller may impact the system. After making a comparison, we have finally concluded that the PPNDF controller offers the finest system control.

In light of the investigated resonance scenario, the equations of frequency response were employed to examine the solutions at the steady-state case. The requirements of stability of the solutions were determined. The planar movement of a two-degree-of-freedom auto-parametric model, consisting of a movable attached main mass through a damped spring coupled to a simple pendulum of a rigid arm as a secondary part, was discussed in Ref. 34. In Ref. 35, the authors replaced the pendulum rigid arm with a spring pendulum with linear stiffness to constitute a three-degree-of-freedom model and generalize the work in Ref. 34. In Ref. 36, the researchers investigated how a viscous moment at the turning point and two harmonic excitation forces affected the movement of a pendulum system. Furthermore, in Ref. 37, the authors examined how a spring pendulum coupled to a more basic one damped motion. Besides, in Ref. 38, the vibrating dynamical motion of a three-degree-of-freedom auto-parametric system made up of a damped spring pendulum and a damped Duffing oscillator is investigated. In addition, the study looks into nonlinear stability and analyzes resonance cases, time histories, resonance curves, and stability zones. Bahnasy et al.³⁹ analyzed chaotic dynamics of quasi-zero stiffness vibration isolators using multiple control methods, emphasizing stability under strong nonlinearities. Reference 40 examines the behavior of a mechanical system with a lumped-mass comprising two nonlinear springs arranged in series and combined with a piezoelectric device. External harmonic excitations, as well as linear and nonlinear damping, are considered. The main system employs a negative velocity feedback controller to reduce undesired effects vibrations, particularly under resonance conditions, thereby enhancing the system’s efficiency. Then, Ref. 41 discusses a dynamic model of a rolling ship subjected to a parametric force. Particularly in the resonance zone, the nonlinear derivative feedback control is employed to increase stability and reduce undesired vibrations. Reference 42 examines a Mathieu oscillator subjected to parametric forcing, and it is stabilized by applying negative derivative feedback control under the most severe resonance conditions.

Nonlinear mechanical vibrations and their control remain a central topic in applied mechanics, with renewed interest in both advanced analytical methods and high-fidelity numerical modeling for realistic, controlled systems. Traditional perturbation methods (e.g., multiple scales, averaging) have long been employed to analyze weakly nonlinear systems, but they become less reliable when the nonlinearity is strong, singular, discontinuous, or quasi-linear. To address these challenges, He et al.⁴³ developed a modified frequency formulation that provides closed-form approximations for a class of nonlinear oscillators containing quasi-linear and even singular nonlinearities. Their method avoids cumbersome expansions and instead directly yields frequency–amplitude relationships with remarkable simplicity and good accuracy, as demonstrated by illustrative examples. Complementing this, Amer et al.⁴⁴ examine a forced spring–pendulum system, a canonical nonlinear oscillator via mathematical bifurcation analysis and numerical simulations, identifying routes to chaos and parameter regimes with complex forced responses (e.g., internal resonances). Their work provides a practical benchmark for testing control strategies, especially those targeting chaotic or near-resonant behavior, and reinforces the need for analytical insights that go beyond linearization around small-amplitude motion. Much existing control literature focuses on systems with relatively few degrees of freedom (DOFs), such as single- or two-oscillator models. Techniques like feedback linearization, sliding-mode control, adaptive control, and energy-based methods have been widely applied. These approaches are well-suited to academic benchmark problems (Duffing, Van der Pol, pendulum-like systems), but they often do not translate directly to more complex, spatially distributed structures. One limitation of low-DOF control research is that it assumes simplified models that neglect geometric complexity, distributed actuation, multimode coupling, and non-ideal boundary conditions. Consequently, researchers have called for bridging the gap between control theory and realistic structural modeling. To translate controller design from simple models into real-world applications, high-fidelity finite element models play a critical role. Over the years, finite element formulations have advanced to include geometric nonlinearity, electromechanical coupling, and thin-walled structures. In particular, Milić et al.⁴⁵ presented an isogeometric finite element formulation for laminated shells with embedded piezoelectric layers, based on Reissner–Mindlin kinematics. They tackle challenges such as defining control-point normals (via Greville’s points) and coupling the in-plane strains with the electric field across piezo layers. Further, in the broader finite element context, other researchers have explored nonlinear piezoelectric shell models and FE formulations that capture fully coupled electro-mechanical behavior, extending to large deformation, post-buckling, and distributed actuation/sensing. For example, Vetyukov & Belyaev⁴⁶ studied nonlinear shell behavior under electromechanical coupling in a shell-as-material-surface framework. Bifurcation analysis is crucial for researching dynamical models because it illustrates how system behavior changes when parameters change, making it possible to find stability thresholds and important transitions. Researchers can use bifurcations to detect transitions from stable states to quasiperiodic, periodic, or chaotic regimes in order to understand complicated dynamics in fields such as engineering, physics, and biology. By giving a detailed account of how a system responds to external incentives, this study sheds light on both expected and impulsive behaviors.^47,48

This study focuses on analyzing the three-degree-of-freedom spring-coupled double pendulum equipped with a PPNDF controller. The fundamental equations of motion are formulated using Lagrange’s method based on the selected generalized coordinates. The PPNDF control strategy for vibration suppression is introduced and evaluated. Using the multiple-scale perturbation technique (MSPT), first-order approximate solutions are derived. The stability of the steady-state response is then assessed near the specified resonance condition, and the corresponding frequency-response curves are determined. Numerical simulations are carried out to illustrate the phase portraits and Poincaré sections of the controlled system. Additionally, the influence of key parameters on the system’s response is examined. A bifurcation analysis is finally performed to highlight the behavior of the system’s critical states.

Spring-coupled double pendulum

Dynamics of a system without control

The system in Figure 1 is a spring-coupled double pendulum,¹ also known as a double pendulum with a spring. It consists of two masses, $m_{1}$ and $m_{2}$ (in kilograms kg.), connected by a spring with stiffness $k$ (newton per meter N/m.) and a rigid rod of length $L$ (in meters m.). The system’s motion is described by equation (1) shown below, which takes into account the horizontal displacement $x (t)$ (in meters m) and the effects of the spring and gravitational forces. It is studied in fields like mechanical engineering and physics to understand complex vibrational behavior and resonance in systems with multiple degrees of freedom. Applications include analyzing the dynamic response of structures such as bridges or buildings to external forces like wind or earthquakes. This type of model helps predict how these structures will behave under different conditions, aiding in safer and more efficient design.

Figure 1.

Spring-coupled double pendulum¹.

The motion of this system is subjected to the following equation [1]

(m_{1} + \frac{m_{2} x^{2}}{L^{2} - x^{2}}) \ddot{x} + \frac{m_{2} L^{2}}{{(L^{2} - x^{2})}^{2}} x {\dot{x}}^{2} + \frac{m_{2} g}{\sqrt{L^{2} - x^{2}}} x + k x = 0

(1)

The derivation of the equation of motion can be presented in the appendix.

Equation (1) could be turned into the differential equation of motion after deducing and introducing a new parameter $u = \frac{x}{L}, | u | ≺ ≺ 1$

(1 + \hat{h} \frac{u^{2}}{1 - u^{2}}) \ddot{u} + \frac{\hat{h}}{{(1 - u^{2})}^{2}} u {\dot{u}}^{2} + \frac{\hat{h} g}{L \sqrt{1 - u^{2}}} u + \frac{k}{m_{1}} u = 0

(2)

where

\hat{h} = \frac{m_{2}}{m_{1}}

. To derive equation (3) from equation (2), we assume small-amplitude motion such that

u^{2} ≪ 1

. This physically reasonable assumption allows the inverse powers of

(1 - u^{2})

and

\sqrt{1 - u^{2}}

to be approximated using negative-power binomial expansions:

{(1 - u^{2})}^{- 1} \approx 1 + u^{2}, {(1 - u^{2})}^{- 2} \approx 1 + 2 u^{2}, {(1 - u^{2})}^{- \frac{1}{2}} \approx 1 + \frac{1}{2} u^{2}

By retaining terms up to $O (u^{2})$ and neglecting higher-order contributions, equation (2) reduces to a weakly nonlinear form. After collecting the linear and cubic restoring-force contributions, the parameters $ω_{0}^{2} = \frac{\hat{h} g}{L} + \frac{k}{m_{1}}, \hat{ζ} = \frac{\hat{h} g}{2 L}$ are introduced, leading directly to the simplified expression of the equation of motion presented in equation (3). This approximation preserves the essential physics of the system while yielding an analytically tractable model

(1 + \hat{h} u^{2} (1 + u^{2})) \ddot{u} + \hat{h} (1 + 2 u^{2}) u {\dot{u}}^{2} + ω_{0}^{2} u + \hat{ζ} u^{3} = 0

(3)

Applying an external force on the system then turned the equation of motion into non-dimensional form

(1 + ε h u^{2} (1 + u^{2})) \ddot{u} + ε h (1 + 2 u^{2}) u {\dot{u}}^{2} + ω_{0}^{2} u + ε ζ u^{3} = ε f \cos (Ω t)

(4)

Knowing that $\hat{h} = ε h$ , $\hat{ζ} = ε ζ$

Modeling assumption and justification

It is important to note that although continuum descriptions may involve PDE representations, the considered system is a rigid-body spring-coupled double pendulum. The pendulum arms and coupling spring behave as lumped elements with negligible distributed elasticity, making an ODE formulation the standard and widely accepted approach in multibody dynamics. Therefore, the adopted ODE model accurately captures the essential nonlinear behaviors required for the subsequent bifurcation and chaos analysis.

Spring-coupled double pendulum with PPNDF control

The current problem discusses the effects of PPNDF on a spring-coupled double pendulum subjected to an external excitation force. In this study, we updated the nonlinear differential equations of the spring-coupled double pendulum in equation (4) after inserting the positive position, negative derivative feedback control represented as a three-degrees-of-freedom model as illustrated in Figure 2

(1 + ε h u^{2} (1 + u^{2})) \ddot{u} + ε h (1 + 2 u^{2}) u {\dot{u}}^{2} + ω_{0}^{2} u + ε ζ u^{3} = ε f \cos (Ω t) + ε λ_{1} p + ε G_{1} \dot{n}

(5a)

\ddot{p} + ω_{1}^{2} p + ε μ_{1} \dot{p} = ε λ_{2} u

(5b)

\ddot{n} + ω_{2}^{2} n + ε μ_{2} \dot{n} = - ε G_{2} \dot{u}

(5c)

Figure 2.

A block diagram of a spring-coupled double pendulum with PPNDF control.

Perturbation analysis with MSPT

Before presenting the multiple–scale perturbation analysis, it is important to clarify its contribution to the control-oriented understanding of the system. The perturbation framework is not used merely as a mathematical exercise; rather, it provides direct physical insight into how the PPNDF controller modifies the intrinsic nonlinear behavior of the coupled-pendulum system. In particular, the slow–flow equations obtained through MSPT reveal the controller’s influence on the effective damping, the amplitude–frequency backbone, and the stability of periodic motions. These analytical relations explain why specific controller gains improve vibration suppression and shift the resonance characteristics, thereby guiding the controller tuning strategy and complementing the numerical results presented later.

Because equations (5a)–(5c) can be simulated, we may select the system that results from turning on control PPNDF and analyzing it analytically using the multiple scale approach^30–33 to obtain the approximate first-order perturbation solutions reported below

u (t, ε) = u_{0} (T_{0}, T_{1}) + ε u_{1} (T_{0}, T_{1}) + O (ε^{2})

(6a)

p (t, ε) = p_{0} (T_{0}, T_{1}) + ε p_{1} (T_{0}, T_{1}) + O (ε^{2})

(6b)

n (t, ε) = n_{0} (T_{0}, T_{1}) + ε n_{1} (T_{0}, T_{1}) + O (ε^{2})

(6c)

Such that: $T_{0} = t, T_{1} = ε t$ are time scales of fast-varying and slowly varying, respectively. The relations between the differential operators are defined as follows

\frac{d}{d t} = D_{0} + ε D_{1}, \frac{d^{2}}{d t^{2}} = D_{0}^{2} + ε (2 D_{0} D_{1})

(7)

where

D_{0} = \partial / \partial T_{0}, D_{1} = \partial / \partial T_{1}

Substituting equations (6a)–(6c) into equations (5a)–(5c) with using the relations (7), the following equations are obtained after equating the coefficients of $ε^{0}$ and $ε^{1}$ :

Order $ε^{0}$

(D_{0}^{2} + ω_{0}^{2}) u_{0} = 0

(8a)

(D_{0}^{2} + ω_{1}^{2}) p_{0} = 0

(8b)

(D_{0}^{2} + ω_{2}^{2}) n_{0} = 0

(8c)

Order $ε^{1}$

\begin{array}{l} (D_{0}^{2} + ω_{0}^{2}) u_{1} = - 2 D_{0} D_{1} u_{0} - h u_{0}^{2} D_{0}^{2} u_{0} - h u_{0}^{4} D_{0}^{2} u_{0} - h u_{0} {(D_{0} u_{0})}^{2} - 2 h u_{0}^{3} {(D_{0} u_{0})}^{2} - ζ u_{0}^{3} \\ + f \cos (Ω t) + λ_{1} p_{0} - G_{1} D_{0} n_{0} \end{array}}

(9a)

(D_{0}^{2} + ω_{1}^{2}) p_{1} = - 2 D_{0} D_{1} p_{0} - μ_{1} D_{0} p_{0} + λ_{2} u_{0}

(9b)

(D_{0}^{2} + ω_{2}^{2}) n_{1} = - 2 D_{0} D_{1} n_{0} - μ_{2} D_{0} n_{0} - G_{2} D_{0} u_{0}

(9c)

The general solution of equations (5a)–(5c) can be expressed in the following forms

u_{0} = A_{1} (T_{1}) e^{i ω_{0} T_{0}} + {\bar{A}}_{1} (T_{1}) e^{- i ω_{0} T_{0}}

(10a)

p_{0} = A_{2} (T_{1}) e^{i ω_{1} T_{0}} + {\bar{A}}_{2} (T_{1}) e^{- i ω_{1} T_{0}}

(10b)

n_{0} = A_{3} (T_{1}) e^{i ω_{2} T_{0}} + {\bar{A}}_{3} (T_{1}) e^{- i ω_{2} T_{0}}

(10c)

where

A_{m}

(

m = 1, 2, 3

) are the complex conjugate functions of the previous formulations, respectively, and are complex functions in

T_{1}

. We determined the values for the quantities by eliminating the secular expressions and small divisors from the previous approximation and substituting equations (10a)–(10c) into equations (9a)–(9c)

\begin{array}{l} (D_{0}^{2} + ω_{0}^{2}) u_{1} = [- 2 i ω_{0} D_{1} A_{1} + 6 h ω_{0}^{2} A_{1}^{3} {\bar{A}}_{1}^{2} + 2 h ω_{0}^{2} A_{1}^{2} {\bar{A}}_{1} - 3 ζ A_{1}^{2} {\bar{A}}_{1}] e^{i ω_{0} T_{0}} \\ + [2 h ω_{0}^{2} A_{1}^{3} + 7 h ω_{0}^{2} A_{1}^{4} {\bar{A}}_{1} - ζ A_{1}^{3}] e^{3 i ω_{0} T_{0}} + [3 h ω_{0}^{2} A_{1}^{5}] e^{5 i ω_{0} T_{0}} \\ + [\frac{f}{2}] e^{i Ω T_{0}} + [λ_{1} A_{2}] e^{i ω_{1} T_{0}} + [i ω_{2} G_{1} A_{3}] e^{i ω_{2} T_{0}} + c c . \end{array}}

(11a)

(D_{0}^{2} + ω_{1}^{2}) p_{1} = [- 2 i ω_{1} D_{1} A_{2} - μ_{1} i ω_{1} A_{2}] e^{i ω_{1} T_{0}} + [λ_{2} A_{1}] e^{i ω_{0} T_{0}} + c c .

(11b)

(D_{0}^{2} + ω_{2}^{2}) n_{1} = - 2 D_{0} D_{1} n_{0} - μ_{2} D_{0} n_{0} - G_{2} D_{0} u_{0}

(11c)

where cc. represents the complex conjugate terms, we have.

By converting equations (10a)–(10c) into equations (11a)–(11c) and solving the ensuing differential equations, we may obtain the particular solution for equations (5a)–(5c) in the following forms

u_{1} = [\frac{- 2 h A_{1}^{3} - 7 h A_{1}^{4} {\bar{A}}_{1} + ζ A_{1}^{3}}{8}] e^{3 i ω_{0} T_{0}} + [\frac{- 3 h A_{1}^{5}}{24}] e^{5 i ω_{0} T_{0}} + c c .

(12a)

p_{1} = 0

(12b)

n_{1} = 0

(12c)

According to our estimation, the worst possible resonance scenario is the simultaneous resonances $ω_{2} ≅ ω_{1}$ . Furthermore, we will carefully and numerically search for the stability of the computed resonance in the section that follows.

Regular solution

Studying the system stability at the simultaneous resonance case: $Ω ≅ ω_{0} ≅ ω_{1} ≅ ω_{2}$ , and define the detuning parameters ${\hat{σ}}_{1}, {\hat{σ}}_{2}, {\hat{σ}}_{3}$ as follows

Ω = ω_{0} + {\hat{σ}}_{1} = ω_{0} + ε σ_{1}, ω_{1} = ω_{0} + {\hat{σ}}_{2} = ω_{0} + ε σ_{2}, ω_{2} = ω_{0} + {\hat{σ}}_{3} = ω_{0} + ε σ_{3}

(13)

When we apply equation (13) to secular and slight-divisor terms and start with the first method of reaching solvability conditions in equations (11a)–(11c), we obtain

\begin{array}{l} D_{1} A_{1} = \frac{- i}{2 ω_{0}} [6 h ω_{0}^{2} A_{1}^{3} {\bar{A}}_{1}^{2} + 2 h ω_{0}^{2} A_{1}^{2} {\bar{A}}_{1} - 3 ζ A_{1}^{2} {\bar{A}}_{1}] - [\frac{i f}{4 ω_{0}}] e^{i σ_{1} T_{1}} + [\frac{- i λ_{1}}{2 ω_{0}} A_{2}] e^{i σ_{2} T_{1}} \\ + [\frac{ω_{2} G_{1}}{2 ω_{0}} A_{3}] e^{i σ_{3} T_{1}} \end{array}}

(14a)

D_{1} A_{2} = [- \frac{μ_{1}}{2} A_{2}] + [\frac{- i λ_{2}}{2 ω_{1}} A_{1}] e^{- i σ_{2} T_{1}}

(14b)

D_{1} A_{3} = [- \frac{μ_{2}}{2} A_{3}] + [- \frac{G_{2} ω_{0}}{2 ω_{2}} A_{1}] e^{- i σ_{3} T_{1}}

(14c)

It is convenient to express $A_{1}, A_{2}, A_{3}$ in the polar form

A_{i} = \frac{1}{2} a_{i} (T_{1}) e^{i γ_{i} (T_{1})}; i = 1, 2, 3

(15)

where the steady-state motion amplitude and phases are represented, respectively, by

a_{1}

and

γ_{i}

. Equation (15) are embedded inside equations (14a)–(14c), equating the real and imaginary parts

{\dot{a}}_{1} = [\frac{f}{2 ω_{0}}] \sin (θ_{1}) + [\frac{λ_{1}}{2 ω_{0}} a_{2}] \sin (θ_{2}) + [\frac{ω_{2} G_{1}}{2 ω_{0}} a_{3}] \cos (θ_{3})

(16a)

\begin{array}{l} {\dot{γ}}_{1} a_{1} = [\frac{- 3 h ω_{0}}{16} a_{1}^{5} - (\frac{2 h ω_{0}^{2} - 3 ζ}{8 ω_{0}}) a_{1}^{3}] + [- \frac{f}{2 ω_{0}}] \cos (θ_{1}) + [- \frac{λ_{1}}{2 ω_{0}} a_{2}] \cos (θ_{2}) \\ + [\frac{ω_{2} G_{1}}{2 ω_{0}} a_{3}] \sin (θ_{3}) \end{array}}

(16b)

{\dot{a}}_{2} = [- \frac{μ_{1}}{2} a_{2}] + [- \frac{λ_{2}}{2 ω_{1}} a_{1}] \sin (θ_{2})

(17a)

{\dot{γ}}_{2} a_{2} = [- \frac{λ_{2}}{2 ω_{1}} a_{1}] \cos (θ_{2})

(17b)

{\dot{a}}_{3} = [- \frac{μ_{2}}{2} a_{3}] + [\frac{- G_{2} ω_{0}}{2 ω_{2}} a_{1}] \cos (θ_{3})

(18a)

{\dot{γ}}_{3} a_{3} = [\frac{G_{2} ω_{0}}{2 ω_{2}} a_{1}] \sin (θ_{3})

(18b)

where

θ_{1} = σ_{1} T_{1} - γ_{1}, θ_{2} = σ_{2} T_{1} + γ_{2} - γ_{1}, θ_{3} = σ_{3} T_{1} + γ_{3} - γ_{1}

Frequency response equations (FREs)

The FREs of the realistic situation ( $a_{1} \neq 0, a_{2} \neq 0, a_{3} \neq 0$ ) can be derived via the following methods in order to achieve the steady-state solution of the spring-coupled double pendulum using PPNDF controllers connected to the fixed point of equations (16a)–(18b) at ${\dot{a}}_{n} = 0$ and ${\dot{θ}}_{n} = 0 (n = 1, 2, 3)$

0 = [\frac{f}{2 ω_{0}}] \sin (θ_{1}) + [\frac{λ_{1}}{2 ω_{0}} a_{2}] \sin (θ_{2}) + [\frac{ω_{2} G_{1}}{2 ω_{0}} a_{3}] \cos (θ_{3})

(19a)

\begin{array}{l} σ_{1} a_{1} = [\frac{- 3 h ω_{0}}{16} a_{1}^{5} - (\frac{2 h ω_{0}^{2} - 3 ζ}{8 ω_{0}}) a_{1}^{3}] + [- \frac{f}{2 ω_{0}}] \cos (θ_{1}) + [- \frac{λ_{1}}{2 ω_{0}} a_{2}] \cos (θ_{2}) \\ + [\frac{ω_{2} G_{1}}{2 ω_{0}} a_{3}] \sin (θ_{3}) \end{array}}

(19b)

0 = [- \frac{μ_{1}}{2} a_{2}] + [- \frac{λ_{2}}{2 ω_{1}} a_{1}] \sin (θ_{2})

(20a)

(σ_{1} - σ_{2}) a_{2} = [- \frac{λ_{2}}{2 ω_{1}} a_{1}] \cos (θ_{2})

(20b)

0 = [- \frac{μ_{2}}{2} a_{3}] + [\frac{- G_{2} ω_{0}}{2 ω_{2}} a_{1}] \cos (θ_{3})

(21a)

(σ_{1} - σ_{3}) a_{3} = [\frac{G_{2} ω_{0}}{2 ω_{2}} a_{1}] \sin (θ_{3})

(21b)

From equations (20a)–(21b), we can conclude that

\sin (θ_{2}) = [- \frac{ω_{1} μ_{1}}{λ_{2} a_{1}} a_{2}]

(22)

\cos (θ_{2}) = [\frac{- 2 ω_{1} (σ_{1} - σ_{2}) a_{2}}{λ_{2} a_{1}}]

(23)

\cos (θ_{3}) = [\frac{- ω_{2} μ_{2}}{G_{2} ω_{0} a_{1}} a_{3}]

(24)

\sin (θ_{3}) = [\frac{2 ω_{2} (σ_{1} - σ_{3}) a_{3}}{G_{2} ω_{0} a_{1}}]

(25)

\sin (θ_{1}) = [\frac{2 ω_{0}}{f}] [(\frac{λ_{1} ω_{1} μ_{1}}{2 ω_{0} λ_{2} a_{1}} a_{2}^{2}) + (\frac{G_{1} ω_{2}^{2} μ_{2}}{2 G_{2} ω_{0}^{2} a_{1}} a_{3}^{2})]

(26)

\cos (θ_{1}) = [\frac{- 2 ω_{0}}{f}] [\begin{array}{l} σ_{1} a_{1} + \frac{3 h ω_{0}}{16} a_{1}^{5} + (\frac{2 h ω_{0}^{2} - 3 ζ}{8 ω_{0}}) a_{1}^{3} - \frac{ω_{1} λ_{1} (σ_{1} - σ_{2}) a_{2}^{2}}{ω_{0} λ_{2} a_{1}} \\ - \frac{G_{1} ω_{2}^{2} (σ_{1} - σ_{3}) a_{3}^{2}}{G_{2} ω_{0}^{2} a_{1}} \end{array}]

(27)

Combining equations (19a)−(21b) and (22)–(27) and summing the squared values yields

[μ_{1}^{2} + 4 {(σ_{1} - σ_{2})}^{2}] ω_{1}^{2} a_{2}^{2} = λ_{2}^{2} a_{1}^{2}

(28)

[μ_{2}^{2} + 4 {(σ_{1} - σ_{3})}^{2}] ω_{2}^{2} a_{3}^{2} = G_{2}^{2} ω_{0}^{2} a_{1}^{2}

(29)

{[(\frac{λ_{1} ω_{1} μ_{1}}{2 ω_{0} λ_{2} a_{1}} a_{2}^{2}) + (\frac{G_{1} ω_{2}^{2} μ_{2}}{2 G_{2} ω_{0}^{2} a_{1}} a_{3}^{2})]}^{2} + {[\begin{array}{l} σ_{1} a_{1} + \frac{3 h ω_{0}}{16} a_{1}^{5} + (\frac{2 h ω_{0}^{2} - 3 ζ}{8 ω_{0}}) a_{1}^{3} \\ - \frac{ω_{1} λ_{1} (σ_{1} - σ_{2}) a_{2}^{2}}{ω_{0} λ_{2} a_{1}} - \frac{G_{1} ω_{2}^{2} (σ_{1} - σ_{3}) a_{3}^{2}}{G_{2} ω_{0}^{2} a_{1}} \end{array}]}^{2} = [\frac{f^{2}}{4 ω_{0}^{2}}]

(30)

Investigation of stability

Starting with the steady-state solution stability analysis, we do the following

\begin{array}{l} a_{m} = a_{m 0} + a_{m 1} \\ θ_{m} = θ_{m 0} + θ_{m 1} \end{array}}; (m = 1, 2, 3)

(31)

While

a_{m 1}

and

θ_{m 1}

are small perturbations, equations (19a)–(21b) apply to

a_{m 0}

θ_{m 0}

. After substituting equation (31) into equations (19a)–(21b) and excluding all terms other than linear ones in

a_{m 1}

and

θ_{m 1}

, we get

\begin{array}{l} {\dot{a}}_{11} = [\frac{f}{2 ω_{0}} \cos (θ_{10})] θ_{11} + [\frac{λ_{1}}{2 ω_{0}} \sin (θ_{20})] a_{21} + [\frac{λ_{1}}{2 ω_{0}} a_{20} \cos (θ_{20})] θ_{21} \\ + [\frac{ω_{2} G_{1}}{2 ω_{0}} \cos (θ_{30})] a_{31} + [- \frac{ω_{2} G_{1}}{2 ω_{0}} a_{30} \sin (θ_{30})] θ_{31} \end{array}}

(32a)

\begin{array}{l} {\dot{θ}}_{11} = [\frac{σ_{1}}{a_{10}} + \frac{15 h ω_{0}}{16} a_{10}^{3} + (\frac{6 h ω_{0}^{2} - 9 ζ}{8 ω_{0}}) a_{10}] a_{11} + [- \frac{f}{2 ω_{0} a_{10}} \sin (θ_{10})] θ_{11} \\ + [\frac{λ_{1}}{2 ω_{0} a_{10}} \cos (θ_{20})] a_{21} + [- \frac{λ_{1}}{2 ω_{0} a_{10}} a_{20} \sin (θ_{20})] θ_{21} + [- \frac{ω_{2} G_{1}}{2 ω_{0} a_{10}} \sin (θ_{30})] a_{31} \\ + [- \frac{ω_{2} G_{1}}{2 ω_{0} a_{10}} a_{30} \cos (θ_{30})] θ_{31} \end{array}}

(32b)

{\dot{a}}_{21} = [- \frac{λ_{2}}{2 ω_{1}} \sin (θ_{20})] a_{11} + [- \frac{μ_{1}}{2}] a_{21} + [- \frac{λ_{2}}{2 ω_{1}} a_{10} \cos (θ_{20})] θ_{21}

(33a)

{\dot{θ}}_{21} = {\dot{θ}}_{11} + [- \frac{λ_{2}}{2 a_{20} ω_{1}} \cos (θ_{2})] a_{11} + [\frac{(σ_{2} - σ_{1})}{a_{20}}] a_{21} + [\frac{λ_{2}}{2 a_{20} ω_{1}} a_{10} \sin (θ_{20})] θ_{21}

(33b)

{\dot{a}}_{31} = [\frac{- G_{2} ω_{0}}{2 ω_{2}} \cos (θ_{30})] a_{11} + [- \frac{μ_{2}}{2}] a_{31} + [\frac{G_{2} ω_{0}}{2 ω_{2}} a_{10} \sin (θ_{30})] θ_{31}

(34a)

{\dot{θ}}_{31} = {\dot{θ}}_{11} + [\frac{G_{2} ω_{0}}{2 ω_{2} a_{30}} \sin (θ_{30})] a_{11} + [\frac{(σ_{3} - σ_{1})}{a_{30}}] a_{31} + [\frac{G_{2} ω_{0}}{2 ω_{2} a_{30}} a_{10} \cos (θ_{30})] θ_{31}

(34b)

The previous-equation system is obtained in the following matrix

{[\begin{array}{l} {\dot{a}}_{11} & {\dot{θ}}_{11} & {\dot{a}}_{21} & {\dot{θ}}_{21} & {\dot{a}}_{31} & {\dot{θ}}_{31} \end{array}]}^{T} = [J] {[\begin{array}{l} a_{11} & θ_{11} & a_{21} & θ_{21} & a_{31} & θ_{31} \end{array}]}^{T}

(35)

where

[J]

can be represented by the right parts of equations (32a)–(34b)

The eigenvalues of $[J]$ assumed through the next equation

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

(36)

In which

Γ_{m} (m = 1, 2, . . ., 8)

are the coefficients of equation (36). Subsequently, the solution of the system with PPNDF is stable if the real part of eigenvalue has a negative sign; unstable otherwise. In order to ensure that all of the roots of equation (36) have negative real part, it applies the Routh–Hurwitz criterion, which states that all of the principal minors and the following determinant D must be positive

D = | \begin{array}{l} Γ_{1} & 1 & 0 & 0 & 0 & 0 \\ Γ_{3} & Γ_{2} & Γ_{1} & 1 & 0 & 0 \\ Γ_{5} & Γ_{4} & Γ_{3} & Γ_{2} & Γ_{1} & 1 \\ 0 & Γ_{6} & Γ_{5} & Γ_{4} & Γ_{3} & Γ_{2} \\ 0 & 0 & 0 & Γ_{6} & Γ_{5} & Γ_{4} \\ 0 & 0 & 0 & 0 & 0 & Γ_{6} \end{array} |

(37)

Although the Multiple-Scale Perturbation Technique (MSPT) is employed in this study, it is worth noting that the homotopy perturbation method (HPM) represents an alternative analytical technique that can effectively handle strong nonlinearities without the need for a small perturbation parameter. MSPT is selected here because the spring-coupled double pendulum exhibits weak to moderate nonlinear behavior within the operating range considered, and the analysis focuses on near-resonant dynamics, for which MSPT naturally eliminates secular terms and provides physically meaningful amplitude–phase modulation equations. Future work may explore the use of HPM to extend the present results to strongly nonlinear or large-amplitude regimes.

To further enhance the analytical framework, it is worth noting that He’s frequency formulation offers an efficient means of estimating the initial frequency and amplitude of nonlinear oscillatory systems. Although the present study relies on the Multiple-Scale Perturbation Technique (MSPT), which already provides accurate near-resonant frequency estimates for weakly to moderately nonlinear dynamics, He’s formulation could be incorporated in future work to generate rapid and accurate initial guesses prior to performing MSPT or alternative perturbation methods such as HPM. This step may reduce computational effort and improve the precision of frequency-response and stability predictions, particularly for strongly nonlinear or large-amplitude motions.

The improvement of spring-coupled double pendulum oscillations with the application of PPNDF

Numerical simulation

The frequency-response and time-history curves are numerically simulated in this section. Equations (5a)–(5c) use the fourth-order Runge–Kutta algorithm at the primary resonance and one-to-one internal resonances to analyze the time history of a spring-coupled double pendulum with and without PPNDF. Figure 3 illustrates this at the selected values

(f = 0.01, ω_{0} ≅ ω_{1} ≅ ω_{2} ≅ Ω = 0.44, λ_{1} = 0.2, λ_{2} = 0.02, G_{1} = 0.5, G_{2} = 0.5, h = 0.01, ζ = 0.049, μ_{1} = 0.003, μ_{2} = 0.4)

Figure 3.

Time history of (a) the spring-coupled double pendulum without dampers. (b) Phase plane of the spring-coupled double pendulum without dampers.

Figures 3(a) and (b) shows the time history and phase plane of the spring-coupled double pendulum system without using any controls, which illustrates a very high amplitude. Therefore, Figures 4(a) and (b) panels show how the system behaves over time history and in the phase plane, both with and without dampers. Without dampers, the oscillations continue indefinitely, but with different dampers, the oscillations gradually diminish with regard to the damper used, eventually stabilizing the system. In Figures 4(c) and (d), a closer review of the dampers’ ability to lessen oscillation amplitude and create a more stable system is highlighted. Also, it has illustrated the differences between the effects of the PPF only, which reduces the amplitude of the spring-coupled double pendulum, NDF only, which reduces the amplitude of the spring-coupled double pendulum more than the PPF control, and when we combine them in the PPNDF controller. Obviously, the best effect comes from applying the PPNDF, which leads to a stable spring-coupled double pendulum, and that is supported when we calculate the efficiency of all of them $E_{a}$ (where $E_{a}$ = the amplitude of the studied system without any controller/the amplitude of the same system combined with controller). After using the PPF, the amplitude had decreased more than eight times since $E_{a}$ equals 8.545, but if we applied the NDF alone, the efficiency was found to be 25.024. The most effective controller was PPNDF, which has the largest $E_{a}$ reached 345.797.

Figure 4.

(a) Spring-coupled double pendulum time history without and with applying different dampers. (b) Spring-coupled double pendulum phase plane without and with applying different dampers. (c) A closer view to illustrate the difference in the effect of different dampers on the time history. (d) A closer view to illustrate the difference in the effect of different dampers on the phase plane.

Figure 5 presents a detailed comparison between the numerical solution obtained using the fourth-order Runge–Kutta method and the perturbation solution derived through the MSPT technique for the spring-coupled double pendulum system subjected to the PPNDF controller. Figure 5(a) illustrates the time response of the primary displacement, where the perturbation solution shows excellent agreement with the numerical results, particularly during the decay of oscillations. Figure 5(b) displays the evolution of the first component associated with the PPNDF controller. Although the numerical solution exhibits high-frequency oscillations, the overall trend remains consistent with the prediction of the perturbation analysis. In Figure 5(c), the second component of the PPNDF response also demonstrates strong consistency between the two approaches, confirming both the accuracy of the MSPT approximation and the effectiveness of the proposed control strategy in suppressing system vibrations. This agreement validates the assumptions adopted in the perturbation analysis and supports the reliability of the PPNDF technique.

Figure 5.

Comparison of perturbation analysis and numerical simulation for the spring-coupled double pendulum system subjected to the PPNDF technique (a) amplitude of the spring-coupled double pendulum system, (b) amplitude of the first part of PPNDF, (c) amplitude of the second part of PPNDF.

Analysis of the results

This section calculates the frequency response curves (FRC) for the practical scenario $a_{1} \neq 0, a_{2} \neq 0,$ and $a_{3} \neq 0$ by solving and plotting the nonlinear equations (10a)–(12c). Figures were performed in 3D to ensure the accuracy of the results and the implications of the parameters.

Figure 6 illustrates how the control signal gain $G_{1}$ affects the system’s response. The first three graphs, Figures 6(a)–(c) show how different parts of the system and controllers respond to these changes, particularly at points of potential resonance. The red curves indicate unstable behavior, where even small perturbations lead to large deviations, while the blue curves indicate stable, predictable responses. Each sub-figure presents the variation of $a_{1}$ , $a_{2}$ , and $a_{3}$ as $G_{1}$ and $σ_{1}$ change, corresponding, respectively, to the spring-coupled double pendulum, the Positive Position Feedback (PPF) controller, and the Negative Derivative Feedback (NDF) controller. Increasing $G_{1}$ reduces all three amplitudes, demonstrating improved vibration suppression and system stability.

Figure 6.

Spring-coupled double pendulum with control: (a–c) vibration amplitudes against $σ_{1}$ at different values of control signal gain $G_{1}$ . (d) Stability study chart in $σ_{1} - G_{1}$ plane.

Furthermore, the system’s response is influenced by the interaction of several parameters: higher feedback gains increase the damping effect and reduce peak amplitudes, while variations in the damping coefficients affect the rate at which oscillations decay. Excitation frequencies determine the resonance behavior, with certain frequencies leading to amplified responses if not properly controlled. Together, these parameters govern the overall dynamic behavior, highlighting the importance of carefully tuning $G_{1}$ , damping coefficients, and operating frequencies to achieve optimal vibration suppression. Figure 6(d) shows the system’s stability: the green areas are stable, and the red area is unstable. This means adjusting $G_{1}$ can either stabilize or destabilize the system.

Figure 7 demonstrates how changing the feedback signal gain $G_{2}$ affects the amplitude responses $a_{1}$ , $a_{2}$ , and $a_{3}$ . Each graph shows how the amplitudes vary with different values of $G_{2}$ and $σ_{1}$ , highlighting shifts in resonance peaks as $G_{2}$ changes. Here, $a_{1}$ corresponds to the spring-coupled double pendulum as presented in Figure 7(a), $a_{2}$ to the PPF controller as presented in Figure 7(b), and $a_{3}$ to the NDF controller as presented in Figure 7(c). Increasing $G_{2}$ leads to a noticeable reduction in $a_{1}$ , a slight decrease in $a_{2}$ , and an increase in $a_{3}$ . This indicates that while higher $G_{2}$ values improve vibration suppression of the main pendulum; they have differing effects on the controllers, enhancing the NDF response but slightly reducing the PPF effect. Figure 7(d) shows a stability map, where the red areas indicate stable conditions, and the green area shows instability. Essentially, tweaking $G_{2}$ influences both the system’s behavior and its stability.

Figure 7.

Spring-coupled double pendulum with control: (a–c) vibration amplitudes against $σ_{1}$ at different values of the feedback signal gain $G_{2}$ . (d) Stability study chart in $σ_{1} - G_{2}$ plane.

Moreover, the system’s response is influenced by the interplay of multiple parameters: feedback gains affect damping efficiency, damping coefficients determine oscillation decay rates, and excitation frequencies govern resonance behavior. Careful tuning of $G_{2}$ along with damping and frequency parameters, is therefore essential for optimal vibration suppression across the system.

Figure 8 illustrates the influence of increasing the control signal gain $λ_{1}$ on the system’s frequency response. As $λ_{1}$ increases, the resonance peaks in the graphs shift, reflecting changes in the dynamic behavior of the system. A slight increase is observed in the amplitudes $a_{1}$ and $a_{3}$ as illustrated in Figures 8(a) and (c), respectively. Whereas in Figure 8(b) $a_{2}$ decreases as $λ_{1}$ becomes larger. This indicates that higher values of $λ_{1}$ amplify the response of the spring-coupled double pendulum and the NDF-controlled motion, while simultaneously introducing a stabilizing effect on the PPF controller, leading to a reduction in $a_{2}$ . The last subfigure, Figure 8(d), shows the system’s stability, with green areas indicating stable conditions and red areas showing instability. This means adjusting $λ_{1}$ can either stabilize or destabilize the system.

Figure 8.

Spring-coupled double pendulum with control: (a–c) vibration amplitudes against $σ_{1}$ at different values of control signal gain $λ_{1}$ . (d) Stability study chart in $σ_{1} - λ_{1}$ plane.

Overall, Figure 8 highlights how tuning $λ_{1}$ modifies the resonance characteristics and differently influences each component of the coupled system.

Furthermore, Figure 9 demonstrates how variations in the feedback signal gain $λ_{2}$ affect the system’s amplitude responses. Sub-figure 9(a) presents the amplitude $a_{1}$ of the spring-coupled double pendulum, sub-figure 9(b) shows the amplitude $a_{2}$ associated with the PPF controller, and sub-figure 9(c) illustrates the amplitude $a_{3}$ corresponding to the NDF controller. As $λ_{2}$ increases, all three amplitudes— $a_{1}$ , $a_{2}$ , and $a_{3}$ —exhibit a slight increase, accompanied by noticeable shifts in the resonance peaks. The final sub-figure 9(d) provides a stability map, indicating the influence of $λ_{2}$ on the system’s dynamic stability; red regions represent stable operating zones, while green regions indicate potential instability. Overall, Figure 9 highlights that tuning $λ_{2}$ modestly amplifies the system response and plays a significant role in shaping the stability boundaries of the spring-coupled double pendulum system.

Figure 9.

Spring-coupled double pendulum with control: (a–c) vibration amplitudes against $σ_{1}$ at different values of the feedback signal gain $λ_{2}$ . (d) Stability study chart in $σ_{1} - λ_{2}$ plane.

Figure 10 provides a detailed examination of how the system responds to variations in the external excitation force $f$ and the detuning parameter $σ_{1}$ . Sub-figures 10(a)–(c) present the amplitudes $a_{1}$ , $a_{2}$ , and $a_{3}$ , corresponding, respectively, to the spring-coupled double pendulum, the PPF controller, and the NDF controller. As the excitation force $f$ increases, all three amplitudes exhibit a noticeable and consistent increase, highlighting the system’s strong sensitivity to higher external forcing. The resonance peaks observed in these curves occur when the excitation frequency approaches the natural frequency of the system, resulting in significantly amplified vibrations.

Figure 10.

Spring-coupled double pendulum with control: (a–c) vibration amplitudes against $σ_{1}$ at different values of the external excitation force amplitude $f$ . (d) Stability study chart in $σ_{1} - f$ plane.

Sub-figure 10(d) provides a stability map showing how different combinations of $f$ and $σ_{1}$ influence the overall stability of the system. The map clearly distinguishes stable operating regions from those where instability may arise, demonstrating how changes in excitation magnitude and system parameters shape the stability boundaries of the spring-coupled double pendulum.

Figures 11 (a)–(c) illustrates how the amplitudes $a_{1}$ , $a_{2}$ , and $a_{3}$ vary with changes in the damping coefficient $μ_{1}$ . Sub-figure 11(a) corresponds to the response of the spring-coupled double pendulum, 11(b) represents the PPF controller response, and 11(c) shows the NDF controller response. As $μ_{1}$ increases, the amplitude $a_{2}$ decreases noticeably, indicating that the PPF controller becomes more effective in suppressing vibrations under higher damping. In contrast, the amplitudes $a_{1}$ and $a_{3}$ exhibit saturation behavior, where further increases in $μ_{1}$ produce minimal changes in their peak values but lead to a noticeable reduction in the bandwidth of the frequency-response curves. Subfigure d represents a stability map that highlights the stability and instability regions of the system $μ_{1}$ .

Figure 11.

Spring-coupled double pendulum with control: (a–c) vibration amplitudes against $σ_{1}$ at different values of the damping coefficient of the PPF control $μ_{1}$ . (d) Stability study chart in $σ_{1} - μ_{1}$ plane.

The red curves in all sub-figures denote unstable regions, where the system becomes highly sensitive or unbounded, whereas the blue curves indicate stable regions with predictable and bounded motion. Overall, Figure 11 highlights how adjusting the damping coefficient $μ_{1}$ shapes both the amplitude characteristics and the stability boundaries of the coupled system.

Figure 12 presents the variation of the amplitudes $a_{1}$ , $a_{2}$ , and $a_{3}$ with changes in the damping coefficient $μ_{2}$ . Sub-figures 12(a)–(c) show the responses of the spring-coupled double pendulum, the PPF controller, and the NDF controller, respectively. As $μ_{2}$ increases, the amplitude $a_{1}$ initially decreases but exhibits a slight increase at higher values of $μ_{2}$ , $a_{2}$ shows a minor increase, while $a_{3}$ decreases. These trends indicate that $μ_{2}$ has a differential effect on each component, modulating the response amplitudes depending on the damping level. Figure 12(d) shows that a higher value of the damping coefficients $μ_{2}$ leads to more stability.

Figure 12.

Spring-coupled double pendulum with control: (a–c) vibration amplitudes against $σ_{1}$ at different values of the damping coefficient of NDF control $μ_{2}$ . (d) Stability study chart in $σ_{1} - μ_{2}$ plane.

The red curves denote unstable regions, where the system response becomes unbounded or highly sensitive, whereas the blue curves indicate stable operating regions with predictable behavior. Overall, Figure 12 highlights the critical role of $μ_{2}$ in controlling both the amplitude characteristics and the stability boundaries of the spring-coupled double pendulum system.

Figure 13 examines the effect of changes in the parameters $ω_{0}, ω_{1},$ and $ω_{2}$ (with $ω_{0} = ω_{1} = ω_{2}$ ) on the amplitudes plotted against the detuning parameter $σ_{1}$ . Sub-figure 13(a) shows the amplitude $a_{1}$ of the spring-coupled double pendulum, 13(b) presents the amplitude $a_{2}$ corresponding to the PPF controller, and 13(c) illustrates the amplitude $a_{3}$ for the NDF controller. As the frequency parameters increase, $a_{1}$ and $a_{3}$ decrease significantly, whereas $a_{2}$ experiences only a slight reduction. These trends indicate that higher natural frequencies reduce the overall response of the system, particularly affecting the main pendulum and the NDF-controlled motion, while the PPF controller is less sensitive. Subfigure 13(d) shows stability depending on $ω_{0}$ and $σ_{1}$ , with the same color-coding for stable and unstable regions.

Figure 13.

Spring-coupled double pendulum with control: (a–c) vibration amplitudes against $σ_{1}$ at different values of natural frequency $ω_{0}$ . (d) Stability study chart in $σ_{1} - ω_{0}$ plane.

The red curves denote unstable regions, while the blue curves indicate stable operating zones, emphasizing how the frequency parameters influence both the amplitude levels and the stability boundaries of the system.

Figures 14 and 15 explore the influence of the nonlinear oscillation coefficient ratio $ζ$ and the damping coefficient of the structural framework $h$ on the system’s frequency response and stability. In each figure, sub-figures (a), (b), and (c) display the amplitudes $a_{1}$ , $a_{2}$ , and $a_{3}$ corresponding, respectively, to the spring-coupled double pendulum, the PPF controller, and the NDF controller. As $ζ$ and $h$ increase, all three amplitudes show a slight increase, accompanied by shifts in the resonance peaks, indicating that these parameters modestly amplify the system’s response.

Figure 14.

Spring-coupled double pendulum with control: (a–c) vibration amplitudes against $σ_{1}$ at different values of the nonlinear oscillation coefficient $ζ$ . (d) Stability study chart in $σ_{1} - ζ$ plane. $ζ$ via $σ_{1}$ .

Figure 15.

Spring-coupled double pendulum with control: (a–c) vibration amplitudes against $σ_{1}$ at different values of the damping coefficient of the structure framework $h$ . (d) Stability study chart in $σ_{1} - h$ plane.

The stability diagrams in sub-figure (d) of both figures reveal regions of stable and unstable behavior, with red areas representing stable zones and green areas indicating potential instability. This analysis demonstrates that careful adjustment of $ζ$ and $h$ can influence both the amplitude response and the stability of the system, highlighting their important role in tuning the dynamic performance of the spring-coupled double pendulum.

Finally, in Figures 16 and 17 illustrate how changing the value of $σ_{2}$ and $σ_{3}$ affects the system’s amplitude responses $a_{1}$ , $a_{2}$ , and $a_{3}$ with respect to $σ_{1}$ . Subfigure (d) shows a stability map where the green regions are unstable, and the red region is stable. Overall, the figure shows how adjustments in $σ_{2}$ and $σ_{3}$ can impact both the stability and behavior of the system’s responses.

Figure 16.

Spring-coupled double pendulum with control: (a–c) vibration amplitudes against $σ_{1}$ at different values of the detuning parameter $σ_{2}$ . (d) Stability study chart in $σ_{1} - σ_{2}$ plane.

Figure 17.

Spring-coupled double pendulum with control: (a–c) vibration amplitudes against $σ_{1}$ at different values of the detuning parameter $σ_{3}$ . (d) Stability study chart in $σ_{1} - σ_{3}$ plane.

Broader applicability of the PPNDF framework

Although the present study focuses on the spring-coupled double pendulum, the proposed PPNDF control strategy and analytical framework are readily extendable to a broad class of nonlinear vibratory systems. Many micromechanical and MEMS oscillators exhibit similar dynamical features, including nonlinear stiffness, modal coupling, and sensitivity to low-frequency excitations, making them strong candidates for PPNDF-based vibration mitigation. The methodology adopted in this work (harmonic balance, stability analysis, and frequency-response characterization) can be directly applied to these systems with only minor modifications to account for microscale damping and geometry-dependent nonlinearities in stiffness. As discussed in Ref. 49, MEMS resonators in particular suffer from low-frequency instability and amplitude fluctuation issues, which could benefit from the robust damping and response-flattening effects introduced by the PPNDF controller. This connection highlights the potential for extending the present approach to microscale devices and motivates future work in this direction.

Effect of parameter uncertainty

Although the main analysis in this study is conducted under nominal system parameters, practical engineering applications inevitably involve deviations in mass, stiffness, and damping. To address this concern, we included a brief robustness discussion regarding the PPNDF controller. Due to its combined positive-position and negative-derivative structure, the controller maintains stable operation under moderate parameter variations, and its qualitative response characteristics (e.g., oscillation reduction and frequency-shaping behavior) remain preserved within typical engineering tolerances. This observation is consistent with the general robustness properties of derivative-based damping enhancement. These remarks highlight that the proposed control approach is not limited to the nominal model and can accommodate practical levels of parameter uncertainty.

Bifurcation for nonlinear dynamics

In this bifurcation analysis, the controller gains are intentionally chosen as $λ_{1} = λ_{2} = 0.02$ and $G_{1} = G_{2} = 0.01$ , which are smaller than those used in the main resonance-control simulations. This selection enables the system to display its natural nonlinear transitions—such as quasi-periodic and chaotic motions—that would otherwise be heavily suppressed by stronger control gains. Consequently, the bifurcation diagrams provide a clearer picture of the system’s potential instability patterns. Notably, even within these quasi-periodic and chaotic regions, the PPNDF controller still manages to reduce the vibration amplitude and delay the emergence of chaos relative to the uncontrolled system, demonstrating its robustness beyond the primary resonance conditions.

The nonlinear system exhibits a rich set of dynamical responses under variation of the excitation frequency $Ω$ , as illustrated by the bifurcation diagram in Figure 18. The bifurcation diagram summarizes these transitions clearly: starting with quasi-periodicity at low frequencies, then passing through a window of stable periodic oscillation, and eventually evolving into complex chaotic behavior at higher forcing frequencies. The observed sequence of dynamical transitions—quasi-periodic → periodic → chaotic—is typical of nonlinear controlled oscillatory systems and highlights the sensitivity of the system to excitation frequency under PPNDF control.

Figure 18.

Bifurcation diagram of $u$ versus $Ω$ with PPNDF control at $λ_{1} = λ_{2} = 0.02; G_{1} = G_{2} = 0.01$ .

At low values of $Ω$ , the system initially follows a small–amplitude response before transitioning into a quasi-periodic regime. This behavior is characterized by the emergence of a toroidal attractor in the phase space and a closed invariant curve in the Poincaré map, as shown in Figures 19(a) and (b) for $Ω = 0.1$ and $Ω = 0.3$ . The smooth closed loop of Poincaré points indicates the coexistence of two incommensurate frequencies, confirming quasi-periodicity.

Figure 19.

Phase portraits and Poincaré maps of the quasi-periodic state at (a) $Ω = 0.1$ .(b) $Ω = 0.3$ .

As $Ω$ increases, the system undergoes a transition to periodic motion. In Figures 20(a) and (b) for $Ω = 0.4$ and $Ω = 0.5$ , the trajectories collapse onto a single closed orbit, and the Poincaré section reduces to a single point, highlighting the establishment of a stable limit cycle. This transition is consistent with the amplitude branch observed in the bifurcation diagram, where the system settles into a regular oscillatory state.

Figure 20.

Phase portraits and Poincaré maps of the periodic state at (a) $Ω = 0.4$ . (b) $Ω = 0.5$ .

Further increase in the forcing frequency leads to a breakdown of periodicity and the onset of chaotic oscillations. The phase portraits at Figures 21(a) and (b) for $Ω = 1.5$ and $Ω = 3$ show a dense, aperiodic trajectory that fills a region of the phase plane, while the Poincaré maps display a scattered cloud of non-repeating points. This loss of structure marks the development of a strange attractor, confirming the presence of chaos. The chaotic regions observed in the Poincaré maps correspond to positive Lyapunov exponents (not shown here), reinforcing the identification of sensitive dependence on initial conditions.

Figure 21.

Phase portraits and Poincaré maps of the Chaotic state at (a) $Ω = 1.5$ .(b) $Ω = 3$ .

Conclusion

This study presented a comprehensive analytical and numerical investigation of the nonlinear dynamics and vibration suppression capabilities of a spring-coupled double pendulum under various control strategies. The main findings of this work are summarized below:

Key Findings:

PPF control achieved a vibration amplitude reduction of approximately 91.79%, confirming its effectiveness as a baseline passive controller.

NDF control provided superior suppression, reducing vibration amplitude by around 96.44%, demonstrating the benefits of incorporating nonlinear damping effects.

The proposed PPNDF controller exhibited the best overall performance, achieving a maximum reduction of about 99.80% in both primary resonance and 1:1 internal resonance scenarios.

Multiple-Scale Perturbation Techniques (MSPT) facilitated analytical derivation of time-dependent amplitudes, enabling a deeper understanding of the system’s nonlinear dynamics and control behavior.

Analytical frequency-response and stability analyses corroborated numerical simulations, validating the accuracy of the proposed approach.

Bifurcation analysis revealed that the system undergoes a sequence of dynamical transitions as the excitation frequency Ω varies, progressing from quasi-periodic motion at low frequencies, to periodic motion, and eventually to chaotic behavior at higher frequencies. This highlights the system’s strong nonlinear characteristics and its sensitivity to excitation.

Limitations:

The current study assumes idealized system parameters and perfect knowledge of the model, neglecting uncertainties, parameter variations, and potential environmental disturbances.

The analysis focuses primarily on a specific spring-coupled double pendulum configuration; generalization to more complex or multi-component systems may require further investigation.

Future Work:

Extending the proposed PPNDF control strategy to micro-scale devices such as MEMS oscillators and other nonlinear resonant systems, where low-frequency vibration suppression is critical.

Investigating controller performance under uncertain or time-varying parameters, including random excitations and environmental disturbances.

Exploring multi-objective control approaches to optimize both vibration suppression and energy efficiency simultaneously.

In summary, the proposed PPNDF controller demonstrates remarkable effectiveness in suppressing vibrations and stabilizing the nonlinear system, while the limitations identified provide clear directions for further research and practical implementation.

Footnotes

Acknowledgments

The researchers would like to thank the Deanship of Graduate Studies and Scientific Research at Qassim University for financial support (QU-APC-2025).

ORCID iD

Ashraf Taha EL-Sayed

Author contributions

S.K.E.: investigation, methodology, formal analysis, reviewing and editing, and funding acquisition. H.S.B.: investigation, methodology, data curation, validation, reviewing, and editing. A.T.E.-S.: conceptualization, resources, methodology, writing—original draft preparation, visualization, reviewing, and editing. A.M.S.: formal analysis, validation, investigation, methodology, data curation, conceptualization, validation, and 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 financial support (QU-APC-2025).

Declaration of conflicting interests

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

Data Availability Statement

All data generated or analyzed during this study are included in this published article.

Appendix

The mathematical derivation of the equation of motion with Lagrange’s method (nonlinear) plus a small-motion linearization can be illustrated as:

Setup and geometry.

Let $x (t)$ be the horizontal displacement of mass $m_{1}$ (to the right positive).

The lower mass $m_{2}$ is constrained to move only vertically; the rigid rod of fixed length $L$ connects the two pins.

With the top pin at $(x, 0)$ and the bottom pin at $(0, y)$ the rod constraint is $\sqrt{x^{2} + y^{2}} = L$ , so $y (t) = - \sqrt{L^{2} - x {(t)}^{2}}$ .

Kinetic and potential energies (38)

T = \frac{1}{2} m_{1} {\dot{x}}^{2} + \frac{1}{2} m_{2} {\dot{y}}^{2}

(39)

V = \frac{1}{2} k x^{2} + m_{2} g y

With $y = - \sqrt{L^{2} - x^{2}}$ one gets $\dot{y} = \frac{x \dot{x}}{\sqrt{L^{2} - x^{2}}}$

Equations (38) and (39) allow for the expression of the Lagrangian function ( $L$ ) in the following way (40)

L = T - V = \frac{1}{2} m_{1} {\dot{x}}^{2} + \frac{1}{2} m_{2} \frac{x^{2} {\dot{x}}^{2}}{(L^{2} - x^{2})} - \frac{1}{2} k x^{2} + m_{2} g \sqrt{L^{2} - x^{2}}

The system equations of motion can be determined in the manner described below using the Euler–Lagrange’s formulation (41)

\frac{d}{d t} (\frac{\partial L}{\partial \dot{x}}) - (\frac{\partial L}{\partial x}) = 0

The fundamental equation of motion may be written as a nonlinear ordinary differential equation in the form (42)

[m_{1} + m_{2} \frac{x^{2}}{(L^{2} - x^{2})}] \ddot{x} + m_{2} \frac{L^{2} x {\dot{x}}^{2}}{{(L^{2} - x^{2})}^{2}} + k x + \frac{m_{2} g x}{\sqrt{L^{2} - x^{2}}} = 0

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