Control and optimization mechanism of an electromagnetic transducer model with nonlinear magnetic coupling

1.   Introduction

Generally, electromagnetic transducers are employed to gauge thickness, calculate an object's rotational speed, and detect defects in materials from steel to various other alloys. Managing the nonlinear interaction between vibrating modes is crucial for advancing nano-mechanical or micro-electromechanical devices. Interconnected oscillators serve as primary models aimed at the performance over various technical, chemical, biological, and physical schemes. In nonlinear electromechanical oscillator organizations, the mechanical part functions as a sensor and is magnetically linked to an electrical part, representing a signal of the observed vibration. The space in the permanent magnet facilitates interaction between the mechanical and electrical elements. Chaos management, vacillations, and the stability of a nonlinear electromechanical structure are examined in [1]. Also, an electromechanical gyrostat system that underwent external perturbation for its chaotic behavior, synchronization, and chaotic characteristics (adaptive control, delayed feedback control) observed in [2]. Yamapi and Bowong [3] utilized a sliding mode organizer to manage the electrostatic transducer classification while exploring the dynamic and chaotic behaviors of a self-sustaining electromechanical system both with and without discontinuity. Siewe et al. [4] employed an electromechanical oscillator method to capture the vertical motion of the earth during an earthquake. The application of slight amplitude damping for managing chaos in the system was also investigated. They discovered that the damping coefficient estimation affects both the chaotic and periodic orbits. The interactions and synchronization of two systems studied in [5,6]: a magnetically linked electrical Rayleigh-Duffing oscillator with linear mechanical oscillators, along with an interconnected self-sustaining electromechanical system featuring various functions. They employed the harmonic balance and averaging methods to identify the amplitudes of the oscillatory states. The dynamics, global bifurcations, and chaotic behavior of a self-sustaining nonlinear electromechanical system exhibiting nonlinear dynamics investigated in [7,8]. Moreover, the impact of elevated nonlinearity values on the behavior and synchronization of interconnected electromechanical systems is examined in [9]. Furthermore, the chaotic dynamics and nonlinear oscillations in an electromechanical seismograph system featuring stiffness that varies over time examined in [10]. Also, different types of active controllers are experimented to minimize system oscillations and determined that negative velocity feedback is the most effective active control for the behavior of the system. The influence of noise factors, coupling coefficients, and restraining quantities on the response of an electromechanical seismograph examined in [11].

A time-varying stiffness nonlinear electromechanical seismograph system's behavior, stability, approximate solutions, and dynamic feedback control were all investigated by Amer [12]. He also contrasted the perturbation solution with the numerical solution. Amer et al. [13] examined the performance of a twin-tail aircraft structure with both cubic and quadratic nonlinearities via an active control procedure. Sayed et al. [14,15,16,17] examined the non-linear dynamic properties of the rectangular plate within combined excitations. Also, they examined three cases of internal and primary resonances (1:2, 1:1, and 1:1:3) and matched the analytical and numerical solutions of the system. The efficiency of various control procedures in lowering the notable vibrations of a beam was examined by Hamed and Amer [18]. The oscillations and stability of the MEMS gyroscope scheme with distinct parametric forces were investigated in [19]. The frequency response equations for the concurrent resonance situation have been generated using the averaging technique. Dynamical systems motivated by parametric and external influences thoroughly examined in the works of [20,21]. The air gap of a permanent magnet, which only generates a uniform radial magnetic field, is assumed to reflect the interaction between mechanical and electrical components in most electromechanical system studies. To the best of our knowledge, the assumption of deterministic electromechanical system dynamics has been the sole method used to investigate the effects of this particular magnetic connection. However, when random disturbances are taken into account, these systems can exhibit remarkable characteristics. Similar mechanical systems have previously been shown in the literature to display a variety of behaviours and intricate chaotic dynamics when subjected to noise [22,23]. One kind of instrument that uses electromagnetic transduction to gather vibration energy is an electromechanical seismograph.

Researchers have recently focused on the behavior of electromechanical schemes and their impact on the energy harvesting challenge. The effectiveness of vibration energy harvesting devices influenced by random ambient excitations was analyzed by Martens et al. [24] through the solution of relevant Fokker-Planck equations. Borowiec et al. [25] demonstrated that the noise element of the force influences the system's stability in their investigation of how random excitation affects the performance of an energy harvester. A stochastic averaging method in [26] was introduced to evaluate the mean square electric voltage of a nonlinear energy harvesting system. Li et al. [27] explored a piezoelectric energy harvester featuring tri-stable potential wells persuaded by external magnetic fields. They claimed that the system can be appropriately designed to enhance the frequency bandwidth for a specific deterministic or stochastic input and achieve a high harvesting efficiency at coherence resonance. The multiple scales method (MMS) [28,29,30,31,32,33] and averaging techniques [34,35] are two perturbation methods commonly employed to analyze the efficiency of parametrically excited models. These methods have proven to remain successful in forecasting the behavior of such systems, especially within the frequency range close to the significant parametric resonance [36]. Traditional MMS are shown to effectively estimate responses in very basic scenarios, such as confined frequency ranges around the central parametric resonance, moderate excitations, and minimal system features [37,38,39,40,41,42,43]. Moreover, it has been shown that several control mechanism approaches can reduce the detrimental vibrations produced by different nonlinear systems in [44,45,46,47,48,49,50,51,52].

This study investigates and manages the nonlinear dynamics and vibration reduction of a model of a nonlinear electromagnetic transducer subjected to parametric and harmonic excitations. The response and stability of the solutions during the most unfavorable resonance cases have analyzed using the perturbation technique [53]. The dynamic response of the sandwiched functionally-graded piezoelectric semiconductor (FGPS) plate with the consideration of the initial electron density is investigated, and the natural frequencies and multi-field coupling are obtained in [54]. Based on the nonlocal piezoelectric semiconductor theory, Fang et al. [55] investigated the transient response of a piezoelectric semiconductor (PS) fiber, and analyzed the bending vibration, electric potential, and concentration of electrons along the nano-fiber with different nonlocal effects. Also, Liu et al. [56] proposed an active disturbance rejection control (ADRC) scheme for the electromagnetic docking of spacecraft in the presence of time-varying delay, fault signals, external disturbances, and elliptical eccentricity. Formerly, Lyu et al. [57] investigated an integrated predictor observer feedback control strategy for the vibration suppression of large-scale spacecraft affected by unbounded input time-delay effects.

The main contributions of this article are summarized as follows:

1) The study introduces a novel control mechanism Nonlinear Proportional-Derivative Velocity Feedback (NPDVF) Controller, which integrates nonlinear first- and second-order filters to improve vibration control in systems involving both mechanical and electrical components. This controller is designed to stabilize and suppress vibrations caused by complex nonlinearities and mixed forces within simultaneous resonance case (${\varOmega }_{2}\cong {\omega }_{1}, {\varOmega }_{4}\cong {\omega }_{2}$).

2) The research utilizes the perturbation method to derive approximate solutions for the system equations up to the second order, allowing for a better understanding of the system's behavior under nonlinear conditions. This technique aids in analyzing the stability and response characteristics of the system.

3) A comprehensive mathematical model is presented, which describes the coupled nonlinear ordinary differential equations governing the interaction between the mechanical and electrical components of the system. The model includes cubic and quadratic nonlinearities, which represent the complex behaviors observed in real-world systems.

4) The proposed NPDVF controller is shown to significantly reduce harmful vibrations, mitigate unstable motion, and stabilize nonlinear bifurcations in the system. Numerical simulations demonstrate its ability to handle a variety of excitation frequencies and forcing magnitudes, eliminating self-excited vibrations and improving structural stability.

5) Extensive numerical simulations conducted using MAPLE and MATLAB validate the performance of the controller. The simulations confirm that the NPDVF controller effectively stabilizes the system under mixed force conditions, and the results are consistent with the perturbation analysis, providing strong support for the controller's effectiveness.

6) The examination of stability and the impact of various framework coefficients were assessed together theoretically and numerically.

7) The various nonlinear controllers that influence the system are compared through numerical methods. The primary finding from the numerical result indicates that the new controller NPDVF, is most effective in diminishing and eradicating the model's excessive oscillations.

2.   Description of the model equations

A seismograph records ground movement during an earthquake. It falls into the category of electromechanical systems known as electromagnetic transducers. The system being examined is the most basic seismograph as it solely captures the vertical component of ground motion and only accommodates low-frequency movements. It, however, exhibits intricate dynamics. This apparatus is illustrated in Figure 1. In the current study, the electrical component of the seismograph includes a linear inductor $L$, a linear capacitor $C$, and a linear resistor $R$, with their voltage and charge conforming to the following equations.

where $q$ stands the instantaneous electrical charge, $\dot{q}$ remains its time derivative ($\dot{q} = \frac{dq}{d\tau } = i$, where $i$ is the current).

The mechanical component consists of a suspended mass; its movement is influenced by the intrinsic forces of the mass-spring system and the natural forces affecting the arrangement. Let $F\left(\tau \right)$ be the force varying with time that influences the frame because of the ground movement. We assume that damping forces (friction, air resistance, etc.) exist, and the corresponding spring exhibits nonlinear with linear stiffness characterized by ${k}_{0}$; parameters ${k}_{1}$ and ${k}_{2}$ describing the nonlinearity of the stiffness based on its type. The mechanical and electrical components engage via the air-gap of a permanent magnet that generates a radial magnetic field $\overrightarrow{B}$. The connection between the magnetic field $B$ and the position of the coil $y$ is considered as:

where ${B}_{0}$ is the greatest intensity that the field $B$ attains occurs, ${y}_{0}$ is the armature's initial position, $y$ remains its oscillation amplitude, and ${y}_{max}$ represents the maximum amplitude. Subsequently, the movement of the mechanical component must consider the connection between the Laplace force and the current indicated by:

While in the electrical part, we must include the Lenz electromotive voltage ${E}_{bemf}$ as :

The entire mathematical model that signifies the physical model in Figure 1 can be derived using Newton's second law and Kirchhoff's laws. It is regulated by the subsequent nonlinear differential equations:

where $y$ is the relative displacement of the mass $m$ with inertial forces $m\ddot{y}$ and damping forces ${\mu }_{0}\dot{y},$ and ${k}_{0}, {k}_{1}, {k}_{2}$ remain linear and nonlinear stiffness of the electromechanical oscillator system. The coupling between the above equations is assured by nonlinear radial magnetic field. The external ground motion is expected to be stochastic or periodic $\left(F\left(\tau \right) = {F}_{0}+{F}_{1}\mathit{cos}({\varOmega }_{1}\tau)+{F}_{2}y\mathit{cos}({\varOmega }_{2}\tau)\right)$ where ${F}_{0}$ is the critical amplitude, ${F}_{1}\mathit{cos}({\varOmega }_{1}\tau)$ is the external force, with amplitude ${F}_{1}$ and ${\varOmega }_{1}$ is the excitation frequency, ${F}_{2}y\mathit{cos}({\varOmega }_{2}\tau)$ is the parametric force, with amplitude ${F}_{2}$ and ${\varOmega }_{2}$ is the excitation frequency. We put Eq (2.5) into dimensionless form by setting: $x = y/l$, $z = q/{Q}_{0}$ where ${Q}_{0}$ is the reference charge and $l$ is the reference length. Let's set ${\omega }_{e} = \sqrt{\frac{1}{LC}}$, ${\omega }_{m} = \sqrt{\frac{{k}_{0}}{m}}$, by the time transformation $t = {\omega }_{e}\tau$.

Dimensionless variables introduced by scaling the system using characteristic quantities. The main goal is to eliminate the physical units and simplify the equations. Then we convert (2.5) into the following

We attain the resulting dimensionless equations for the present electromechanical seismograph as:

Assume

Where, $\varepsilon$ is often used as a small perturbation parameter in the analysis of nonlinear systems. The idea is to treat $\varepsilon$ as a small parameter and perform an asymptotic expansion or perturbation analysis. It ensures stability, smooths out nonlinearities, helps with perturbation analysis, and is crucial for ensuring that numerical methods work effectively.

So, Eq (2.6) will be written as

The first oscillator $x$ (mechanical part) is a forced Duffing oscillator related with nonlinear coupling term, and the second one $z$ (electrical part) is a linear damped oscillator with nonlinear coupling term. $\dot{x}$, $\dot{z}$, $\ddot{x}$ and $\ddot{z}$ are the first and second derivative with respect to time $t$, ${\mu }_{m}$ and ${\mu }_{e}$ are linear damping coefficients, ${\lambda }_{1}, {\lambda }_{2}$ are non-linear parameters, $\varepsilon$ is a small perturbation parameter where $0 < \varepsilon < < 1$, ${f}_{0}, {f}_{1}, {f}_{2}, {f}_{3}, {f}_{4}$, are the excitation forces amplitudes (${f}_{0}, {f}_{1}, {f}_{2}$ are mixed mechanical excitations and ${f}_{3}, {f}_{4}$ are mixed electrical excitations), ${\omega }_{1}, {\omega }_{2}$ stand the natural frequencies and ${\varOmega }_{1}, {\varOmega }_{2}, {\varOmega }_{3}, {\varOmega }_{4}$ remain the excitation frequencies, ${\gamma }_{j}$ and ${\beta }_{j}$ $(j = \mathrm{0, 1}, 2)$ are the coupling terms.

The modified and investigated system [11] after adding mixed excitations with new controller technique is in the following form [11]:

${F}_{1c}\left(t\right), {F}_{2c}\left(t\right)$ are the control inputs to reduce the vibrations that happen at principal simultaneous resonance case (${\varOmega }_{2}\cong {\omega }_{1}, {\varOmega }_{4}\cong {\omega }_{2}$) and this optimal controller is called Nonlinear Proportional-Derivative (NPD) within Negative Cubic Velocity Feedback Controller (NCVFC) to be a resonant innovative controller (NPDVF) as in the form:

where $-({p}_{1}x+\dot{{d}_{1}x}), -({p}_{2}z+\dot{{d}_{2}z})$ are the linear control forces, and $-({\alpha }_{1}{x}^{3}+{\alpha }_{2}{x}^{2}\dot{x}+{\alpha }_{3}x{\dot{x}}^{2})$, $-({\alpha }_{4}{z}^{3}+{\alpha }_{5}{z}^{2}\dot{z}+{\alpha }_{6}z{\dot{z}}^{2})$ are the non-linear control forces, and ${G}_{1}, {G}_{2}$ stand the gain amounts. Figure 2 depicts the block chart of the model within the novel controller.

3.   The mathematical model study

The perturbation technique [53] is investigated here to get the following approximate solutions as follows:

Where, ${T}_{0} = t$ and ${T}_{1} = \varepsilon t$ represent time scales.

The derivatives are presented as:

Substituting Eq (3.1) into Eq (2.8) within Eqs (2.9, 3.2) and simply solving the expressions of the order $O\left({\varepsilon }^{0}\right)$ and $O\left({\varepsilon }^{1}\right)$ to be in the following relation:

Connecting terms of the same order of $\varepsilon$ in Eq (3.3) to get:

The solution for Eq (3.4) is

Substituting Eq (3.6) into Eq (3.5) yields

where $cc$ represents the complex conjugate of the preceding parameters, $NST$ signifies non-secular terms. By means of primary parametric resonance case ${\varOmega }_{2}\cong {\omega }_{1}, {\varOmega }_{4}\cong {\omega }_{2}$, the frequency detuning parameters ${\sigma }_{1}, {\sigma }_{2}$ for the parametric excitation frequency considered by

In Eq (3.7), the solvability conditions produce when terms of secular terms are cancelled via Eq (3.8). Bearing in mind Eq (3.8), removing the secular terms in Eq (3.7) we get:

To distinct the averaging conditions that manage the elements of Eq (3.9), let definite ${A}_{1}, {B}_{1}$ and ${\stackrel{̄}{A}}_{1}, {\stackrel{̄}{B}}_{1}$ are expressed in the next polar expressions

where ${a}_{1}, {a}_{2}$ and ${\theta }_{1}, {\theta }_{2}$ stand the steady-state amplitudes and phases, respectively. Substitute Eq (3.10) in Eq (3.9), we acquire:

where ${\gamma }_{1} = {\sigma }_{1}{T}_{1}-2{\theta }_{1}$, ${\gamma }_{2} = {\sigma }_{2}{T}_{1}-2{\theta }_{2}$, then separating real and imaginary elements:

For steady-state responses (${a}_{1}^{\text{'}} = {a}_{2}^{\text{'}} = {\gamma }_{1}^{\text{'}} = {\gamma }_{2}^{\text{'}} = 0$), the periodic solution resultant to Eqs (3.12) and (3.13) are given as:

The Newton-Raphson method and MATLAB software are used to determine the steady-state answers from the algebraic equations. The Lyapunov first approach is used to determine the right-hand side eigenvalues of the Jacobian matrix at Eqs. (3.15) to calculate the stability of the steady-state shell system.

where

To determine the controlled model's stable zones, calculate the following determinant in the previous matrix.

Then,

Where, $\lambda$ designates the Jacobian matrix's eigenvalue,

are the coefficient of Eq (3.18).

Routh-Hurwitz criterion is used to examine the stability of equilibrium solutions via discovering sufficient and necessary requirements to be stable are:

Furthermore, to conclude the nature of the roots of Eq (3.18), we define the following discriminants $\varDelta$, ${\varDelta }_{0}$, ${\varDelta }_{1}$, ${\varDelta }_{2}$, and ${\varDelta }_{3}$ [58,59] as:

By joining the conditions Eq (3.19) with each one of the following possible cases, one can establish the sort of roots of Eq (3.18) as:

a) If $\varDelta < 0$ then the equation has two different real and two complex conjugate roots.

b) If $\varDelta > 0$, then either the four roots are all complex conjugate or real according to the subsequent two cases:

ⅰ. If ${\varDelta }_{0} < 0$ and ${\varDelta }_{3} < 0$ then all four roots are different and real.

ⅱ. If ${\varDelta }_{0} > 0$ or if ${\varDelta }_{3} > 0$ then the roots are two pairs of complex conjugates.

c) If $\varDelta = 0$ then the system has multiple roots according to the following four cases:

ⅰ. If ${\varDelta }_{0} < 0$, ${\varDelta }_{3} < 0$ and ${\varDelta }_{2}\ne 0$, there are two real simple roots and a real double roots

ⅱ. If ${\varDelta }_{3} > 0$ or (${\varDelta }_{0} > 0$ and (${\varDelta }_{3}\ne 0$ or ${\varDelta }_{1}\ne 0$)), there are two complex conjugate roots and two real equal roots.

ⅲ. If ${\varDelta }_{2} = 0$ and ${\varDelta }_{3}\ne 0$, there are three real equal roots and one real different root.

ⅳ. If ${\varDelta }_{3} = 0$, at that time:

1- If ${\varDelta }_{0} < 0$, there are two real double roots.

2- If ${\varDelta }_{0} > 0$ and ${\varDelta }_{1} = 0$, the roots be present two double complex conjugate roots.

3- If ${\varDelta }_{2} = 0$, all four roots are equal to $-\frac{{r}_{1}}{4}$.

4.   The optimum control parameters

The mechanism of the NPDVF controller at the measured simultaneous resonance can be clarified with the aid of Eqs (3.12) and (3.13). It is clear from Eqs (3.12) and (3.13) that the addition of the NPDVF controller to the considered framework has modified the linear damping terms ${\mu }_{m}$ and ${\mu }_{e}$ to the controlled terms ${\mu }_{m-control}$ and ${\mu }_{e-control}$. Moreover, the detuning parameter ${\sigma }_{1}$ and ${\sigma }_{2}$ is modified to ${\sigma }_{1-control}$ and ${\sigma }_{2-control}$, where ${\mu }_{m-control}$, ${\mu }_{e-control}$, ${\sigma }_{1-control}$, and ${\sigma }_{2-control}$ are given as follows:

It is clear from Eqs (4.1)–(4.4) that ${\mu }_{m-control}$, ${\mu }_{e-control}$, ${\sigma }_{1-control}$, and ${\sigma }_{2-control}$ are periodic functions on the controller, where the controlled system has the equivalent linear damping coefficients ${\mu }_{m-control} = -\frac{1}{2}\left({\mu }_{m}+{d}_{1}\right)$, ${\mu }_{e-control} = -\frac{1}{2}\left({\mu }_{e}+{d}_{2}\right)$ and the detuning parameters ${\sigma }_{1-control} = \left({\sigma }_{1}-\frac{{p}_{1}}{2{\omega }_{1}}\right) = {\varOmega }_{2}-\left({\omega }_{1}+\frac{{p}_{1}}{2{\omega }_{1}}\right)$, ${\sigma }_{2-control} = \left({\sigma }_{2}-\frac{{p}_{2}}{2{\omega }_{2}}\right) = {\varOmega }_{4}-\left({\omega }_{2}+\frac{{p}_{2}}{2{\omega }_{2}}\right)$. This means that the linear control forces ${p}_{1}$ and ${p}_{2}$ are responsible for changing the system natural frequencies ${\omega }_{1}$ and ${\omega }_{2}$, while the velocity gain ${d}_{1}$ and ${d}_{2}$ is responsible for modifying the system linear damping coefficients ${\mu }_{m}$ and ${\mu }_{e}$. Consequently, to develop the vibration suppression efficiency of the measured scheme, the linear control forces ${p}_{1}, {p}_{2}$ and ${d}_{1}, {d}_{2}$ should be designated in a way that maximizes the objective function ${\mu }_{m-control}$, ${\mu }_{e-control}$, ${\sigma }_{1-control}$ and ${\sigma }_{2-control}$. By comparing the obtained results in Figures 8(b) and 9(a) with the objective function given by Eqs (4.1)–(4.4), we can notice that the best vibration suppression condition has occurred at the maximum values of function ${\mu }_{m-control}$, ${\mu }_{e-control}$, ${\sigma }_{1-control}$ and ${\sigma }_{2-control}$ as summarized in Table 1.

The table 1 provides a list or summary of the important variables that were considered in the design process. The choice of each parameter was guided by the system's functional requirements and manufacturing constraints to avoid unstable regions of the system. This approach avoids pushing the system into regions that might otherwise lead to increased costs, manufacturing difficulties, or operational instability. The parameters listed in Tab. 1 are critical to the overall design of the system. Each of these variables was carefully chosen to balance the system's functional requirements with the practical limitations imposed by manufacturing capabilities.

To improve the system's response, Eqs (4.1)–(4.4) introduce more sophisticated feedback mechanisms, specifically designed to manage these nonlinearities and provide better stabilization and vibration suppression. The enhanced NPDVF controller improves the system's ability to avoid resonance conditions or mitigate their effects. The improvement of the NPDCVF controller represented by Eqs (4.1)–(4.4) is essential for addressing the nonlinear dynamics, feedback instability, and resonance issues in systems with both mechanical and electrical components. The combination of these equations allows the NPDVF controller to anticipate future disturbances, adjust for system nonlinearities, and optimize control actions in real time.

5.   Results and discussions

5.1.   Numerical simulation with time history

The nonlinear dynamical structure was just demonstrated by Eq (3.10) through (3.14) above. The MATLAB®18 computer programmer was then used to numerically simulate three distinct control strategies (PD, NCVF, and NPDVF) to identify which controller would minimise the destructive vibrations caused during work of the model. Figures 3–6 are plotted to appear the time history of the worst resonance case before and after using various controllers via the following values of the parameters:

with the initial conditions $x\left(0\right) = 0.5, \dot{x}\left(0\right) = 0, z\left(0\right) = 0.5, \dot{z}\left(0\right) = 0.$

Figure 3 presents the basic steady-state amplitudes $x\left(t\right)$ and $z\left(t\right)$ without controller at the simultaneous resonance situation ${\varOmega }_{2}\cong {\omega }_{1}, {\varOmega }_{4}\cong {\omega }_{2}$ as the worst resonance case of the system. Figures 4–6 show the results of adding the measured controllers (PD-NCVF-NPDCVF), enabling you to indicate the best one for reducing the high vibration amplitudes. After checking, we found that the best one controller is NPDVF as it reduces the vibrations at the measured resonance in a short time. The new controller NPDCVFC is the most efficient, according to these diagrams (3–6). The controller for the observed model's active vibration is developed in this part (refer to Figure 6). The NPD and NCVFC algorithms are used for active vibration control, as mentioned earlier. With this controller rule, the depreciation rate's are enhanced for this structural model. It is obvious that the infinite norm of vibration amplitudes might be reduced with this straightforward method. The outcomes demonstrate how well the optimization strategy reduced vibrations and how quickly suitably positioned actuators and sensors were able to reduce vibrations in the model. Thus, we will select and statistically evaluate the controlled model in order to study it and examine the impact of various controlled parameters on the system.

5.2.   Simulation of stability and the influence of various controlled model factors

This part examined the effects of different parameters used the regulated model in Eqs (3.14-3.15), numerically illustrated the stable and unstable areas. The beneficial case is examined to acquire a significant quantity of parameter effects. As illustrated in Figures 7–26, all curves display only stable segments without any instability areas when NPDVF controller is applied to the system. This provides an additional justification for adding this supplementary controller to the system, which is a beneficial outcome for any vibrating system. Solid curves indicate stable reactions. The solid line () illustrates the stable regions. Figures 7 and 19 describe the performance of the amplitude-frequency response steady-state response curves ${a}_{1}$ versus ${\sigma }_{1}$ and ${a}_{2}$ versus ${\sigma }_{2}$, respectively, with nonstable regions, which shows only stable zones. As the linear damping coefficients ${\mu }_{m}, {\mu }_{e}$ decreased the amplitudes ${a}_{1}, {a}_{2}$ are increased and the stability region is increased as appeared in Figures 8, 20. By way of the excitation frequencies ${\omega }_{1}, {\omega }_{2}$ are increased the steady-state amplitudes ${a}_{1}, {a}_{2}$ are increased and all regions are stable as exposed in Figures 9, 10, 21. Moreover, the diagrams of the nonlinear control coefficient ${\alpha }_{2}$ is slightly monotonic decreasing with no instability regions as depicted in Figure 11. Moreover, when the amplitudes ${a}_{1}, {a}_{2}$ are decreased the nonlinear control parameter ${\alpha }_{3}$ is increased as illustrated in Figures 12, 22. Also, the nonlinear coefficient ${\alpha }_{4}$ is monotonic decreasing in which when it decreased, the amplitude ${a}_{1}$ increased as shown in Figure 13. Also, as the parametric force coefficients ${f}_{2}, {f}_{4}$ are increased, then the steady-state amplitudes ${a}_{1}, {a}_{2}$ are increased with increasing in stability regions as displayed in Figures 14, 15 and 23. Besides, the curve of the linear control force coefficients ${p}_{1}, {p}_{2}$ are shifted to right (S.R) when the values of parameters ${p}_{1}, {p}_{2}$ are increased with no instability regions as graphed in Figures 16 and 24. On other hand, when the linear control force coefficients ${d}_{1}$ is increased, the steady-state amplitudes ${a}_{1}, {a}_{2}$ are increased and all zones are stable as labelled in Figures 17 and 25. In the last, as the gain coefficients ${G}_{1}, {G}_{2}$ increased the steady-state amplitudes ${a}_{1}, {a}_{2}$ are small with stability regions which are presented in Figures 18 and 26.

As listed in the figures, in which, M.I refers to monotonic increasing and M.D refers to monotonic decreasing

Therefore, Table 2 provides a detailed account of the effects of various parameters of the controlled system with various effects on the system to appear the stable regions.

5.3.   Comparison of the same model with more recent studies

An electromechanical seismograph model associated electric circuit model system akin to Eq (2.7) was examined in [9,10,11]. However, they applied the multiple time scale process within the principal parametric resonance item to study the behaviour of the structure under mixed excitations, eliminating the need for a controller. The development of the model given in [11] is examined in the current work. I add a variety of control strategies to the vibrating structure system's modified system in order to determine which one reduces the framework structure's risk of vibration. Additionally, the upgraded system's new controller NPDVF is examined in this article. The results of this research show that the novel controller less than the other controllers decreases the high vibrational amplitude of the model exposed to parametric stimulation inside the simultaneous resonance, as shown in subsection 5.1. The perturbation approach is used to aid in the acquisition of analytical solutions. Plotting of the frequency response graphs occurs at different framework parameter levels. We end with a numerical validation of the obtained results. The comparisons show that the current approach produces findings that are remarkably similar to those found in [11] and that the discrepancies are less than 1%.

5.4.   Comparison between the proposed NPDVF control and other nonlinear controllers of nonlinear systems

A comparison between nonlinear controllers with the proposed NPDVF controller is presented in a summary Table 3 that compares the performance of the different controllers based on the criteria of robustness, computational cost, and control signal amplitude. The table should provide a clear visual comparison that illustrates the advantages of the proposed NPDVF controller.

5.5.   Disturbance unavoidably of main control coefficients of the proposed controlled model

The time response of control inputs is shown in Figures 27–32 where the control input is also used to eliminate the effect of disturbance mainly at the stable stage. The control inputs is not only regulating the system's behavior but is also designed to mitigate the effect of external disturbances, particularly during the stable stage of the system's operation. These figures will show how the control input varies over time as the system responds to a disturbance. Each figure shows how the control input changes over time and responds to disturbances, providing a clear visualization of the controller's effectiveness in rejecting disturbances. This suggests that the system can quickly react to external disturbances, with the linear control force ${p}_{1} = 2.3$ stabilizing at a specific value after some initial disturbance. The system reaches stability within a short time, indicating a fast disturbance rejection process, and the disturbance effect is effectively mitigated as in Figure 27. Also, Figure 28 shows that the system achieves a disturbance-free state after a brief period. The disturbance input ${d}_{1} = 2.5$ ​has been neutralized quickly, indicating that the controller effectively nullifies the impact of the disturbance on the system in a short time. The nonlinear control gain ${G}_{1} = 0.5$ is an important factor in reducing disturbances. When this gain is set to 0.5, it appears that the system is better able to suppress disturbances. This indicates that the nonlinear controller is tuned to counteract the disturbances effectively, ensuring that the system can reach a steady state despite the presence of external influences as in Figure 29. Additionally, the control parameters ${p}_{2}$​ and ${d}_{2}$ appear to be carefully selected to ensure the system reacts appropriately to disturbances while maintaining stability. The values of ${p}_{2} = 1.9$ and ${d}_{2} = 0.5$ are likely optimized for quick disturbance rejection and minimal control effort. These parameters ensure that the system reaches the desired state quickly, further reducing the influence of external disturbances as shown in Figures 30 and 31. Finally, When ${G}_{1} = 0.4$ is applied, the system is better able to reduce or even eliminate the impact of external disturbances. The system's response, particularly in terms of the control input, should show a clear improvement after this nonlinear gain is activated, resulting in a smoother and more stable control input over time as illustrated in Figure 32. By effectively control input parameters such as ${p}_{1}, {p}_{2}, {d}_{1}, {d}_{2}, {G}_{1, }$ and ${G}_{2}$​, the system is able to reject disturbances in a timely manner and ensure robust performance.

6.   Conclusions

This study introduces a novel nonlinear proportional-derivative cubic velocity feedback (NPDVF) Controller that effectively manages vibrations in systems with coupled mechanical and electrical components. The controller's performance has been validated through numerical simulations and perturbation methods, demonstrating its ability to reduce vibrations, stabilize nonlinear motions, and suppress unwanted oscillations in the presence of mixed forces. A number of controller design approaches (PD control, NCVF control, and NPDVF as an innovative control method) were assessed in order to ascertain which one best minimizes high amplitude vibrations during the simultaneous resonance case ${\varOmega }_{2}\cong {\omega }_{1}, {\varOmega }_{4}\cong {\omega }_{2}$. The solution of the studied controlled model can be approximated using the perturbation approach. The key advantages of the NPDVF controller, including its robustness to parameter uncertainties and external disturbances, low computational cost, and efficient control signal amplitude, make it a promising solution for nonlinear vibration control systems. A comprehensive comparison with traditional control strategies, such as PD control, sliding mode control (SMC), and model predictive control (MPC), highlights the superior performance of the NPDVF controller in several aspects. Specifically, the NPDVF controller outperforms others in terms of computational efficiency and robustness, while also maintaining a smaller control signal amplitude, which reduces actuator wear and improves energy efficiency. These results suggest that the NPDVF controller is an effective and practical solution for real-time vibration control in complex systems. This study advances our understanding of the control dynamics in nonlinear models with combined excitations via perturbation technique for controlling chaotic behavior in such models. Future work could investigate the impact of time delay on the effectiveness of the NPDVF controller. Specifically, it would be beneficial to analyze how time delay influences the system's stability, vibration reduction, and bifurcation control.

Use of Generative-AI tools declaration

The author declares he has not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The author extends his appreciation to Prince Sattam bin Abdulaziz University for funding this research work through the project number (PSAU/ 2024/01/29249).

Conflict of interest

The author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.