Vibration Reduction and Stability Investigation of Van Der Pol–Mathieu–Duffing Oscillator via the Nonlinear Saturation Controller

Abstract

This study investigates the effect of a nonlinear saturation controller (NSC) on the van der Pol–Mathieu–Duffing oscillator (VMDO). The oscillator is a single degree of freedom (DOF) system. It is driven by an external force. It is described by a nonlinear differential equation (DE). The multiple-scale perturbation method (MSPT) is applied. It gives second-order analytical solutions. The first indirect Lyapunov method is used. It provides the frequency–response equation. It also shows the stability conditions. Internal resonance is included. The analysis considers steady-state responses. It studies simultaneous primary resonance with a 1:2 internal resonance (${\Lambda }_1\cong {\varpi }_1$ and ${\varpi }_1\cong 2{\varpi }_2$). Time–response simulations are presented. They show controlled and uncontrolled systems. Numerical solutions (NSs) are obtained with the fourth-order Runge–Kutta method (RK-4). They are compared with the approximate analytical solution (AS). The agreement is strong. It confirms the perturbation method. It shows that the method captures the main system dynamics.

1. Introduction

In the 1920s, Dutch researcher Balthasar van der Pol, who was researching electrical circuits, invented the famous nonlinear system known as the van der Pol oscillator. The VMDO is a fundamental model in nonlinear dynamics, renowned for its ability to capture complex oscillatory behaviors across diverse fields. It uniquely combines nonlinear damping and nonlinear stiffness. This synergy allows it to accurately model phenomena such as heartbeats, bridge vibrations, and electrical circuit instabilities. Recently, vibration control in mechanical structures has become a major research topic, as it is essential for structural safety and efficiency. Therefore, this research aims to develop and analyze a nonlinear control strategy for suppressing VMDO vibrations. Xu et al. employed time-delayed linear and nonlinear time-delayed position feedback control motions of a van der Pol in [1]. Vibration is controlled by employing a positive position feedback (PPF) controller, and an approximate solution is provided through the MSPT approach [2]. Two types of van der Pol oscillators with fractional-order terms were introduced in [3], where the averaging method was applied to obtain approximate solutions. In these cases, the damping coefficient was increased to nearly its maximum value, while the additional stiffness coefficient was reduced to almost zero. The effects of two periodic excitations with widely different frequencies on the nonlinear response of a bistable VMDO were analyzed in [4]. It was shown that, by adjusting the high-frequency drives and parametric oscillation intensities, a symmetric bistable potential could be transformed into a symmetric monostable potential. Proportional–derivative control and negative velocity feedback (NVC) were applied in [5,6,7,8] to regulate different systems. The dynamic features of a VMDO under two periodic excitations and a distributed time delay were investigated in [9]. A pitchfork bifurcation was observed, induced by high-frequency internal excitation and low-frequency external stimulation, and the fractal behavior of the system was analyzed when the two excitation frequencies coincided. The suppression of hysteresis in a forced nonlinear self-sustained oscillator operating near primary resonance was explored in [10]. To investigate the inhibition effect, rapid forcing was applied to the oscillator. A perturbation-based analytical method was employed to identify the influence zone, the quasi-periodic control domain, and the hysteresis region. Time-delay techniques have been employed in several studies to suppress vibrations in nonlinear systems, owing to the active and delayed spring characteristics of the control mechanism [11,12,13,14]. The van der Pol equation was examined in [15] with a significant delay introduced into the cubic nonlinearity, where the delay factor was sufficiently large. The dynamics of the van der Pol equation under such delays were found to be complex and highly variable.

The impact of coupling coefficient changes and insertion loss on the extraordinary point displacement in a two-coupled degree of freedom system was examined in the work in [16]. According to the investigation, adjusting the nonlinearity coefficient decreased the resonance frequency shift at the exceptional point as well as the oscillation amplitude. The authors of [17] utilized a cubic negative velocity controller (CNVC) to control vibrations and estimate the solution through an averaging perturbation approach, with an emphasis on a mass-damper-spring model. To investigate nonlinear mode coupling between micro- and nano-beam resonators, well-regulated and reproducible experiments were provided in [18]. Other micro- and nanostructures could benefit from this experimental method to better understand their nonlinear interactions and take advantage of them for more sensitive and quieter responses. Three types of nonlinear interactions between the first and third bending modes of vibrations of slightly curved beams (arches) using electrothermal tuning and electrostatic excitation were illustrated: two–one internal resonance, three–one internal resonance, and mode veering (near crossing). The experimental process was carried out in air and at ambient temperature, is very flexible and reproducible, and does not require precise or sophisticated manufacturing.

Ref. [19] utilized NSC with a van der Pol oscillator. They also used perturbation and direct numerical integration solutions to investigate the impact of feedback gains, the primary goal of Warminski et al. A flexible beam with MFC actuators was to be used with specially designed controllers [20]. Using the perturbation technique, mathematical solutions for the beam using NSC were found. In terms of minimizing system vibration, PPF and NSC controllers are the most effective. An approximation for nonlinear oscillators was obtained by Nayfeh et al. [21] and Mickens [22]. Motamid [23] combined the homotopy perturbation method with the Laplace transformation for parametric DEs, and an approximated bounded solution was generated. In contrast to automated frequency dampers, the van der Pol oscillator is a well-known nonlinear oscillator that has been extensively studied and has implications that are still being addressed today. MSPT is used to create the mean equations that compute the amplitudes and phases for the first-order approximation solutions. The nonlinear damper can effectively reduce the amplitude of the principal resonance response. In order to solve the Cauchy problem of the van der Pol equation in the complex domain, [24] employed the analytical approximate technique. For both simple and complex scenarios, analytical continuation is possible with these approximations. To manage the computational process and increase the precision of the end results, they examined the impact of variance in the problem’s original data.

Orlov and Chichurin [25] investigated the unstable and dynamically stable behavior of the system. Furthermore, he provided statistical evidence that the oscillator’s amplitude rises when the stability requirements are not fulfilled. It is numerically shown that the nonlinear receiver is effective in lowering the nonlinear vibrations of the primary nonlinear oscillator under primary resonance conditions. A thorough parametric analysis was conducted in [26] to assess how important design elements affected the overall stability of the system. Complex nonlinear dynamics, including disconnected resonance curves and periodic stability to chaotic behavior transitions, were discovered. Additionally, several control strategies have been studied and proven to reduce the dangerous vibrations produced in various nonlinear systems [27,28,29,30]. The fourth-order Runge–Kutta approach yielded numerical findings that further validated the precision of these responses. Resonance examples were used to determine the characteristic exponents and solvability requirements. Additionally, they were employed to show the frequency–response curves. By looking at the stability and instability ranges, the nonlinear stability analysis was carried out. Authors of [31,32,33,34] demonstrated a novel method for solving differential delay equations using the MSPT approach that allows for precise approximation of solutions over extended periods of time.

Alluhydan et al. [35] indicated the effect of CNVC on the vibration of a quarter-car model and compared proportional derivative (PD), linear negative velocity control (LNVC), PPF, negative derivative feedback (NDF), and CNVC. Authors of [36] demonstrated the many zones of stability and instability of a dynamic system within frequency–response curves, as well as the act of changing parameters on the system’s behavior. Arena [37] used the asymptotic method of MSPT; an ordered hierarchy of linear perturbation equations governs the periodic cell’s nonlinear dynamics. In order to investigate the linear and nonlinear dispersion properties, the Floquet–Bloch theory is iteratively used at each order of the perturbation equations due to its linearity and spatial periodicity. Razzaq et al. [38] looked at primary issues with the employed method when it is applied to small-scale systems, like the incapacity to qualify acceleration and restoring forces, and they estimated the system’s restoring forces and nonlinearities using a combination of analytical methods, experimental data, and Lagrange polynomial interpolation. Erturk et al. [39] found the series solution of the coupled asymmetric van der Pol oscillator in two dimensions and presented the NS by the Runge–Kutta method. Abohamer et al. [40] examined the AS after adding NVC and negative cubic velocity feedback controllers to the van der Pol–Methieu–Duffing oscillator using the MSPT. Wei [41] used the two categories from the wake oscillator and one degree of freedom model, which depend on the van der Pol oscillator, and studied the stability of the system from an energy perspective. Taylor expansion was used to solve the DE. The results of nonlinear van der Pol oscillators were compared in three ways: ode15, ode45, and the fourth-order Runge–Kutta, as shown in [42]. A straightforward but efficient iteration process for finding the van der Pol equation’s solution is suggested in [43]. This process is a strong tool for figuring out a nonlinear equation of motion periodic solution. Furthermore, a comparison between the obtained solution and those obtained by the RK-4 was made, which demonstrated the precision of the iteration procedure applied.

Ref. [44] solved the nonlinear (VMDO) using a Morlet waveform neural network (MWNN) as a smart computing technique, simplified the intricate mathematical formulas employed in conventional numerical procedures to solve DE, and reduced the number of time-consuming stages.

The main advantage of employing NSC for the VMDO lies in its ability to effectively suppress nonlinear vibrations that arise from the combined effects of self-excitation and nonlinear stiffness. By introducing a saturation mechanism, the controller prevents excessive amplitude growth, reduces the risk of bifurcations and chaotic behavior, and enhances the overall stability of the system. Furthermore, NSC provides a practical means of vibration suppression without requiring excessive control effort, making it energy-efficient and suitable for real-world engineering applications. This approach not only improves structural reliability but also extends the operational lifetime of mechanical and aerospace systems subjected to complex dynamic excitations.

With the use of a suitable control technique, this study seeks to provide basic guidelines for removing high vibrations of the framework. In order to actively reduce excessive vibration for the nonlinear dynamical system, the NSC is taken into account. The perturbation technique produced an analytical conclusion that was almost simultaneous for both the primary and internal resonance cases. The perturbation approach utilizing the NSC process yields the FREs, stability analysis, and mathematical answers. Plotting and reporting of the effects of each parameter on the vibrating and NSC systems is performed, and the impacts of several parameters and the controller on the system are simulated employing MATLAB R2023b software.

2. Governing Equations and Approximate Solutions

The VMDO is a complex nonlinear dynamical system that combines features of the VMDO. This combination enables the modeling of a wide range of vibrational phenomena across various scientific and engineering disciplines. A van der Pol–Duffing oscillator’s equation of motion is represented by the following DE [45], which can be modified as follows:

$$\ddot{\chi }+\epsilon {\alpha }_1\dot{\chi }-\epsilon \gamma {\chi }^2\dot{\chi }+{\varpi }_1^2\chi +\epsilon \lambda {\chi }^3+\epsilon \beta {\chi }^5=\epsilon f_1\cos ({\Lambda }_{1}t)+\epsilon \chi f_2\cos ({\Lambda }_{2}t)$$

(1)

where $\chi$ denotes the displacement of the VMDO. $\dot{\chi }$ and $\ddot{\chi }$ represent the velocity and acceleration, respectively. The parameter ${\varpi }_1$ is the system’s natural frequency, and ε is a small perturbation parameter. The coefficient ${\alpha }_1$ corresponds to the linear damping. The parameters $\beta$, $\gamma$, and $\lambda$ represent the nonlinearities terms. The excitation frequencies and amplitudes are ${\Lambda }_1$, ${\Lambda }_2$ and $f_1$,$f_2$.

One possible way to express the DE of motion that describes the oscillations of the van der Pol with an NSC controller in this manner is as follows:

$$\ddot{\chi }+\epsilon {\alpha }_1\dot{\chi }-\epsilon \gamma {\chi }^2\dot{\chi }+{\varpi }_1^2\chi +\epsilon \lambda {\chi }^3+\epsilon \beta {\chi }^5=\epsilon f_1\cos ({\Lambda }_{1}t)+\epsilon \chi f_2\cos ({\Lambda }_{2}t)+\epsilon G_1{\Upsilon }^2$$

(2)

$$\ddot{\Upsilon }+\epsilon {\alpha }_2\dot{\Upsilon }+{\varpi }_2^2\Upsilon =\epsilon G_2\chi \Upsilon$$

(3)

where $\Upsilon$ represents the displacement of NSC. ${\varpi }_2$ is the NSC controller’s inherent frequency, and $G_1$ and $G_2$ are the control and feedback signals.

Figure 1 depicts the van der Pol–Duffing oscillator model with NSC in the following flowchart:

3. Perturbation Analysis

We used the MSPT [31,32,33] as follows to obtain the approximate answer up to the first approximation:

$$\left.\begin{array}{l}\chi \left(t;\epsilon \right)={\chi }_0\left(T_0,T_1\right)+\epsilon {\chi }_1\left(T_0,T_1\right)+O\left({\epsilon }^2\right)\\ \Upsilon \left(t;\epsilon \right)={\Upsilon }_0(T_0,T_1)+\epsilon {\Upsilon }_1\left(T_0,T_1\right)+O\left({\epsilon }^2\right)\end{array}\right\}$$

(4)

where $T_0=t$ and $T_1=\epsilon t$. The derivatives have the following forms:

$$\left.\begin{array}{l}{\displaystyle \frac{d}{dt}}=D_0+\epsilon D_1+\dots \\ {\displaystyle \frac{d^2}{d{t}^2}}=D_0^2+2\epsilon D_0{D}_1+\dots \end{array}\right\}$$

(5)

Substituting from (4) and (5) in (2) and (3), then comparing coefficients that are roughly equivalent to the power of $\epsilon$, we obtain:

$$O\left({\epsilon }^0\right)$$

$$\left(D_0^2+{\varpi }_1^2\right){\chi }_0=0$$

(6)

$$\left(D_0^2+{\varpi }_2^2\right){\Upsilon }_0=0$$

(7)

$$O\left({\epsilon }^1\right)$$

$$\left.\begin{array}{c}\left(D_0^2+{\varpi }_1^2\right){\chi }_1=-2D_0{D}_1{\chi }_0-{\alpha }_1{D}_0{\chi }_0+\gamma {\chi }_0^2(D_0{\chi }_0)-\lambda {\chi }_0^3-\beta {\chi }_0^5\\ +f_1\cos ({\Lambda }_1{T}_0)+{\chi }_0 f_2\cos ({\Lambda }_2{T}_0)+G_1{\Upsilon }_0^2\end{array}\right\}$$

(8)

$$\left(D_0^2+{\varpi }_2^2\right){\Upsilon }_1=-2D_0{D}_1{\Upsilon }_0-{\alpha }_2{D}_0{\Upsilon }_0+G_2{\chi }_0{\Upsilon }_0$$

(9)

Solving (6) and (7), we get the following:

$${\chi }_0(T_0,T_1)=A(T_1) e^{i{\varpi }_1{T}_0}+\overline{A}(T_1) e^{i{\varpi }_1{T}_0}$$

(10)

$${\Upsilon }_0(T_0,T_1)=B(T_1) e^{i{\varpi }_2{T}_0}+\overline{B}(T_1) e^{i{\varpi }_2{T}_0}$$

(11)

where $A$ and $B$ are complex functions in $T_1$. Substituting Equations (10) and (11) in Equations (8) and (9), we have

$$\left.\begin{array}{l}\left(D_0^2+{\varpi }_1^2\right){\chi }_1=\left(-2i{\varpi }_1(D_{1}A)-i{\alpha }_1{\varpi }_{1}A+i\gamma {\varpi }_1{A}^2\overline{A}-3\lambda A^2\overline{A}-10\beta A^3{\overline{A}}^2\right)e^{i{\varpi }_1{T}_0}\\ -\beta A^5{\mathrm{e}}^{5i{\varpi }_1{T}_0}+G_1{B}^2{e}^{2i{\varpi }_2{T}_0}{+\mathrm{G}}_{1}B\overline{B}+\left(i\gamma {\varpi }_1{A}^3-\lambda A^3-5\beta A^4\overline{A}\right){\mathrm{e}}^{3i{\varpi }_1{T}_0}\\ +{\displaystyle \frac{f_1}{2}}{\mathrm{e}}^{i{\Lambda }_1{T}_0}+{\displaystyle \frac{f_2}{2}}{\mathrm{e}}^{i({\Lambda }_2+{\varpi }_1)T_0}+{\displaystyle \frac{f_2}{2}}{\mathrm{e}}^{i({\Lambda }_2-{\varpi }_1)T_0}+C.C.\end{array}\right\}$$

(12)

$$\left(D_0^2+{\varpi }_2^2\right){\Upsilon }_1=\left(-2i{\varpi }_2(D_{1}B)-i{\alpha }_2{\varpi }_{2}B\right){\mathrm{e}}^{i{\varpi }_2{T}_0}+G_{2}AB{e}^{i({\varpi }_1+{\varpi }_2)T_0}+G_{2}A\overline{B}{e}^{i({\varpi }_1-{\varpi }_2)T_0}+C.C.$$

(13)

C.C. is the complex conjugate. After neglecting the secular terms in Equations (12) and (13), the first AS can be written as

$${\chi }_1=E_1{e}^{5i{\varpi }_1{T}_0}+E_2{e}^{3i{\varpi }_1{T}_0}+E_3{e}^{i{\varpi }_1{T}_0}+E_4{e}^{i({\Lambda }_2+{\varpi }_1)T_0}+E_5{e}^{i({\Lambda }_2-{\varpi }_1)T_0}+E_6{e}^{2i{\varpi }_2{T}_0}+E_7+C.C.$$

(14)

$${\Upsilon }_1=E_8{e}^{i({\varpi }_1+{\varpi }_2)T_0}+E_9{e}^{i({\varpi }_1-{\varpi }_2)T_0}+C.C.$$

(15)

where $E_n(n=1,2,\dots ,9)$ are complex functions in $T_1$ and are given in Appendix A.

From the solutions found above. The following resonance instances were determined:

  (i)

Primary resonance: ${\Lambda }_1\cong {\varpi }_1$ or ${\Lambda }_2\cong {\varpi }_1$.

 (ii)

Internal resonance: ${\varpi }_1\cong 2{\varpi }_2$.

(iii)

Simultaneous resonance: Internal and primary resonance.

This research considers the simultaneous resonance situation (${\Lambda }_1\cong {\varpi }_1$ and ${\varpi }_1\cong 2{\varpi }_2$) to be the worst resonance case. In the next part, we will analyze and numerically examine the stability of this instance, respectively.

4. The Periodic Solution

The system stability is examined using the most damaging simultaneous resonance scenario (${\Lambda }_1\cong {\varpi }_1$ and ${\varpi }_1\cong 2{\varpi }_2$) from the first-order approximation. The detuning parameters ${\sigma }_1$ and ${\sigma }_2$ are obtained by

$${\Lambda }_1={\varpi }_1+\epsilon {\sigma }_1, {\varpi }_1=2{\varpi }_2+\epsilon {\sigma }_2$$

(16)

Substituting Equation (16) into (12) and (13), we have

$$2i{\varpi }_1{D}_{1}A=-i{\alpha }_1{\varpi }_{1}A+i\gamma {\varpi }_1{A}^2\overline{A}-3\lambda A^2\overline{A}-10\beta A^3{\overline{A}}^2+{\displaystyle \frac{f_1}{2}}{e}^{i{\sigma }_1{T}_1}+G_1{B}^2{e}^{-i{\sigma }_2{T}_1}$$

(17)

$$2i{\varpi }_2{D}_{1}B=-i{\alpha }_2{\varpi }_{2}B+G_{2}A\overline{B}{e}^{i{\sigma }_2{T}_1}$$

(18)

Exchanging all $A(T_1)$ and $B(T_1)$ by the polar form as

$$\left.\begin{array}{l}A(T_1)={\displaystyle \frac{1}{2}}a(T_1)e^{i{\Theta }_1(T_1)}, D_1(A(T_1))={\displaystyle \frac{1}{2}}(\dot{a}(T_1)+ia(T_1){\dot{\Theta }}_1(T_1))e^{i{\Theta }_1(T_1)}\\ B(T_1)={\displaystyle \frac{1}{2}}b(T_1)e^{i{\Theta }_2(T_1)},D_1(B(T_1))={\displaystyle \frac{1}{2}}(\dot{b}(T_1)+ib(T_1){\dot{\Theta }}_2(T_1))e^{i{\Theta }_2(T_1)}\end{array}\right\}$$

(19)

where the motion’s steady scenario phases are presented as ${\Theta }_1$ and ${\Theta }_2$, while $a$ and $b$ are the motion’s steady scenario amplitudes.

Substituting Equation (19) into Equations (17) and (18), we get

$$\dot{a}={\displaystyle \frac{-{\alpha }_{1}a}{2}}+{\displaystyle \frac{\gamma a^3}{8}}+{\displaystyle \frac{f_1}{2{\varpi }_1}}\sin ({\Phi }_1)+{\displaystyle \frac{G_1{b}^2}{4{\varpi }_1}}\sin ({\Phi }_2)$$

(20)

$$a{\dot{\Theta }}_1={\displaystyle \frac{3\lambda a^3}{8{\varpi }_1}}+{\displaystyle \frac{5\beta a^5}{16{\varpi }_1}}-{\displaystyle \frac{f_1}{2{\varpi }_1}}\cos ({\Phi }_1)-{\displaystyle \frac{G_1{b}^2}{4{\varpi }_1}}\cos ({\Phi }_2)$$

(21)

$$\dot{b}={\displaystyle \frac{-{\alpha }_{2}b}{2}}-{\displaystyle \frac{G_{2}ab}{4{\omega }_2}}\sin ({\Phi }_2)$$

(22)

$$b{\dot{\Theta }}_2=-{\displaystyle \frac{G_{2}ab}{4{\varpi }_2}}\cos ({\Phi }_2)$$

(23)

where ${\Phi }_1={\sigma }_1{T}_1-{\Theta }_1$ and ${\Phi }_2=2{\Theta }_2-{\sigma }_2{T}_1-{\Theta }_1$.

4.1. The Frequency–Response Equations (FREs)

For obtaining the steady scenario solution, we put $\dot{a}=\dot{b}={\dot{\Phi }}_1={\dot{\Phi }}_2=0$ into Equations (20)–(23), and we get

$${\displaystyle \frac{{\alpha }_{1}a}{2}}-{\displaystyle \frac{\gamma a^3}{8}}={\displaystyle \frac{f_1}{2{\varpi }_1}}\sin ({\Phi }_1)+{\displaystyle \frac{G_1{b}^2}{4{\varpi }_1}}\sin ({\Phi }_2)$$

(24)

$$-{\sigma }_{1}a+{\displaystyle \frac{3\lambda a^3}{8{\varpi }_1}}+{\displaystyle \frac{5\beta a^5}{16{\varpi }_1}}={\displaystyle \frac{f_1}{2{\varpi }_1}}\cos ({\Phi }_1)+{\displaystyle \frac{G_1{b}^2}{4{\varpi }_1}}\cos ({\Phi }_2)$$

(25)

$$-{\alpha }_2={\displaystyle \frac{G_{2}a}{2{\varpi }_2}}\sin ({\Phi }_2)$$

(26)

$$({\sigma }_1+{\sigma }_2)=-{\displaystyle \frac{G_{2}a}{2{\varpi }_2}}\cos ({\Phi }_2)$$

(27)

Equations (26) and (27) can be substituted into Equations (24) and (25), and the results can then be squared and added to obtain

$${\left({\displaystyle \frac{{\alpha }_{1}a}{2}}-{\displaystyle \frac{\gamma a^3}{8}}+{\displaystyle \frac{G_1{\alpha }_2{\varpi }_2{b}^2}{2G_2{\varpi }_{1}a}}\right)}^2+{\left(-{\sigma }_{1}a+{\displaystyle \frac{3\lambda a^3}{8{\varpi }_1}}+{\displaystyle \frac{5\beta a^5}{16{\varpi }_1}}+{\displaystyle \frac{G_1({\sigma }_1+{\sigma }_2){\varpi }_2{b}^2}{2G_2{\varpi }_{1}a}}\right)}^2={\displaystyle \frac{f_1^2}{4{\varpi }_1^2}}$$

(28)

Squaring and adding Equations (26) and (27), we get

$${\alpha }_2^2+{({\sigma }_1+{\sigma }_2)}^2={\displaystyle \frac{G_2^2{a}^2}{4{\varpi }_2^2}}$$

(29)

Then, the FREs used are Equations (28) and (29), respectively.

4.2. Stability Analysis at the Fixed Point

Starting with the following steps to determine the steady scenario solution, permit

$$a=a_0+a_1,b=b_0+b_1,{\Phi }_1={\Phi }_{10}+{\Phi }_{11},{\Phi }_2={\Phi }_{20}+{\Phi }_{21}$$

(30)

where $a_0$,$b_0$,${\Phi }_{10}$, and ${\Phi }_{20}$ are solutions of (24)–(27) and $a_1$,$b_1$,${\Phi }_{11}$, and ${\Phi }_{21}$ are considered as small perturbations. Then, from (20) to (23), we get

$${\dot{a}}_1=(-{\displaystyle \frac{{\alpha }_1}{2}}+{\displaystyle \frac{3\gamma }{8}}{a}_0^2)a_1+({\displaystyle \frac{f}{2{\varpi }_1}}\cos {\Phi }_{10}){\Phi }_{11}+({\displaystyle \frac{G_1{b}_0}{2{\varpi }_1}}\sin {\Phi }_{20})b_1+({\displaystyle \frac{G_1{b}_0^2}{4{\varpi }_1}}\cos {\Phi }_{20}){\Phi }_{21}$$

(31)

$${\dot{\Phi }}_{11}=({\displaystyle \frac{-3\lambda a_0}{4{\varpi }_1}}-{\displaystyle \frac{5a_0^3}{4{\varpi }_1}})a_1-({\displaystyle \frac{f}{2{\varpi }_1{a}_0}}\sin {\Phi }_{10}){\Phi }_{11}-({\displaystyle \frac{G_1{b}_0}{2{\varpi }_1{a}_0}}\cos {\Phi }_{20})b_1-({\displaystyle \frac{G_1{b}_0^2}{4{\varpi }_1{a}_0}}\sin {\Phi }_{20}){\Phi }_{21}$$

(32)

$${\dot{b}}_1=-({\displaystyle \frac{G_2{b}_0}{4{\varpi }_2}}\sin {\Phi }_{20})a_1-({\displaystyle \frac{{\alpha }_2}{2}}+{\displaystyle \frac{G_2{a}_0}{4{\varpi }_2}}\sin {\Phi }_{20})b_1-({\displaystyle \frac{G_2{a}_0{b}_0}{4{\varpi }_2}}\cos {\Phi }_{20}){\Phi }_{21}$$

(33)

$$\left.\begin{array}{l}{\dot{\Phi }}_{21}=({\displaystyle \frac{-3\lambda a_0}{4{\varpi }_1}}-{\displaystyle \frac{5a_0^3}{4{\varpi }_1}}-{\displaystyle \frac{G_2}{2{\varpi }_2}}\cos {\Phi }_{20})a_1-({\displaystyle \frac{f}{2{\varpi }_1{a}_0}}\sin {\Phi }_{10}){\Phi }_{11}-({\displaystyle \frac{G_1{b}_0}{2{\varpi }_1{a}_0}}\cos {\Phi }_{20})b_1\\ \hspace{1em}\hspace{1em}-({\displaystyle \frac{G_1{b}_0^2}{4{\varpi }_1{a}_0}}\sin {\Phi }_{20}-{\displaystyle \frac{G_2{a}_0}{2{\varpi }_2}}\sin {\Phi }_{20}){\Phi }_{21}\end{array}\right\}$$

(34)

We put Equations (31)–(34) in a matrix form:

$${[{\dot{a}}_1{\dot{\Phi }}_{11}{\dot{b}}_1{\dot{\Phi }}_{21}]}^T=[J]{[a_1{\Phi }_{11}{b}_1{\Phi }_{21}]}^T$$

(35)

where $[J]$ is the Jacobin matrix of the Equations (31)–(34). The eigenvalues of $[J]$ can be written as

$$\left|\begin{array}{cccc}{\delta }_{11}-\lambda & {\delta }_{12}& {\delta }_{13}& {\delta }_{14}\\ {\delta }_{21}& {\delta }_{22}-\lambda & {\delta }_{23}& {\delta }_{24}\\ {\delta }_{31}& 0& {\delta }_{33}-\lambda & {\delta }_{34}\\ {\delta }_{41}& {\delta }_{42}& {\delta }_{43}& {\delta }_{44}-\lambda \end{array}\right|=0$$

(36)

${\delta }_{ij}\left(i,j=1,2,3,4\right)$, denotes the coefficient found in Equations (31)–(34). Hence, (36) can be rewritten as follows:

$${\lambda }^4+H_1{\lambda }^3+H_2{\lambda }^2+H_3\lambda +H_4=0$$

(37)

Here, $H_i(i=1,2,3,4)$ and is explained in Appendix A. The real parts of the eigenvalues must be negative for the solution to be stable; otherwise, it is unstable. It is necessary to meet the Routh–Hurwitz criterion for

$$H_1>0,H_1{H}_2-H_3>0,H_3(H_1{H}_2-H_3)-H_1^2{H}_4>0,H_4>0$$

(38)

5. Findings and Discussion

The system’s output was measured using MATLAB software. In addition to examining the system’s stability, the behavior of the controlled system was shown to be influenced by several characteristics, utilizing the multiple-scale technique. This section examines the system’s steady scenario behavior in the presence of 1:2 resonance. It also thoroughly investigates the controller for different controller settings under the simultaneous resonance condition (${\Lambda }_1\cong {\varpi }_1$ and ${\varpi }_1\cong 2{\varpi }_2$).

5.1. Numerical Solution with Time History

At simultaneous resonance, the van der Pol system with the NSC controller that was supplied (${\Lambda }_1\cong {\varpi }_1$ and ${\varpi }_1\cong 2{\varpi }_2$) is numerically solved in this part using the RK-4th-order method using the MATLAB (R2014a) computer program (package ode45). The specific values of the equation parameters are as follows: ${\alpha }_1=0.04, {\varpi }_1=1, \lambda =3.5, \beta =2.5, \gamma =0.04; f_1=0.02, f_2=0.05,$${\Lambda }_1={\varpi }_1, {\Lambda }_2=2.$ The time history of the model without a control mechanism is displayed in Figure 2, which shows that the steady-state amplitude of the system without a controller is within the initial values $\chi (0)=0,\dot{\chi }(0)=0,\Upsilon (0)=0.2 \mathrm{and} \dot{\Upsilon }(0)=0$. Figure 3 displays the phase-plane and time history for the nonlinear system with NSC under conditions (${\Lambda }_1\cong {\varpi }_1$ and ${\varpi }_1\cong 2{\varpi }_2$) with parameters $G_1=1.5,G_2=0.6,{\alpha }_2=0.008,{\varpi }_2=0.5$. At t = 500, the controller and main system reach saturation, respectively. Figure 3 shows that the amplitude of the system with NSC decreased to roughly 68.59% of the value without NSC.

5.2. The Effects of Various Parameters

This section examined the steady situation amplitude in the case of 1:2 resonance with respect to the regulated system. We have numerically solved the FREs (28) and (29) using the identical values for the parameters. The results are shown graphically as the steady scenario amplitude a for the system and the NSC controller b against the detuning setting. Following the addition of the NSC, the outputs are shown in Figure 4 as steady scenario amplitudes a and b against the detuning. As shown in

Table 1. The values system parameters on the dynamic response.

Symbol	Description	Value 1 (Basic)	Value 2	Value 3	Figures
$f_1$	Excitation force	0.02	0.017	0.06	Figure 5
${\alpha }_1$	Damping coefficient of the system	0.04	0.008	0.08	Figure 6
${\alpha }_2$	Damping coefficient	0.008	0.02	0.003	Figure 7
$G_1$	Control gain	1.5	0.64	2.6	Figure 8
$G_2$	Feedback control gain	0.6	0.55	0.4	Figure 9
${\varpi }_1=2{\varpi }_2$	Nature frequency	1	1.1	0.94	Figure 10
$\lambda$	Nonlinear term	3.5	2.9	4.63	Figure 11
$\gamma$	Nonlinear term	0.04	1.4	2.4	Figure 12
$\beta$	Nonlinear term	2.5	0.0001	10	Figure 13
${\sigma }_2$	Detuning parameter	0	0.05	−0.03	Figure 14

, three representative values were all allocated to each system parameter to examine how they affected the dynamic response. The force amplitude $f_1$ increased, causing the unstable region to diminish in both system and NSC amplitudes, as shown in Figure 5. As Figure 5a–c illustrate, the system remains unchanged as the force develops. As the values of the damping coefficients (${\alpha }_1$ and ${\alpha }_2$) grew, the unstable zone expanded as shown in Figure 6 and Figure 7. Figure 8 demonstrates how increasing the control signal gain $G_1$ values led to a decrease in the unstable region in the steady scenario amplitude for system a and an increase in the steady scenario for controller b. Figure 9 explains how the primary system’s unstable zone decreased as the control feedback $G_2$ signal’s values increased. On the other hand, the controller’s steady-state amplitude is increased. Figure 10 demonstrates how the values of ${\varpi }_1$ increase the steady scenario amplitudes of the controller and the system. Figure 11, Figure 12 and Figure 13, in that order, illustrate that the unstable region increased by increasing the values of the nonlinear parameters $\lambda$, $\gamma$, and $\beta$. In Figure 14, the steady scenario amplitude for the controller developed as the values of the detuning parameter ${\sigma }_2$ increased, but the steady scenario amplitude for the system shifted to the right.

5.3. Effectiveness of External Excitations

Figure 15 shows the system’s external excitation force increased. Although the amplitude $\chi \left(t\right)$ is increased before the addition of the NSC, the effect is not evident at this time due to the tiny values, as illustrated in Figure 15a, because $\chi \left(t\right)$ has an amplitude between 0 and 0.2. When the NSC controller is added, as shown in Figure 15b, the system’s amplitude decreases, and the amplitude of $\chi \left(t\right)$ falls between 0 and 0.06. As a result, we concluded that the NSC controller effectively reduces the external excitation force. Additionally, the van der Pol motion is described in Figure 16 by increasing the external excitation force. Although the amplitude $\chi \left(t\right)$ is increased before the addition of the NSC, the effect is not evident at this time due to the tiny values, as illustrated in Figure 16a, because $\chi \left(t\right)$ has an amplitude between 0 and 0.198. When the NSC controller is added, as seen in Figure 16b, the system’s amplitude decreases, and the amplitude of $\chi \left(t\right)$ falls between 0 and 0.0592. As a result, we concluded that the NSC controller effectively reduces the external excitation force.

5.4. Important Compassion Results on the System

Figure 17 presents a graphical comparison between the approximate analysis of Equations (20)–(23) and the numerical simulation results of Equations (2) and (3). The dashed lines represent the modulation of amplitudes $a$ and $b$ for the full coordinates $\chi$ and $\Upsilon$, respectively. The solid lines indicate the time response of the vibrations, obtained through numerical modeling using the NSC controller. The figure further illustrates that the AS based on the multiple-scales perturbation technique (MSPT) closely matches the numerical results derived from the Runge–Kutta 4 (RK-4) method. By calculating the errors between AS and NS, we can incorporate both an absolute error (${\zeta }_r$) analysis and the root mean square error (RMSE). Furthermore, this validation has been included in

Table 2. Absolute error (${\mathit{\zeta }}_{\mathit{r}}$) between the NS and the AS.

	With NSC Control			Without Control
T	AS	NS	${\mathit{\zeta }}_{\mathit{r}}$	AS	NS	${\mathit{\zeta }}_{\mathit{r}}$
800	0.0546	0.04895	0.0057	0.1472	0.1481	0.0009
850	0.0163	0.02103	0.0047	0.1217	0.1481	0.0264
900	0.0711	0.07972	0.0086	0.1891	0.1763	0.0128
950	0.065	0.06736	0.0024	0.1262	0.1279	0.0017
1000	0.0552	0.05149	0.0037	0.1560	0.1498	0.0062
1050	0.0575	0.06017	0.0027	0.1866	0.1873	0.0007
1100	0.0594	0.05239	0.0070	0.2043	0.2317	0.0274
1150	0.0568	0.05812	0.0013	0.2058	0.2113	0.0055
1200	0.0505	0.06102	0.0105	0.1928	0.1897	0.0031
1250	0.0417	0.05016	0.0085	0.1664	0.1672	0.0008
1300	0.0318	0.03214	0.0003	0.1287	0.1297	0.0010
1350	0.0218	0.02316	0.0014	0.1822	0.1831	0.0009
1400	0.0111	0.01213	0.0010	0.2301	0.2325	0.0024
1450	0.0086	0.01003	0.0014	0.2241	0.2226	0.0015
1500	0.0061	0.00832	0.0022	0.1769	0.1783	0.0014
	RMSE = 0.0052			RMSE = 0.0107

, which illustrates the MSPT method’s accuracy and convergence characteristics. These modifications improve the clarity of the comparison between the two systems and strengthen the validity of our conclusions.

Figure 18 illustrates the response of the van der Pol–Mathieu–Duffing oscillator with and without control, using low-gain values. Initially, the system exhibits nonlinear oscillations without any control during the time interval $t\in \left[0,250\right]$. At $t=250$, a conventional integral resonant controller (IRC) is applied and remains active until $t=500$. Then, at $t=500$, the IRC controller is deactivated and replaced immediately by the NSC controller, which continues to operate until $t=1500$. As observed in the figure, the IRC controller leads to a slight reduction in vibration amplitude during the interval $t\in \left[250,500\right]$. However, switching to the NSC controller at $t=500$ results in a more effective suppression of the system’s vibrations.

The DE of motion that describes the oscillations of the van der Pol with the IRC controller in this manner follows

$$\ddot{\chi }+\epsilon {\alpha }_1\dot{\chi }-\epsilon \gamma {\chi }^2\dot{\chi }+{\varpi }_1^2\chi +\epsilon \lambda {\chi }^3+\epsilon \beta {\chi }^5=\epsilon f_1\cos ({\Lambda }_{1}t)+\epsilon \chi f_2\cos ({\Lambda }_{2}t)+\epsilon G_{1}z$$

(39)

$$\dot{z}+\eta z=G_2\chi$$

(40)

To achieve the novelty of this work, a comparison is made between the present work and the previous ones in [15,40], as presented in

Table 3. Comparison between the current work and previous ones in [15,40].

No.	Comparison Items	The Current Study	The Examined Work in [15]	The Investigated Work in [40]
1	DOF	1 DOF	1 DOF	1 DOF
2	AS	AS are acquired using MSPT	The nonlocal dynamics procedure with a large delay	The AS are obtained by MSPT
3	Resonance case	Simultaneous primary and internal	Not investigated	Subharmonic
4	NS	NS are obtained from RK-4	The numerical simulation is not achieved	The NS are achieved
5	Stability area	The stability and instability areas are investigated according to the amplitude’s resonance curves	Amplitude’s resonance curves are not presented	The stability boundaries are drawn
6	Parameters’ effect	The parameters affecting amplitudes are plotted	Not mapped out
7	Feedback control	NSC control is used	Does not exist	Displacement–velocity feedback controller is used
8	Diagram of the time history of a solution and phase plan	The time history and phase plan are drawn for all cases with and without control	Not presented	Not graphed
9	Controller effectiveness	The controller’s effectiveness is investigated	Does not exist	Not investigated

.

6. Conclusions

By adding a nonlinear saturation controller, the control performance of a van der Pol–Mathieu–Duffing oscillator is improved. When the external excitation force is applied to the primary system, the expected synchronous resonance situation is near; the proposed nonlinear controller NSC has been investigated using a method of different time scales. The stability of the steady-state solution was examined and quantitatively verified using FREs and perturbation analysis. Numerous experiments are conducted to assess the consequences of different controller and system settings. The primary subjects of the numerical findings are the system’s performance and the impact of various parameters. The vibrating system’s amplitude is suppressed from about 0.207 to roughly 0.0601, and the vibrations are decreased by roughly 70.97% compared to their uncontrolled value. The NSC effectiveness ${\mathrm{E}}_{\mathrm{a}}$ (${\mathrm{E}}_{\mathrm{a}}$ = steady-state amplitude of the main system without controller/steady-state amplitude of the main system with controller) is close to 3.44. The measured resonance case is (${\Lambda }_1\cong {\varpi }_1$ and ${\varpi }_1\cong 2{\varpi }_2$) the worst one.

(1)

Using the NSC technique is the way to drastically reduce the vibration amplitudes of the system.

(2)

The unstable region decreased by increasing the excitation external force amplitude $f_1$ in both the system and NSC amplitudes.

(3)

The values of the natural frequency increase the steady-state amplitudes of the controller and main system. Furthermore, by raising the damping coefficient values ${\alpha }_1$ and ${\alpha }_2$, the unstable zone grew.

(4)

The system’s amplitude fell in the unstable region while the values of the control signal gain $G_1$ increased in the steady state.

(5)

The steady scenario for the controller increased, but the unstable region for the system decreased by developing the values of the control feedback signal $G_2$.

This study focuses primarily on a specific resonance instance and is restricted to weakly nonlinear oscillations. The analysis lacks experimental validation and is purely theoretical and numerical. The model will be expanded to different resonance settings in further studies, and experimental investigations will be conducted to confirm the results.