Nonlinear dynamics,stability,and vibration suppression of magnetically levitated systems under a combined control strategy

Abstract

This study investigates a non-ideal magnetic levitation system (NIMLS) for energy harvesting, with particular emphasis on the influence of the oscillating core mass. The system is excited by an electrodynamic shaker, leading to strongly nonlinear oscillatory behavior governed by a coupled equation resembling the Duffing oscillator. Perturbation techniques are employed to analyze the system dynamics under the combined action of a negative velocity and negative acceleration controller (NVC+NAC), focusing on primary resonance and 1:1 internal resonance condition. Analytical solutions are derived to describe the system response, while MATLAB-based numerical simulations are used to validate the theoretical results. The stability of the system under primary resonance is examined using the Routh–Hurwitz criterion. Furthermore, the effects of structural constraints and system parameters on the dynamic performance of the primary system equipped with the NVC+NAC controller are investigated. A comparative analysis between analytical and numerical solutions confirms the accuracy of the perturbation-based approximations and demonstrates the effectiveness of the proposed controller in suppressing vibration amplitudes.

Keywords

magnetically levitated system multiple time scale technique NVC+NAC nonlinear vibration control resonance case stability frequency response

1. Introduction

Recent advances in vibration-based energy harvesting have focused on converting ambient mechanical vibrations into electrical energy through innovative device designs. Khatun et al.¹ investigated ambient vibration sources and optimized resonant frequencies using proof mass systems, providing guidelines for efficient energy conversion at low excitation levels. Hussain et al.² performed numerical and experimental analyses of nonlinear broadband monostable and bistable energy harvesters, demonstrating significant enhancement in harvested energy through system nonlinearity. Xu et al.³ explored magnetic levitating bistable electromagnetic harvesters, proposing strategies to improve dynamic performance under varying excitation conditions. Francis et al.⁴ examined the effect of the levitated mass’s weight on system oscillations and energy efficiency, highlighting critical design parameters for stable operation. Ribeiro et al.⁵ analyzed nonlinear dynamics under frequency-varying excitations and proposed control strategies to maximize energy extraction. Hajradinović and Zuković⁶ studied vibro-impact systems with non-ideal excitation, providing insight into transient and steady-state motion. Rocha et al.⁷ demonstrated practical energy harvesting from a magnetic levitation system excited by an electrodynamical shaker, validating theoretical models through experimental implementation.

In the domain of vibration control, Amer et al.⁸ applied PD control to active magnetic bearing systems, showing improved stability under high-speed rotational conditions. Alluhydan et al.⁹ investigated time-delayed damping in a Duffing oscillator, demonstrating effective vibration suppression through tailored control schemes. Jamshidi and Collette¹⁰ optimized negative derivative feedback controllers for collocated systems, achieving enhanced system stability and reduced resonant amplitudes. References 8 and 11 presented hybrid Rayleigh–Van der Pol–Duffing oscillator models with PD controllers, demonstrating significant vibration reduction in nonlinear dynamic systems. Further, Amer et al.¹² employed negative cubic velocity feedback to reduce vibrations in vertical conveyor systems, confirming controller efficacy through both analytical and numerical studies. Hamed et al.¹³ applied PD controllers to wind turbine towers, optimizing energy transfer and structural stability. Bauomy and El-Sayed¹⁴ demonstrated the role of nonlinear PD controllers in laminated shells, highlighting their applicability in smart structural systems. Hamed et al.¹⁵ extended PD control to vertically excited transport systems, verifying performance improvements under variable loading conditions.

Positive position feedback (PPF) and nonlinear integral PPF (NIPPF) controllers have also been extensively explored. Alluhydan et al.¹⁶ implemented PPF controllers in hybrid electric vehicle generators, demonstrating enhanced vibration suppression and improved energy efficiency. The impact of NIPPF controllers on forced and self-excited nonlinear beams was investigated by,¹⁷ showing a reduction of flutter-induced vibrations. Omidi and Nima¹⁸ applied multimodal modified PPF controllers to collocated structures, achieving precise mitigation of resonant oscillations. Abdelhafez and Nassar¹⁹ utilized PPF control to suppress high-amplitude vibrations in nonlinear beams, while²⁰ verified the effectiveness of PPF in principal resonance excitation scenarios. Zhu et al.²¹ developed adaptive feedforward PPF strategies for piezoelectric cantilever beams, demonstrating robust vibration control under variable operating conditions.

Nonlinear dynamic analysis has been central in understanding complex vibration phenomena. Fyrillas and Szeri²² analyzed ultra- and subharmonic resonances, providing foundational insights into nonlinear resonance behavior. Fischer and Eberhard²³ applied predictive control to cutting processes, illustrating real-time mitigation of vibratory disturbances. Bauomy and El-Sayed²⁴ examined the effect of specialized vibration controllers on cracked beams, contributing to structural reliability studies.

Analytical methods such as the Method of Multiple Scales and perturbation techniques have been widely employed. Kevorkian and Cole²⁵ introduced the multiple scales method for solving ordinary differential equations in nonlinear systems, while²⁶ provided comprehensive perturbation approaches for dynamic analysis. Dukkipati²⁷ utilized MATLAB-based solutions to facilitate vibration problem-solving in engineering applications. Bek et al.²⁸ conducted an asymptotic analysis on a 3DOF dynamical system near resonance, offering a precise prediction of system behavior. References 29 and 30 systematically analyzed stability and nonlinear responses in multi-degree-of-freedom oscillators, including double pendulums and damped spring pendula, highlighting resonance and bifurcation phenomena. Amer et al.³¹ and Abohamer et al.³² extended these studies to coupled oscillators with piezoelectric harvesters and Duffing systems under sinusoidal excitation, providing detailed insights into chaotic and stability behaviors.

The current aims in this work are to apply the combined NVC and NAC controller to suppress vibrations in the primary system under external excitations. Time responses were evaluated using the fourth-order Runge–Kutta method, and approximate solutions were obtained via MTST. Stability under primary and 1:1 internal resonances was analyzed, with MATLAB simulations confirming the controller’s effectiveness in vibration reduction.

1.1. Motivation and Novelty

Although strong nonlinearities and multi-physical coupling have been widely investigated in nonlinear vibration research, the present work introduces several distinct contributions relative to existing studies. For instance, the self-powered low-frequency nonlinear vibration isolation system reported in Ref. 33 focuses primarily on passive vibration mitigation through nonlinear stiffness tailoring. In contrast, the proposed magnetic levitation system incorporates active negative velocity and negative acceleration feedback (NVC–NAC) to simultaneously regulate nonlinear dynamics and enhance energy harvesting performance. Similarly, the nonlinear analysis of piezoelectric functionally graded nanobeams under combined hygro-magneto-electro-thermo-mechanical loading in Ref. 34 emphasizes multi-field coupling effects at the nanoscale. The present study instead considers an electrodynamically excited magnetic levitation harvester and investigates how feedback-induced nonlinear modulation influences bifurcation structure, multistability, and harvested power. Moreover, while the generalized van der Pol–type model developed in Ref. 35 captures multistable vortex-induced vibrations in bridge deck systems, that approach is phenomenological in nature. By contrast, the current work derives a physically grounded nonlinear model of a magnetic levitation system and demonstrates how controlled multistability can be strategically exploited to improve energy harvesting efficiency. In addition, recent investigations have explored broadband dynamic responses and vibration control in magnetic and electromechanical systems. The time-delayed nonlinear magnetic levitation system in Ref. 36 achieves broadband characteristics via delay-induced nonlinearity, primarily targeting enhanced harvesting bandwidth. The HTS maglev system analyzed in Ref. 37 focuses on electro-mechanical coupling performance under electromagnetic interactions. Furthermore, the active rotary inertia driver proposed in Ref. 38 addresses flutter suppression in large-scale bridge structures through active control mechanisms. In contrast to these studies, the present work integrates NVC and NAC control within a physically derived magnetic levitation framework to realize controlled multistability, broadband vibration suppression, and enhanced robustness against excitation frequency variations. Therefore, the novelty lies in combining physically derived nonlinear modeling with dual feedback control to simultaneously achieve active vibration suppression and improved energy harvesting performance. To clarify the claims of broadband vibration suppression and robustness, broadband performance is herein defined as the ability of the system to maintain effective vibration suppression and energy harvesting over a wide excitation frequency range. In the present study, this broadband behavior is quantified based on the −3 dB bandwidth criterion. Robustness refers to the capability of the proposed controlled magnetic levitation system to preserve stable dynamic performance under excitation frequency variations. These definitions are substantiated and quantitatively demonstrated in the Results section.

2. Mathematical model

2.1. System dynamics without a controller

The system exhibits Duffing-like nonlinear characteristics, which help guide the qualitative interpretation of its response. To analyze the magnetically levitated system with integrated energy harvesting, we employ a mathematical model based on a simplified two-degree-of-freedom harmonic oscillator.⁷ Harmonic oscillators are characterized by their ability to exhibit sinusoidal motion under a restoring force proportional to displacement, and in this study, they provide a simplified framework to represent the system’s dynamic behavior. The mathematical derivation of the system equations is presented in the appendix.

\ddot{y} + ε μ_{1} \dot{y} + {ω_{1}}^{2} y - ε α_{1} y^{2} + ε α_{2} y^{3} + ε γ_{1} \dot{q} = 0

(1)

\ddot{q} + ε μ_{2} \dot{q} + {ω_{2}}^{2} q - ε γ_{2} \dot{y} = ε f \cos Ω t

(2)

Here, the external excitation is taken as

f (t) = f \cos Ω t

,where

f

denotes the excitation amplitude and

Ω

excitation frequency.

2.2. Impact of (NVC+NAC) control on system dynamics

In Figure 1, an electro-dynamical shaker excites the levitated object, which is suspended by a magnetic field generated by electromagnets or permanent magnets to counteract gravity, thus enabling contactless suspension. The energy harvesting circuit converts the vibrational motion into electrical energy through electromagnetic induction or piezoelectric conversion. The weight of the levitated object significantly affects the magnetic field strength required for levitation, as well as the system’s stability and energy harvesting efficiency. The upper part of the figure is adapted from,⁷ the middle part presents the governing equations of the system, and the lower part illustrates the integration of the nonlinear vibration controller (NVC) and the nonlinear absorber component (NAC) connected to the main structure.

\ddot{y} + ε μ_{1} \dot{y} + {ω_{1}}^{2} y - ε α_{1} y^{2} + ε α_{2} y^{3} + ε γ_{1} \dot{q} = - ε g_{1} \dot{y} - ε g_{2} \ddot{y}

(3)

\ddot{q} + ε μ_{2} \dot{q} + {ω_{2}}^{2} q - ε γ_{2} \dot{y} = ε f \cos Ω t - ε g_{3} \dot{q} - ε g_{4} \ddot{q}

(4)

Figure 1.

The schematic flowchart of the system incorporates (NVC+NAC) controllers.

It should be noted that the coefficients $α_{1}, α_{2}, γ_{1}, γ_{2} a n d ε$ are not fully independent parameters. They are interrelated through the material and geometric characteristics of the system, in accordance with the formulation presented in Ref. 7. Physically, this interdependence arises because these coefficients collectively represent the stiffness, damping, and magnetic coupling properties of the levitated structure. Variations in the material modulus $ε$ or the geometric configuration directly influence the associated nonlinear stiffness $α_{1}, α_{2}$ and electromagnetic coupling terms $γ_{1}, γ_{2}$ , leading to coupled effects that govern the system’s overall dynamic behavior. The terms $ε μ_{1}, ε μ_{2}$ are introduced to facilitate the application of the Multiple Time Scale method. Although $ε = 1$ it does not alter the actual values of the damping factors $μ_{1}$ and $μ_{2}$ , but serves as a mathematical device to separate time scales and simplify the perturbation analysis. Acceleration feedback is introduced mathematically and can be linked to concepts such as virtual mass or inertial amplification. Its practical implementation is limited by sensor noise, and the current study assumes ideal sensing conditions. Experimental validation is needed to assess real-world feasibility and robustness.

3. Mathematical treatment

3.1. Approximate analytical solution

In this section, (MTSM) is employed to derive a first-order approximate solution for the nonlinear dynamical system controlled by the (NVC+NAC) controller.^25–27 Using the multiple scales method, where the fast scale is $T_{0}$ and the slow scale is $T_{1} = ε t$ , the dependent variables are expanded in powers of the small parameter $ε$ .

\begin{array}{l} t = T_{0} + T_{1} + . . . \\ y (t, ε) = y_{0} (T_{0}, T_{1}) + ε y_{1} (T_{0}, T_{1}) + O (ε^{2}) \\ q (t, ε) = q_{0} (T_{0}, T_{1}) + ε q_{1} (T_{0}, T_{1}) + O (ε^{2}) \end{array}}

(5)

where the fast scale represents the rapid oscillation of the system, while the slow scale describes the modulation of amplitude and phase over time. Using these time scales, the time derivatives can be expressed accordingly in the subsequent equations.

\begin{array}{l} \frac{d}{d t} = D_{0} + ε D_{1} + ε^{2} D_{2} + . . . \\ \frac{d^{2}}{d t^{2}} = D_{0}^{2} + 2 ε D_{0} D_{1} + . . . \end{array}} D_{j} = \frac{\partial}{\partial T_{j}} (j = 0, 1)

(6)

Note:

({D_{0}}^{2} + {ω_{1}}^{2})

denotes a composite differential operator acting on the response variable, where

D_{0} = \frac{\partial}{\partial T_{0}}

is the derivative with respect to the fast time scale

T_{0}

and

{ω_{1}}^{2}

is a constant coefficient. The “+” here is operator notation used in the multiple time scales perturbation formulation, not a sim not a simple algebraic sum. Implanting equations (5) and (6) in equations (3) & (4) such that:

({D_{0}}^{2} + {ω_{1}}^{2}) y_{0} + ε ({D_{0}}^{2} + {ω_{1}}^{2}) y_{1} = ε (\begin{array}{l} - 2 D_{0} D_{1} y_{0} - μ_{1} D_{0} y_{0} + α_{1} {y_{0}}^{2} - α_{2} {y_{0}}^{3} \\ - γ_{1} D_{0} q_{0} - g_{1} D_{0} y_{0} - g_{2} {D_{0}}^{2} y_{0} \end{array}) + O (ε^{2})

(7)

({D_{0}}^{2} + {ω_{2}}^{2}) q_{0} + ε ({D_{0}}^{2} + {ω_{2}}^{2}) q_{1} = ε (\begin{array}{l} - 2 D_{0} D_{1} q_{0} - μ_{2} D_{0} q_{0} + γ_{2} D_{0} y_{0} \\ + f \cos Ω t - g_{3} D_{0} q_{0} - g_{4} {D_{0}}^{2} q_{0} \end{array}) + O (ε^{2})

(8)

Matching the coefficients of corresponding powers of $ε$ :

O (ε^{0})

({D_{0}}^{2} + {ω_{1}}^{2}) y_{0} = 0

(9)

({D_{0}}^{2} + {ω_{2}}^{2}) q_{0} = 0

(10)

O (ε)

({D_{0}}^{2} + {ω_{1}}^{2}) y_{1} = - 2 D_{0} D_{1} y_{0} - μ_{1} D_{0} y_{0} + α_{1} {y_{0}}^{2} - α_{2} {y_{0}}^{3} - γ_{1} D_{0} q_{0} - g_{1} D_{0} y_{0} - g_{2} {D_{0}}^{2} y_{0}

(11)

({D_{0}}^{2} + {ω_{2}}^{2}) q_{1} = - 2 D_{0} D_{1} q_{0} - μ_{2} D_{0} q_{0} + γ_{2} D_{0} y_{0} + f \cos Ω t - g_{3} D_{0} q_{0} - g_{4} {D_{0}}^{2} q_{0}

(12)

From eq. (9) & (10) Solving the homogeneous differential equations, we get:

y_{0} (T_{0}, T_{1}) = A (T_{1}) e^{i ω_{1} T_{0}} + \bar{A} (T_{1}) e^{- i ω_{1} T_{0}}

(13)

q_{0} (T_{0}, T_{1}) = B (T_{1}) e^{i ω_{2} T_{0}} + \bar{B} (T_{1}) e^{- i ω_{2} T_{0}}

(14)

Differentiate equations (13) and (14) with respect to t and substitute the results into equations (11) and (12):

({D_{0}}^{2} + {ω_{1}}^{2}) y_{1} = (- 2 i ω_{1} D A - μ_{1} i ω_{1} A - 3 α_{2} A^{2} \bar{A} - g_{1} i ω_{1} A + g_{2} {ω_{1}}^{2} A) e^{i ω_{1} T_{0}} + - γ_{1} i ω_{2} B e^{i ω_{2} T_{0}} + 2 α_{1} A \bar{A} + α_{1} A^{2} e^{2 i ω_{1} T_{0}} - α_{2} A^{3} e^{3 i ω_{1} T_{0}} + C C .

(15)

({D_{0}}^{2} + {ω_{2}}^{2}) q_{1} = (- 2 i ω_{2} D B - μ_{2} i ω_{2} B - g_{3} i ω_{2} B - g_{4} {ω_{2}}^{2} B) e^{i ω_{2} T_{0}} + γ_{2} i ω_{1} A e^{i ω_{1} T_{0}} + \frac{f}{2} e^{i Ω T_{0}} + C C .

(16)

By applying this condition in Equations (15) and (16), we can delete the new secular terms as follows.

Where CC. represents the collection of complex conjugated parts. After removing secular words, take the following forms:

(- 2 i ω_{1} D A - μ_{1} i ω_{1} A - 3 α_{2} A^{2} \bar{A} - g_{1} i ω_{1} A + g_{2} {ω_{1}}^{2} A) e^{i ω_{1} T_{0}} - γ_{1} i ω_{2} B e^{i ω_{2} T_{0}} = 0

(17)

(- 2 i ω_{2} D B - μ_{2} i ω_{2} B - g_{3} i ω_{2} B - g_{4} {ω_{2}}^{2} B) e^{i ω_{2} T_{0}} + γ_{2} i ω_{1} A e^{i ω_{1} T_{0}} + \frac{f}{2} e^{i Ω T_{0}} = 0

(18)

From the initial estimate, we identified the following resonance cases:

i. PR: $Ω ≅ ω_{1}$

ii. IR $ω_{2} = ω_{1}$

3.2. Episodic resolutions

In this section, the selected tone situation: $Ω ≅ ω_{2}, ω_{2} ≅ ω_{1}$ used to deliberate the solubility settings, we shall present detuning parameters $σ_{1} & σ_{2}$ so that:

\begin{array}{l} Ω = ω_{2} + ε σ_{2} \\ ω_{1} = ω_{2} + ε σ_{1} \end{array}}

(19)

Incorporating equation (19) along with the secular and small division terms from equations (17) and (18) to establish the solvability criteria as follows:

(- 2 i ω_{1} D A - μ_{1} i ω_{1} A - 3 α_{2} A^{2} \bar{A} - g_{1} i ω_{1} A + g_{2} {ω_{1}}^{2} A) - γ_{1} i ω_{2} B e^{i σ_{1} T_{1}} = 0

(20)

(- 2 i ω_{2} D B - μ_{2} i ω_{2} B - g_{3} i ω_{2} B - g_{4} {ω_{2}}^{2} B) + γ_{2} i ω_{1} A e^{- i σ_{1} T_{1}} + \frac{f}{2} e^{i σ_{2} T_{1}} = 0

(21)

To examine the solution of equations (20) and (21), we express A and B in polar form as follows:

A (T_{1}) = \frac{1}{2} a_{1} (T_{1}) e^{i θ_{1} (T_{1})}, D_{1} A (T_{1}) = \frac{1}{2} ({\dot{a}}_{1} (T_{1}) + i a_{1} {\dot{θ}}_{1} (T_{1})) e^{i θ_{1} (T_{1})}

(22)

B (T_{1}) = \frac{1}{2} a_{2} (T_{1}) e^{i θ_{2} (T_{1})}, D_{1} B (T_{1}) = \frac{1}{2} ({\dot{a}}_{2} (T_{1}) + i a_{2} {\dot{θ}}_{2} (T_{1})) e^{i θ_{2} (T_{1})}

(23)

Where

a_{1}

and

a_{2}

represent the phases and amplitudes of both the system and the controller once they have reached a stable, steady-state condition. This characterizes the long-term oscillatory behavior after transient effects have dissipated. Additionally,

ϕ_{1}

and

ϕ_{2}

denote the phases of the signal. Substituting equations (22) and (23) into (20) and (21) results in the following amplitude-phase evolution equations:

{\dot{a}}_{1} = - \frac{1}{2} μ_{1} a_{1} - \frac{1}{2} g_{1} a_{1} - \frac{γ_{1} ω_{2}}{2 ω_{1}} a_{2} \cos ϕ_{1}

(24)

a_{1} {\dot{θ}}_{1} = \frac{3}{8 ω_{1}} α_{2} {a_{1}}^{3} - \frac{1}{2} g_{2} ω_{1} a_{1} - \frac{γ_{1} ω_{2}}{2 ω_{1}} a_{2} \sin ϕ_{1}

(25)

{\dot{a}}_{2} = - \frac{1}{2} μ_{2} a_{2} - \frac{1}{2} g_{3} a_{2} + \frac{f}{2 ω_{2}} \sin ϕ_{2} + \frac{γ_{2} ω_{1}}{2 ω_{2}} a_{1} \cos ϕ_{1}

(26)

a_{2} {\dot{θ}}_{2} = - \frac{1}{2} g_{4} ω_{2} a_{2} - \frac{γ_{2} ω_{1}}{2 ω_{2}} a_{1} \sin ϕ_{1} - \frac{f}{2 ω_{2}} \cos ϕ_{2}

(27)

Whare,

ϕ_{2} = σ_{2} T_{1} - θ_{2}

ϕ_{1} = σ_{1} T_{1} + θ_{2} - θ_{1}

. Back to the foremost arrangement restrictions, we consume the succeeding equivalences:

{\dot{a}}_{1} = - \frac{1}{2} μ_{1} a_{1} - \frac{1}{2} g_{1} a_{1} - \frac{γ_{1} ω_{2}}{2 ω_{1}} a_{2} \cos ϕ_{1}

(28)

a_{1} {\dot{ϕ}}_{1} = (σ_{1} + σ_{2}) a_{1} - a_{1} {\dot{ϕ}}_{2} - \frac{3}{8 ω_{1}} α_{2} {a_{1}}^{3} + \frac{1}{2} g_{2} ω_{1} a_{1} + \frac{γ_{1} ω_{2}}{2 ω_{1}} a_{2} \sin ϕ_{1}

(29)

{\dot{a}}_{2} = - \frac{1}{2} μ_{2} a_{2} - \frac{1}{2} g_{3} a_{2} + \frac{f}{2 ω_{2}} \sin ϕ_{2} + \frac{γ_{2} ω_{1}}{2 ω_{2}} a_{1} \cos ϕ_{1}

(30)

a_{2} {\dot{ϕ}}_{2} = σ_{2} a_{2} + \frac{1}{2} g_{4} ω_{2} a_{2} + \frac{γ_{2} ω_{1}}{2 ω_{2}} a_{1} \sin ϕ_{1} + \frac{f}{2 ω_{2}} \cos ϕ_{2}

(31)

3.3. Frequency response equation

Equations (28) to (31) may possess a fixed point corresponding to a steady-state solution, which can be determined by setting the time derivatives equal to zero.: ${\dot{a}}_{1} = {\dot{a}}_{2} = {\dot{ϕ}}_{1} = {\dot{ϕ}}_{2} = 0$ .

\frac{γ_{1} ω_{2}}{2 ω_{1}} a_{2} \cos ϕ_{1} = - \frac{1}{2} μ_{1} a_{1} - \frac{1}{2} g_{1} a_{1}

(32)

- \frac{γ_{1} ω_{2}}{2 ω_{1}} a_{2} \sin ϕ_{1} = (σ_{1} + σ_{2}) a_{1} - \frac{3}{8 ω_{1}} α_{2} {a_{1}}^{3} + \frac{1}{2} g_{2} ω_{1} a_{1}

(33)

- \frac{f}{2 ω_{2}} \sin ϕ_{2} = - \frac{1}{2} μ_{2} a_{2} - \frac{1}{2} g_{3} a_{2} + \frac{γ_{2} {ω_{1}}^{2} a_{1}}{γ_{1} {ω_{2}}^{2} a_{2}} (- \frac{1}{2} μ_{1} a_{1} - \frac{1}{2} g_{1} a_{1})

(34)

- \frac{f}{2 ω_{2}} \cos ϕ_{2} = σ_{2} a_{2} + \frac{1}{2} g_{4} ω_{2} a_{2} + \frac{γ_{2} {ω_{1}}^{2} a_{1}}{γ_{1} {ω_{2}}^{2} a_{2}} [(σ_{1} + σ_{2}) a_{1} - \frac{3}{8 ω_{1}} α_{2} {a_{1}}^{3} + \frac{1}{2} g_{2} ω_{1} a_{1}]

(35)

By squaring Equations (34) and (35) and summing both sides, we obtain the following equation:

{(\frac{γ_{1} ω_{2} a_{2}}{2 ω_{1}})}^{2} = {(- \frac{1}{2} μ_{1} a_{1} - \frac{1}{2} g_{1} a_{1})}^{2} + {((σ_{1} + σ_{2}) a_{1} - \frac{3}{8 ω_{1}} α_{2} {a_{1}}^{3} + \frac{1}{2} g_{2} ω_{1} a_{1})}^{2}

(36)

\begin{array}{l} {(\frac{f}{2 ω_{2}})}^{2} = {(- \frac{1}{2} μ_{2} a_{2} - \frac{1}{2} g_{3} a_{2} + \frac{γ_{2} {ω_{1}}^{2} a_{1}}{γ_{1} {ω_{2}}^{2} a_{2}} (- \frac{1}{2} μ_{1} a_{1} - \frac{1}{2} g_{1} a_{1}))}^{2} \\ + {(σ_{2} a_{2} + \frac{1}{2} g_{4} ω_{2} a_{2} + \frac{γ_{2} {ω_{1}}^{2} a_{1}}{γ_{1} {ω_{2}}^{2} a_{2}} ((σ_{1} + σ_{2}) a_{1} - \frac{3}{8 ω_{1}} α_{2} {a_{1}}^{3} + \frac{1}{2} g_{2} ω_{1} a_{1}))}^{2} \end{array}}

(37)

Equations describing the system’s frequency response are used to study the behavior of steady-state solutions under typical operating conditions ( $a_{1} \neq 0, a_{2} \neq 0$ ).

3.4. Constancy examination through linearizing the overhead structure

To assess the stability of the equilibrium solution, the eigenvalues of the Jacobian matrix corresponding to the given equations were analyzed. If all eigenvalues possess negative real parts, the equilibrium is asymptotically stable. Conversely, if at least one eigenvalue has a positive real part, the equilibrium becomes unstable. Stability analysis focuses on the behavior of small perturbations around steady-state solutions $a_{10}, a_{20}, ϕ_{10}$ and $ϕ_{20}$ . Accordingly, we assume that:

\begin{array}{l} a_{1} = a_{11} + a_{10}, a_{2} = a_{21} + a_{20}, ϕ_{1} = ϕ_{11} + ϕ_{10}, ϕ_{2} = ϕ_{21} + ϕ_{20}, \\ {\dot{a}}_{1} = {\dot{a}}_{11}, {\dot{a}}_{2} = {\dot{a}}_{21}, {\dot{ϕ}}_{1} = {\dot{ϕ}}_{11}, {\dot{ϕ}}_{2} = {\dot{ϕ}}_{21} . \end{array}}

(38)

where

a_{10}, a_{20}, ϕ_{10}

and

ϕ_{20}

satisfy equations (28) and (31),

a_{11}, a_{21}, ϕ_{11}

and

ϕ_{21}

represent small perturbations relative to

a_{10}, a_{20}, ϕ_{10}

and

ϕ_{20}

. By substituting equation (38) into equations (28)–(31), expanding for small

a_{11}, a_{21}, ϕ_{11}

and

ϕ_{21}

, and retaining only the linear terms, we obtain:

{\dot{a}}_{11} = r_{11} a_{11} + r_{12} ϕ_{11} + r_{13} a_{21} + r_{14} ϕ_{21}

(39)

{\dot{ϕ}}_{11} = r_{21} a_{11} + r_{22} ϕ_{11} + r_{23} a_{21} + r_{24} ϕ_{21}

(40)

{\dot{a}}_{21} = r_{31} a_{11} + r_{32} ϕ_{11} + r_{33} a_{21} + r_{34} ϕ_{21}

(41)

{\dot{ϕ}}_{21} = r_{41} a_{11} + r_{42} ϕ_{11} + r_{43} a_{21} + r_{44} ϕ_{21}

(42)

where

r_{i j}, i = 1, 2, 3, 4

and

j = 1, 2, 3, 4

are provided in the Appendix.

Calculations (39) through (42) can be expressed in the following matrix form:

{[{\dot{a}}_{11} {\dot{ϕ}}_{11} {\dot{a}}_{21} {\dot{ϕ}}_{21}]}^{T} = [J] {[a_{11} ϕ_{11} a_{21} ϕ_{21}]}^{T}

(43)

[J] = [\begin{array}{l} r_{11} r_{12} r_{13} r_{14} \\ r_{21} r_{22} r_{23} r_{24} \\ r_{31} r_{32} r_{33} r_{34} \\ r_{41} r_{42} r_{43} r_{44} \end{array}]

(44)

[J]

is the Jacobian matrix.

As defined in the Appendix, the elements $r_{i j}$ of the Jacobian in Eq. (44) are obtained as the partial derivatives of the system outputs with respect to its inputs, representing the rate of change of each output relative to each input. Therefore, the stability of the steady-state solutions is governed by the eigenvalues of the Jacobian matrix. The corresponding eigenvalue equation can be derived as follows:

| \begin{array}{l} r_{11} - λ r_{12} r_{13} r_{14} \\ r_{21} r_{22} - λ r_{23} r_{24} \\ r_{31} r_{32} r_{33} - λ r_{34} \\ r_{41} r_{42} r_{43} r_{44} - λ \end{array} | = 0

(45)

wherever the following polynomial’s roots are located:

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

(46)

The appendix contains pigeon-holed quantities from equation (46). Aimed to abovementioned classification’s key to be firm, the Routh-Hurwitz criterion³⁹ must be met. This means:

Γ_{1} > 0, Γ_{1} Γ_{2} - Γ_{3} > 0, Γ_{3} (Γ_{1} Γ_{2} - Γ_{3}) - Γ_{1}^{2} Γ_{4} > 0, Γ_{4} > 0

(47)

4. Results and discussions

4.1. Time history presentation without and with (NVC+NAC) controller

This study looks at the stability of an enforced self-excited nonlinear ray system with the (NIMLS). The system’s behavior is numerically investigated using the fourth-order Runge-Kutta method (MATLAB’s ode45 occupation). The outcomes remain obtainable realistically, graphing steady-state profusions against detuning parameters $σ_{1}, σ_{2}$ , with defined system parameter values⁷:

(μ_{1} = 0.1; μ_{2} = 0.1; α_{1} = 0.5; α_{2} = 0.7; γ_{1} = 0.05; γ_{2} = 1.5; Ω = 1; ω_{1} = ω_{2} = 1; ε = 1; f = 5; g_{1} = 3; g_{2} = 3; g_{3} = 2; g_{4} = 2)

(48)

Figure 2 Steady-state amplitudes and the corresponding Poincaré section diagrams of the system before and after applying the combined NVC and NAC controllers at the most critical excitation frequencies (approximately 1.7 and 42.3, respectively). The addition of NVC and NAC controllers considerably reduces system amplitudes to 0.04 and 1.2, respectively. This displays the efficacy of the NVC+NAC controllers. Ea. = 43, Ea. = 36, and vibrations are decreased by roughly 97.8% associated with their rate deprived of the regulator. The numerical integration for the middle group of images was performed over a dimensionless time interval of 0–1000 with a step size of 5, clearly showing the gradual attenuation of oscillations as the velocity approaches zero.

Figure 2.

The major system’s amplitude before (blue) and after (red) the addition of the NVC+NAC controller.

4.2. Frequency response curves (FRC)

The response amplitude of the system is governed by both the detuning parameter $σ_{1}, σ_{2}$ and the excitation amplitude $f$ . Equations (28) and (31) are solved numerically to obtain the steady-state amplitude responses of the main structure equipped with the combined controller (NVC+NAC) as functions of the detuning parameter $σ_{1}, σ_{2}$ . Figures 3 and 4 depict the frequency response (FR) curves of the first and second modes, respectively. In Figure 3, the second mode detuning parameter $σ_{2}$ is intentionally set to zero in order to isolate the dynamic behavior of the first mode and eliminate modal coupling effects. The solid curves represent stable steady-state solutions. Notably, the slight discontinuities observed in the FR curves correspond to jump phenomena, which arise due to the coexistence of multiple solution branches inherent to the nonlinear dynamics of the magnetically levitated system. These abrupt transitions occur when the system response switches between stable branches as the excitation frequency crosses critical detuning values. The presence of such jump phenomena is a characteristic feature of nonlinear resonance and reflects the influence of nonlinear stiffness and modal interaction. The minimum vibration amplitudes of the primary structure $σ_{2} = 0$ are achieved at specific detuning values for both modes, $σ_{1} = 0$ demonstrating the effectiveness of the proposed controller in suppressing vibrations under primary and 1:1 internal resonance condition. Furthermore, increasing the harmonic excitation force leads to the emergence of multiple resonance peaks, indicating a multi resonance phenomenon associated with enhanced modal interactions.

Figure 3.

Resonance curves for the main system with (NVC+NAC) controller (parameter as in equation (48)).

Figure 4.

Frequency retort curve illustrating system amplitude variation across changed values of external force f on the focal structure with (NVC+NAC) organizer $f = 7, f = 6, f = 5$ .

Figures 5–8 demonstrate the influence of the velocity feedback gain $g_{1}$ and the acceleration feedback gain $g_{2}$ on system behavior. In Figure 5, increasing $g_{1}$ results in a monotonic decrease in the vibration amplitude of the first mode. In Figure 6, increasing $g_{2}$ causes a leftward shift in the response curves. Figures 7 and 8 show $g_{3} a n d g_{4}$ similar trends for the second mode, confirming that higher negative feedback gains enhance the controller’s vibration suppression performance.

Figure 5.

Frequency retort curve illustrating system amplitude variation across changed values of rheostat indication expansion $g_{1}$ on system with (NVC+NAC) supervisor $(g_{1} = 2; g_{1} = 3; g_{1} = 4)$ .

Figure 6.

Frequency retort curve illustrating system amplitude variation across changed values of rheostat signal gain $g_{2}$ on system with (NVC+NAC) regulator $(g_{2} = 3; g_{2} = 2; g_{2} = 1)$ .

Figure 7.

Incidence retort arc illustrating system plenty variation across changed values of relation $g_{3}$ on system with (NVC+NAC) regulator $(g_{3} = 2; g_{3} = 3; g_{4} = 4)$ .

Figure 8.

The regularity comeback arch depicts the variation in system amplitude corresponding to different values of the rheostat gain. $g_{4}$ on classification with (NVC+NAC) controller $(g_{4} = 2; g_{4} = 3; g_{4} = 4)$ .

Figures 9 and 10 illustrate the impact of the damping coefficient $μ_{1}, μ_{2}$ , where higher damping values lead to a reduction in amplitude responses. Figures 11 and 12 show that as the external excitation amplitude $γ_{1}, γ_{2}$ increases, the amplitude of the controlled system also increases. These results highlight the importance of optimizing controller parameters under different excitation conditions. The slight discontinuities observed in the curves correspond to jump phenomena inherent to the nonlinear characteristics of the magnetically levitated system. Such behavior is typical in nonlinear dynamic systems.

Figure 9.

Incidence retort arc illustrating structure amplitude variation across changed values of the hampering factor $μ_{1}$ on the system with (NVC+NAC) regulator $(μ_{1} = 0.1, 0.5, 1)$ .

Figure 10.

The incidence rejoinder bow shows how the structure’s breadth varies with changed ideals of the damping coefficient $μ_{2}$ on the system with (NVC+NAC) controller $(μ_{2} = 0.1, 0.3, 0.5)$ .

Figure 11.

Frequency retort arc illustrating system amplitude difference across changed values of nonlinear parameter $γ_{1}$ on system with (NVC+NAC) supervisor $(γ_{1} = 0.05, 0.2, 0.5)$ .

Figure 12.

The occurrence reply arc depicts variation in classification corresponding to diverse ethics of the nonlinear parameter $γ_{2}$ on the system with (NVC+NAC) controller $(γ_{2} = 1.5, 1, 0.5)$ .

Figure 13 compares three distinct values of the system parameter $σ_{2}$ , showing that the minimum vibration amplitude occurs when $σ_{1} = σ_{2} = 0$ , indicating optimal controller tuning. Figure 14 demonstrates that increasing the natural frequency $ω_{1}, ω_{2}$ leads to a reduction in the amplitude of the controlled system, confirming the stabilizing effect of system stiffness.

Figure 13.

Result of changeable $σ_{2}$ on the system with (NVC+NAC) controller $(σ_{2} = - 0.5, 0, 0.5)$ .

Figure 14.

Outcome of changing $ω_{1} = ω_{2}$ on the system with (NVC+NAC) controller $(ω_{1} = ω_{2} = 1, 1.5, 2)$ .

Finally, Figure 15 presents the time-domain responses of the uncontrolled and controlled systems under different excitation conditions $f = 5, f = 0.5, f = 50$ . The results clearly show that the closed-loop $f = 5$ The configuration achieves minimal oscillations and faster convergence compared with the open-loop case. The broadband performance of the proposed controlled magnetic levitation system is quantitatively evaluated using the −3 dB bandwidth criterion. As shown in Figure 3(a), the peak response reaches approximately 0.027, and the corresponding −3 dB threshold is estimated as 0.019. The system maintains a response level above this threshold over an excitation frequency range of Ω ∈ [−2, 1.5], indicating a broadband dynamic behavior rather than a narrow resonance. Moreover, the smooth variation of the response curve and the absence of abrupt jumps or instability across this frequency range demonstrate the robustness of the proposed negative velocity and acceleration control strategy against excitation frequency variations.

Figure 15.

Outcome among the changed kinds of force $f$ on an uncontrolled system $(f = 5, f = 0.5, f = 50)$ .

5. Comparison

5.1. Evaluation of the time history beforehand and afterward the controller

Figure 16 compares the vibration suppression performance of different control strategies. The combined controller (NVC+NAC) consistently achieves the lowest vibration amplitudes across all configurations. Compared to the uncontrolled system and single-controller cases, it provides significant vibration attenuation in both modes, demonstrating superior efficiency and robustness.

Figure 16.

Outcome among the changed kinds of controllers.

Figure 17 and Table 1 highlight the controller’s effectiveness under various resonance scenarios. Primary and internal resonance cases generate the most severe vibrations in the uncontrolled system. With the (NVC+NAC) controller, vibration amplitudes in the first and second modes are significantly reduced, with reductions exceeding 90% in some cases. Table 1 further quantifies these improvements, showing notable decreases in both vibration energy and response amplitudes.

Figure 17.

Outcome among the changed kinds of Resonance cases.

Table 1.

Outcome of different types of resonance cases on the system.

Resonance cases	First mode of the system (y)		Second mode of the system (q)		Ea
Resonance cases	Without	With	Without	With	First	Second
$Ω = 1, ω_{1} = ω_{2} = 1$	1.7	0.04	42.3	1.2	43	36
$Ω = 3, ω_{1} = ω_{2} = 1$	0.01	0.005	0.6	0.1	2	6
$Ω = \frac{1}{3}, ω_{1} = ω_{2} = 1$	0.11	0.06	5.6	5.1	2	1

Primary resonance occurs when the excitation frequency approaches the system’s natural frequency, while internal resonance results from modal coupling at specific frequency ratios. The simultaneous presence of both resonances produces the largest dynamic responses, representing a worst-case operating condition. These scenarios are particularly relevant for flexible mechanical and aerospace structures operating near critical speed ranges, where modal interactions are significant.

5.2. Evaluation among trepidation resolution and numerical reproduction

Figures 18 and 19 show an adjacent contract between numerical and estimated findings aimed at equally distributing the uncontrolled and controlled systems using the (NVC+NAC) controller. This substantial association verifies both techniques’ correctness and dependability in recording system behavior under a variety of control settings. The alignment observed in these comparisons demonstrates the steadfastness and effectiveness of the numerical and approximation procedures in analyzing arrangement dynamics.

Figure 18.

Vibration amplitude of the uncontrolled main system.

Figure 19.

Contrast between the numerical solution (ـــــــــ) and the approximate solution (………).

5.3. Assessment through previous work

Reference 7 explores energy harvesting technology, which generates electricity for small devices using magnetic oscillations. It focuses on a (NIMLS), evaluating its energy-harvesting potential while considering the weight of the oscillating central block. The device consists of a magnetic levitation setup linked to an electrodynamic shaker that excites the base, inducing oscillations. The resulting model is a highly nonlinear coupled equation similar to the Duffing oscillator. The Runge-Kutta method is employed to analyze the motion of the central block and the harvested power. The study presents phase planes, Poincaré maps, and parametric variations. Analyzing the system’s time history and key performance metrics demonstrates the usefulness of the applied (NVC+NAC) regulator in minimizing sensations. Findings indicate a significant reduction in vibration amplitude—approximately 97%—with a control effort of around 43 and 36. A stout association between numerical and approximate explanations confirms the accuracy of the approach. This comparison further demonstrates that the proposed (NVC+NAC) controller outperforms existing approaches reported in the literature, both in terms of vibration attenuation and control efficiency.

To clearly position the present study within the existing body of literature, Table 2 provides a structured comparison with closely related works. The comparison highlights differences in system configuration, source of nonlinearity, modeling approach, and control strategy. This summary clarifies the distinct contribution of the proposed magnetic levitation system incorporating NVC–NAC feedback in contrast to previously reported nonlinear vibration and energy harvesting systems.

Table 2.

Comparative analysis between the present magnetic levitation system and related nonlinear vibration and energy harvesting studies.

Reference	System type	Modeling approach	Control method	Dynamic analysis scope	Key differences from the present study
Lu et al. (2024)³³	Nonlinear vibration isolation system	Lumped nonlinear isolator model	Passive	Frequency response and isolation performance	Does not investigate bifurcation structure, chaos, or active feedback-based regulation.
Marzbanrad et al. (2018)³⁴	Piezoelectric FG nanobeam	Continuum-based nonlinear beam model	No active control	Nonlinear vibration response under multi-field loading	Focuses on nanoscale multi-physical effects without bifurcation-based control of macroscopic systems.
Cui et al. (2024)³⁵	Bridge deck VIV system	Generalized van der Pol phenomenological model	No explicit feedback control	Multistability and nonlinear oscillations	Employs phenomenological modeling without physically derived magnetic levitation dynamics or dual feedback stabilization.
Present Study	Magnetic levitation energy harvester	Physically derived nonlinear magnetic levitation model	Active NVC–NAC feedback	Frequency response and Bifurcation diagrams, chaos classification	Integrates dual feedback control with bifurcation-based regulation to achieve controlled multistability and enhanced energy harvesting.

6. Chaotic behavior and bifurcation analysis

This section investigates chaotic behavior as an essential aspect of understanding the nonlinear dynamic response of the system. As the bifurcation parameter varies, the system undergoes a sequence of qualitative transitions, evolving from periodic motion to quasi-periodic oscillations, followed by the emergence of chaotic-like behavior. With further variation of the parameter, the system recovers a quasi-periodic response before ultimately settling back into a stable periodic state. This non-monotonic transition scenario reflects the complex interplay between nonlinearity, damping, and control effects and is a characteristic feature of many nonlinear dynamical systems.

The analysis is carried out using the dimensionless governing equations (3) and (4), which are reformulated into an equivalent set of first-order state-space equations for numerical implementation. The resulting system is examined using MATLAB to generate bifurcation diagrams, phase portraits, and Poincaré maps that provide complementary insight into the system dynamics.

As illustrated in Figure 20, the bifurcation diagram captures the evolution of the system response with respect to variations in the bifurcation parameter and highlights critical transitions associated with changes in system stability. The appearance of irregular and aperiodic responses following more regular oscillatory motion suggests the onset of chaotic-like dynamics, commonly associated with mechanisms such as frequency interaction and the breakdown of invariant structures in phase space.

Figure 20.

Bifurcation diagrams of $y (t)$ and $q (t)$ vs $Ω$ for $g_{1} = g_{2} = g_{3} = g_{4} = 0.01$ .

As clearly illustrated in Figure 21, the bifurcation diagram exhibits a wide spectrum of nonlinear dynamic responses as the bifurcation parameter $Ω$ varies over a broad range. For small values of the parameter (0 ≤ $Ω$ ≤ 0.4), the system response is dominated by quasi-periodic oscillations, as shown in Figure 21. This behavior is characterized by the coexistence of multiple incommensurate frequencies, indicating the presence of a torus in the phase space and reflecting a weakly modulated oscillatory motion induced by nonlinear interactions between the primary structure and the attached subsystem.

Figure 21.

Phase portraits and Poincare maps of the periodic state at $Ω = 0.15$ .

At specific parameter values, namely $Ω$ = 0.5 and $Ω$ = 1, the system undergoes a qualitative transition to stable periodic oscillations, forming narrow periodic windows embedded within the quasi-periodic regime, as depicted in Figure 22. These periodic windows arise due to frequency locking phenomena, where the interacting frequencies become synchronized, resulting in regular and repeatable motion.

Figure 22.

Phase portraits and Poincare maps of the periodic state at $Ω = 0.5$ .

With a further increase in the bifurcation parameter (0.6 ≤ $Ω$ ≤ 1.22), the system predominantly maintains a quasi-periodic response, as illustrated in Figure 23. In this region, the nonlinear coupling remains sufficiently strong to prevent full synchronization, while the system preserves bounded and non-divergent trajectories in the phase space.

Figure 23.

Phase portraits and Poincare maps of the periodic state at $Ω = 0.7$ .

A pronounced transition to chaotic-like dynamics is observed within the narrow interval 1.23 ≤ $Ω$ ≤ 1.26, as shown in Figure 24. This regime is characterized by a highly irregular response, extreme sensitivity to initial conditions, and a dense distribution of points in the bifurcation diagram, suggesting a breakdown of the invariant torus and the onset of chaotic motion due to strong nonlinear effects.

Figure 24.

Phase portraits and Poincare maps of the periodic state at $Ω = 1.26$ .

Beyond this chaotic region, the system re-enters a quasi-periodic state for 1.27 ≤ $Ω$ ≤ 1.29, as depicted in Figure 25. This recovery indicates that the chaotic attractor loses stability and is replaced by a torus-like structure, highlighting the complex interplay between nonlinearity and damping in governing the system dynamics.

Figure 25.

Phase portraits and Poincare maps of the periodic state at $Ω = 1.28$ .

Finally, for larger values of the bifurcation parameter (2 ≤ $Ω$ ≤ 3), the system converges to stable periodic oscillations, as shown in Figure 26. In this regime, dissipative effects and control-induced stabilization dominate the system response, leading to regular, repeatable oscillations and enhanced dynamic stability.

Figure 26.

Phase portraits and Poincare maps of the periodic state at $Ω = 2$ .

7. Conclusions

This study investigated the vibration control of NIMLS interacting with another body, introducing strong nonlinearities and complex dynamic behavior. The system was modeled using coupled differential equations, and approximate analytical solutions were obtained, alongside numerical simulations, for validation and visualization of the system responses. Frequency response analysis provided insights into the dynamic characteristics and effectiveness of the proposed control strategy, while sensitivity analysis examined the influence of parameter variations on performance and supported controller optimization. Stability analysis ensured reliable system operation and prevented uncontrolled oscillations. The key findings of this study are summarized as follows:

➢ The simultaneous occurrence of primary and internal resonances represents $Ω ≅ ω_{2}, ω_{2} ≅ ω_{1}$ one of the most severe conditions in the system.

➢ Implementation of NVC+NAC reduced the vibration amplitude by approximately 97% compared to the uncontrolled case.

➢ The combined controller NVC+NAC demonstrated effectiveness between 36–43, successfully suppressing vibrations in the primary structure.

➢ Controlled systems exhibited enhanced stability under increasing external excitation forces $f$ .

➢ Increasing the negative linear velocity coefficient $g_{1}, g_{3}$ led to a reduction in the main structure’s response amplitude, while higher negative linear acceleration coefficients $g_{4}$ further decreased the primary response.

➢ The resonance peak shifted leftward with increasing NVC+NAC values $g_{2}$ , and the amplitude of the primary system decreased monotonically with both damping coefficient $μ_{1}, μ_{2}$ and natural frequency $ω_{1} = ω_{2}$ .

➢ Numerical and approximate analytical solutions were in close agreement, with frequency response curves showing strong consistency between the analytical method (FRC) and the 4th-order Runge–Kutta simulations.

➢ Specific delayed coupling conditions resulted in the smallest vibration amplitudes $σ_{1} = σ_{2}$ , indicating optimal control performance.

➢ The mathematical model of relative displacement under the (NVC+NAC) strategy exhibited only slight peak overshoot, confirming the controller’s accuracy and responsiveness.

➢ Integration of NVC and NAC controllers substantially enhances vibration suppression, achieving faster stabilization and lower peak responses than conventional methods.

➢ System parameters, including the levitated core’s mass and magnetic field strength, directly influence energy harvesting efficiency, which is maximized under controlled vibrations.

In conclusion, the proposed NVC+NAC controller effectively mitigates complex nonlinear vibrations in magnetically levitated systems, ensuring both vibration suppression and dynamic stability under varying excitations. These results provide valuable insights for the design and operation of nonlinear magnetic levitation systems. Future work will extend this approach to multi-degree-of-freedom and time-delay systems and explore adaptive and intelligent control schemes to further enhance performance and robustness. It should be noted that the stability conclusions presented here are primarily based on time histories and frequency response analysis. Formal nonlinear stability verification (e.g., Lyapunov or bifurcation methods) is not performed in this study. Therefore, the reported “enhanced stability” refers to the observed vibration suppression in simulations. Future work may include rigorous nonlinear stability analysis to fully validate these findings.

7.1. Future work

Applying some different approximate methods like the harmonic method, given the challenges of traditional control methods, we will explore the effectiveness of fractional-order feedback control for suppressing vibrations in the nonlinear dynamical system. Applying some different numerical methods like spectral method. Make comparisons between theoretical research and experimental results. Make simulation models.

Footnotes

ORCID iD

Ashraf Taha EL-Sayed

Author contributions

A.T.E.-S: Conceptualization, Investigation, resources, software, methodology, writing—original draft preparation, visualization, reviewing, and editing. R. K. H.: Investigation, Methodology, Formal Analysis, reviewing and editing, and funding acquisition. Y.A.A.: Investigation, methodology, data curation, validation, reviewing, and editing. M.A.E.: Methodology, software, validation, data curation, reviewing, and editing. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIU-DDRSP2603).

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 generalized coordinate of the middle oscillator block is Y, and the electrical part of the shaker is an analogy force voltage based on the law of Kirchhoff loops, where Q is given by the instantaneous electrical charge.

The kinetic energy is given by the motion of the middle block and the inductance of the shaker

T = \frac{1}{2} [m {\dot{Y}}^{2} + L_{s} {\dot{Q}}^{2}]

where.

L_s represent the inductance of the electrodynamical device

The potential energy is given by the gravitational potential energy of the middle block

V = m g Y + \frac{1}{2} Y^{2} (K_{m a g b} - K_{m a g t}) + S \dot{Q} Y + \frac{1}{2} \frac{Q^{2}}{C_{s}}

C_{s}

represent the capacitance of the electrodynamical device

The Rayleigh–Ritz procedure

D = \frac{1}{2} [c_{m} {\dot{Y}}^{2} + c_{3} {\dot{Y}}^{2} + R_{s} {\dot{Q}}^{2}]

R_{s}

represent the resistance of the electrodynamical device

The Lagrangian function is obtained

L = T - V = \frac{1}{2} [m {\dot{Y}}^{2} + L_{s} {\dot{Q}}^{2}] - m g Y + \frac{1}{2} Y^{2} (K_{m a g b} - K_{m a g t}) + S \dot{Q} Y + \frac{1}{2} \frac{Q^{2}}{C_{s}}

The equations of motion are derived by applying Lagrange’s equations.

\frac{d}{d t} (\frac{\partial L}{\partial \dot{Y}}) - (\frac{\partial L}{\partial Y}) + (\frac{\partial D}{\partial \dot{Y}}) = 0

\frac{d}{d t} (\frac{\partial L}{\partial \dot{Q}}) - (\frac{\partial L}{\partial Q}) + (\frac{\partial D}{\partial \dot{Q}}) = F_{e x t}

m \ddot{Y} + m g + Y (K_{m a g b} - K_{m a g t}) + S \dot{Q} + c_{m} \dot{Y} + c_{3} \dot{Y} = 0

L_{s} \ddot{Q} - S \dot{Y} + \frac{Q}{C_{s}} + R_{s} \dot{Q} = F \cos (Ω t)

Let:

F (Y) = k_{1} Y + k_{3} Y^{3}

m g = k_{1} Y_{0} + k_{3} Y_{0}^{3}

Y = y - Y_{0}, Q = q

\dot{Y} = \dot{y}, \ddot{Y} = \ddot{y}

Yields.

m \ddot{y} + (k_{1} + 3 k_{3} Y_{0}^{2}) y - 3 k_{3} Y_{0} y^{2} + k_{3} y^{3} + S \dot{q} + (c_{m} + c_{3}) \dot{y} = 0

L_{s} \ddot{q} - S \dot{y} + \frac{q}{C_{s}} + R_{s} \dot{q} = F \cos (Ω t)

Then

\ddot{y} + μ_{1} \dot{y} + {ω_{1}}^{2} y - α_{1} y^{2} + α_{2} y^{3} + γ_{1} \dot{q} = 0

\ddot{q} + μ_{2} \dot{q} + {ω_{2}}^{2} q - γ_{2} \dot{y} = f \cos Ω t

where

μ_{1} = \frac{(c_{m} + c_{3})}{m}, ω_{1}^{2} = \frac{(k_{1} + 3 k_{3} Y_{0}^{2})}{m}, α_{1} = \frac{3 k_{3} Y_{0}}{m}, α_{2} = \frac{k_{3}}{m}, γ_{1} = \frac{S}{m}

μ_{2} = \frac{R_{s}}{L_{s}}, ω_{2}^{2} = \frac{1}{L_{s} C_{s}}, γ_{2} = \frac{S}{L_{s}}, f = \frac{F}{L_{s}}

r_{11} = - \frac{1}{2} μ_{1} - \frac{1}{2} g_{1}, r_{12} = \frac{γ_{1} ω_{2}}{2 ω_{1}} a_{20} \sin ϕ_{10}, r_{13} = - \frac{γ_{1} ω_{2}}{2 ω_{1}} \cos ϕ_{10}, r_{14} = 0

r_{21} = \frac{(σ_{1} + σ_{2})}{a_{10}} - \frac{9}{8 ω_{1}} α_{2} a_{10} - \frac{g_{2} ω_{1}}{2 a_{10}} - r_{41}, r_{22} = \frac{γ_{1} ω_{2}}{2 ω_{1} a_{10}} a_{20} \cos ϕ_{10} - r_{42}, r_{23} = \frac{γ_{1} ω_{2}}{2 ω_{1} a_{10}} \sin ϕ_{10} - r_{43}, r_{24} = - r_{44}, r_{31} = \frac{γ_{2} ω_{1}}{2 ω_{2}} \cos ϕ_{10}, r_{32} = - \frac{γ_{2} ω_{1}}{2 ω_{2}} a_{10} \sin ϕ_{10}, r_{33} = - \frac{1}{2} μ_{2} - \frac{1}{2} g_{3}, r_{34} = \frac{f}{2 ω_{2}} \cos ϕ_{20}, r_{41} = \frac{γ_{2} ω_{1}}{2 ω_{2} a_{20}} \sin ϕ_{10}, r_{42} = \frac{γ_{2} ω_{1}}{2 ω_{2} a_{20}} a_{10} \cos ϕ_{10}, r_{43} = \frac{σ_{2}}{a_{20}} + \frac{1}{2 a_{20}} g_{4} ω_{2}, r_{44} = - \frac{f}{2 ω_{2} a_{20}} \sin ϕ_{20}

Γ_{1} = - (r_{11} + r_{22} + r_{33} + r_{44}), Γ_{2} = r_{22} (r_{11} + r_{33} + r_{44}) + r_{44} (r_{11} + r_{33}) + r_{11} r_{33} - r_{12} r_{21} - r_{13} r_{31} - r_{14} r_{41} - r_{24} r_{42} - r_{34} r_{43},

Γ_{3} = r_{11} (r_{24} r_{42} + r_{34} r_{43} - r_{22} (r_{33} + r_{44}) - r_{33} r_{44}) + r_{22} (r_{13} r_{31} + r_{14} r_{41} - r_{33} r_{44} + r_{34} r_{43}) + r_{33} (r_{12} r_{21} + r_{14} r_{41} + r_{24} r_{42}) + r_{44} (r_{12} r_{21} + r_{13} r_{31}) + r_{12} (r_{23} r_{31} + r_{24} r_{41}) + r_{14} (r_{21} r_{42} + r_{31} r_{43}) + r_{34} (r_{13} r_{41} + r_{23} r_{42}),

Γ_{4} = r_{11} (r_{22} (r_{33} r_{44} - r_{34} r_{43}) - r_{42} (r_{24} r_{33} + r_{23} r_{34})) - r_{22} (r_{41} (r_{14} r_{33} + r_{13} r_{34}) + r_{31} (r_{13} r_{44} + r_{14} r_{43})) - r_{33} (r_{12} (r_{21} r_{44} + r_{24} r_{41}) + r_{14} r_{21} r_{42}) - r_{12} (r_{31} (r_{23} r_{44} + r_{24} r_{43}) - r_{34} (r_{21} r_{43} - r_{23} r_{41})) + r_{42} (r_{31} (r_{13} r_{24} - r_{14} r_{23}) - r_{13} r_{21} r_{34})

References

Khatun

Sharma

Sarma

. Investigation of ambient vibration sources for direct energy harvesting by optimizing resonant frequency using proof mass. Measurement Science and Technology 2024; 35(5): 055101.

Hussain

Cheng

Cao

, et al. Numerical analysis and experimental validation of nonlinear broadband monostable and bistable energy harvesters. Communications in Nonlinear Science and Numerical Simulation 2024; 131: 107808.

Leng

, et al. Nonlinear dynamics and performance enhancement strategies for the magnetic levitating Bistable electromagnetic energy harvester. Journal of Vibration Engineering & Technologies 2024; 12(3): 3963–3976.

Francis

Kandil

ElSaid

, et al. Investigation of the weight effect on the oscillations of a magnetically-levitated body coupled to an energy harvester. Journal of Physics: Conference Series 2024; 2793(1): 012003.

Ribeiro

Balthazar

Daum

, et al. Nonlinear dynamics behaviour and its control under frequency-varying excitations for energy harvesting. International Journal of Nonlinear Dynamics and Control 2023; 2(3): 219–238.

Hajradinović

Zuković

. Numerical analysis of steady state and transient motion of a vibro-impact system with non-ideal excitation with a nonlinear spring. In: Advanced Technologies, Systems, and Applications VI. Springer, 2022, pp. 575–590.

Rocha

Balthazar

Tusset

, et al. On a non-ideal magnetic levitation system: nonlinear dynamical behavior and energy harvesting analyses. Nonlinear Dynamics 2019; 95(4): 3423–3438.

Amer

Abd El-Salam

El-Sayed

. Behavior of a Hybrid Rayleigh-Van der Pol-Duffing Oscillator with a PD Controller. Journal of applied research and technology 2022; 20(1): 58–67.

Alluhydan

Moatimid

Galal

, et al. Inspection of a Time-Delayed Excited Damping Duffing Oscillator. Axioms 2024; 13(6): 416.

10.

Jamshidi

Collette

. Optimal negative derivative feedback controller design for collocated systems based on H2 and H∞ method. Mechanical Systems and Signal Processing 2022; 181: 109497.

11.

Amer

Abd El-Salam

. Negative Derivative Feedback Controller for Repressing Vibrations of the Hybrid Rayleigh-Van der Pol-Duffing Oscillator. Nonlinear Phenomena in Complex Systems 2022; 25(3): 217–228.

12.

Amer

El-Sayed

Ahmed

EEE

. Vibration reduction of vertical conveyor system via negative cubic velocity feedback under external and parametric excitations. Journal of Mechanical Science and Technology 2022; 36(2): 543–551.

13.

Hamed

Aly

Saleh

, et al. Vibration performance, stability and energy transfer of wind turbine tower via PD controller. Computers, Materials & Continua 2020; 64(2): 871–886.

14.

Bauomy

El-Sayed

. Act of nonlinear proportional derivative controller for MFC laminated shell. Physica Scripta 2020; 95(9): 095210.

15.

Hamed

Alotaibi

El-Zahar

. Nonlinear vibrations analysis and dynamic responses of a vertical conveyor system controlled by a proportional derivative controller. IEEE Access 2020; 8: 119082–119093.

16.

Alluhydan

Amer

El-Sayed

, et al. Controlling the Generator in a Series of Hybrid Electric Vehicles Using a Positive Position Feedback Controller. Applied Sciences 2024; 14(16): 7215.

17.

Alluhydan

Amer

El-Sayed

, et al. The Impact of the Nonlinear Integral Positive Position Feedback (NIPPF) Controller on the Forced and Self-Excited Nonlinear Beam Flutter Phenomenon. Symmetry 2024; 16(9): 1143.

18.

Omidi

Nima Mahmoodi

. Multimode modified positive position feedback to control a collocated structure. Journal of Dynamic Systems, Measurement, and Control 2015; 137(5): 051003.

19.

Abdelhafez

Nassar

. Suppression of vibrations of a forced and self-excited nonlinear beam by using positive position feedback controller PPF. British Journal of Mathematics & Computer Science 2016; 17(4): 1–19.

20.

Jun

. Positive position feedback control for high-amplitude vibration of a flexible beam to a principal resonance excitation. Shock and Vibration 2010; 17(2): 187–203.

21.

Zhu

Yue

Liu

, et al. Active vibration control for piezoelectric cantilever beam: an adaptive feedforward control method. Smart Materials and Structures 2017; 26(4): 047003.

22.

Fyrillas

Szeri

. Control of ultra-and subharmonic resonances. Journal of Nonlinear Science 1998; 8: 131–159.

23.

Fischer

Eberhard

. Controlling vibrations of a cutting process using predictive control. Computational Mechanics 2014; 54: 21–31.

24.

Bauomy

El-Sayed

. Outcome of special vibration controller techniques linked to a cracked beam. Applied Mathematical Modelling 2018; 37: 266–287.

25.

Kevorkian

Cole

. The method of multiple scales for ordinary differential equations. In: Multiple Scale and Singular Perturbation Methods. Springer, 1996, pp. 267–409.

26.

Nayfeh

. Perturbation Methods. Wiley, 1973.

27.

Dukkipati

. Solving vibration analysis problems using MATLAB. New Age International Pvt Ltd Publishers, 2007.

28.

Bek

Amer

Almahalawy

, et al. The asymptotic analysis for the motion of 3DOF dynamical system close to resonances. Alexandria Engineering Journal 2021; 60(4): 3539–3551.

29.

Amer

Starosta

Elameer

, et al. Analyzing the stability for the motion of an unstretched double pendulum near resonance. Applied Sciences 2021; 11(20): 9520.

30.

Amer

Bek

Hassan

. The dynamical analysis for the motion of a harmonically two degrees of freedom damped spring pendulum in an elliptic trajectory. Alexandria Engineering Journal 2022; 61(2): 1715–1733.

31.

Amer

Abdelhfeez

Elbaz

, et al. Investigation of the dynamical analysis, stability, and bifurcation for a connected damped oscillator with a piezoelectric harvester. Journal of Vibration Engineering & Technologies 2025; 13(2): 155.

32.

Abohamer

Amer

Arab

, et al. Analyzing the chaotic and stability behavior of a duffing oscillator excited by a sinusoidal external force. Journal of Low Frequency Noise, Vibration and Active Control 2025; 44(2): 969–986.

33.

Hao

, et al. An investigation of a self-powered low-frequency nonlinear vibration isolation system. Engineering Structures 2024; 315: 118395.

34.

Marzbanrad

Ebrahimi-Nejad

Shaghaghi

, et al. Nonlinear vibration analysis of piezoelectric functionally graded nanobeam exposed to combined hygro-magneto-electro-thermo-mechanical loading. Materials Research Express 2018; 5(7): 075022.

35.

Cui

Zhao

, et al. A generalized van der Pol nonlinear model of vortex-induced vibrations of bridge decks with multistability. Nonlinear Dynamics 2024; 112(1): 259–272.

36.

Firoozy

Ebrahimi-Nejad

. Broadband energy harvesting from time-delayed nonlinear oscillations of magnetic levitation. Journal of Intelligent Material Systems and Structures 2020; 31(5): 737–755.

37.

Zheng

Nie

Jing

, et al. Potential and electro-mechanical coupling analysis of a novel HTS maglev system employing double-sided homopolar linear synchronous motor. IEEE Transactions on Intelligent Transportation Systems 2024; 25(10): 13573–13583.

38.

Zhang

. The active rotary inertia driver system for flutter vibration control of bridges and various promising applications. Science China Technological Sciences 2023; 66(2): 390–405.

39.

Routh

. A treatise on the stability of a given state of motion, particularly steady motion. Macmillan and Company, 1877.