A chaotic behavior and stability analysis on quasi-zero stiffness vibration isolators with multi-control methodologies

Abstract

A quasi-zero stiffness vibration isolator (QZSVI) is used in applications like precision instruments, aerospace, microelectronics manufacturing, and seismic isolation to protect sensitive equipment from low-frequency vibrations. Their key advantage lies in achieving near-zero stiffness, allowing for highly effective vibration attenuation while maintaining system stability. These passive systems are cost-effective and reliable, offering superior vibration isolation without the need for external power or active control. This work proposes the use of negative displacement, velocity, and cubic velocity feedback control techniques to enhance the QZSVI’s isolation performance. We found that the composite negative velocity and cubic velocity control (NVFC + NCVFC) is more effective with low cost compared to other types of controller (its effectiveness is about 94.8%). The approximate solutions (AS) of the controlling system of equations of motion (EOM) are acquired using a multiple-scales procedure (MSP) up to the second order, and it is subsequently validated numerically through the Runge–Kutta method (RKM) from the fourth-order. Modulation equations (ME) are obtained by exploring resonance instances and solvability conditions. Time history graphs and frequency response curves, generated via MATLAB and Wolfram Mathematica 13.2, are presented to analyze stability and steady-state solutions. It is investigated how altering the parameters affects the system amplitude. Poincaré maps, Lyapunov exponent spectra (LEs), and bifurcation diagrams are presented to illustrate the system’s diverse behavior patterns. Furthermore, the transmissibility of force, displacement, and acceleration is computed and displayed. A QZSVI minimizes low-frequency vibrations, making it ideal for precision applications in metrology, automotive, aerospace, civil engineering, medical equipment, and renewable energy. It achieves superior damping, ensuring high stability and precision.

Introduction

The QZSVI is sophisticated passive vibration isolation equipment that is intended to reduce vibrations, particularly at low frequencies where conventional isolators sometimes have difficulties. Classical isolators, like dampers or springs, frequently have to choose between vibration isolation and weight-carrying capability. In order to handle big loads, a higher stiffness is required; nevertheless, this decreases the system’s capacity to isolate low-frequency oscillations. To get past this restriction, QZSVI systems combine linear and nonlinear components, such as buckling beams or pre-compressed springs, and provide virtually “flat” stiffness characteristics around the position of equilibrium. As a result, the system is able to support loads efficiently and provides excellent isolation at lower frequencies. Extended exposure to vibration and impact can lead to discomfort and fatigue in the human body in vehicle engineering. In the fields of mechanical and architectural engineering, excessive vibrations can cause structural damage, making vibration control a critical focus. Linear passive vibration isolators are frequently employed due to their affordability and effectiveness in managing vibrations. Vibration control is vital in engineering to reduce harmful effects. Metamaterials and origami-based structures offer unique properties like energy absorption and negative stiffness, making them effective solutions. Though distinct, their overlap has led to combined concepts like “origami metamaterials,” blending mechanical and structural innovations. ¹

A bio-inspired limb-like construction with asymmetry bars and springs is studied in Ref. 2 for passive vibration isolation. Flexible quasi-zero, zero, or negative stiffness can be achieved by adjusting the system’s nonlinear stiffness through structural factors. The asymmetric design and spring ratios improve stiffness tuning and loading capacity. Its benefits over current isolators are confirmed by experiments, providing a novel approach to designing vibration control. In Ref. 3, the authors looked at how applying higher-order expansions improved these reactions. In Ref. 4, a three-degrees-of-freedom (DOF) system impacted by harmonic excitation forces was created by analyzing the movement of a tuned absorber attached to a 2DOF pendulum mechanism. The fundamental resonance is examined, and the system’s stability is analyzed utilizing the function of frequency response and the phase plane approach.

In Ref. 5, it is suggested to use a displacement–velocity feedback regulation technique to improve QZSVI’s isolating capability. In the electronically controlled QZSVI mechanism, the delay in time is taken into account. All potential improvements to optimize the QZSVI for better overall performance are outlined in Ref. 6, along with a few engineering examples demonstrating its applications. The dynamic model in Ref. 7 was constructed using LE, and new indicators for the QZSVI are developed and compared with other isolators, considering stiffness linearity and the region’s width. The theoretical results are then confirmed through the establishment of an experimental setup. The authors study the worst resonance case of a Duffing oscillator under feedback control using MSP in Ref. 8. The effects of time delay on amplitude and stability analysis were examined. In Ref. 9, a shearer magnetic braking system with multi-excitation forces is studied and solved by MSP. The worst resonance case, with NVFC showing 99.95% control effectiveness, outperforming NCVFC at 96.7%. In Ref. 10, the NVFC is used to reduce vibrations in the 2DOF system. Bifurcation diagrams, Lyapunov exponents, and piezoelectric EH performance are investigated. In Ref. 11, the behavior of an elastic pendulum under harmonic forcing is examined, with its pivot point following a circular trajectory at a steady angular velocity. In a dynamical pendulum system, where the suspension point follows a Lissajous curve trajectory, the energy transfer between various modes is depicted in Ref. 12. In Ref. 13, the authors conducted an analysis of a suspended mass connected to a nonlinear damped spring pendulum, one end of which was anchored to a fixed location. Both transverse and radial forces exerted on the spring have an impact on the motion. Two distinct strategies involving three different time frames are used in Ref. 14 to enhance the findings of Ref. 13. These scales are thought to be necessary for lowering errors. In these investigations, the EOM were estimated using the MSP Ref. 15. The vibrating movement of a damped pendulum of a rigid body at its equilibrium point is examined numerically in Ref. 16. One of the energy harvesting technologies that transforms vibratory movement into electric energy is an electromagnetic device. In Ref. 17, the vibratory movement of a connected dynamical system to this device is explored. In Ref. 18, the authors examined how a spring pendulum coupled to a more basic one-damped motion. In Ref. 19, the researchers investigated how a viscous moment at the turning point and two harmonic excitation forces affected the movement of a pendulum system. In Ref. 20, authors examined the pivot movement of a basic pendulum that was traveling on an elliptical path and had a stiff arm attached to a longitudinal absorber. By adding regional averages and replacing rapid scale inputs with localized periodic-in-time problems, the authors ²¹ were able to decouple the rapid and sluggish scales. By breaking down errors into averaging error, slow scale error, and fast scale error, they were able to create an a posteriori error estimator through the use of the dual-weighted residual approach. The writers verified the error estimator’s accuracy and demonstrated how to use it for adaptive control of a numerical MSP.

In Ref. 22, the nonlinear oscillation of a dual pendulum with 2DOF that is not stretched and whose suspended point travels in an elliptic path at a constant angular velocity is studied. These pendulums are connected with varying masses and vary in length. The EOM are derived by applying Lagrange’s equations (LE). Up to the third level of approximation, the desired AS are found using the MSP. The planar movement of a distinct 3DOF triple pendulum system is explored in Ref. 23, featuring a double rigid pendulum linked to an unstretched pendulum that is suspended from a fixed point. The controlling system of EOM are developed by using LE. Utilizing the MSP, higher-order AS of these equations are obtained. On the other hand, using the Galerkin method, time and displacement variables are uncoupled, and the resulting ordinary differential equation (DE) governing the web’s behavior is solved using the MSP. In Ref. 24, the authors explored the creation of a nonlinear model for the meniscus force acting on the trolling mode atomic force microscopy (TR-AFM) nanoneedle and presented an analytical solution by applying the MSP technique to the distributed-parameter model of the TR-AFM resonator. A nonlinear railway car with dual bogies was the subject of a bifurcation analysis in Ref. 25 to examine how the interaction between the bogies affects the hunting behavior of the vehicle. By means of an asymptotic growth of a minor perturbation parameter through the MSP, solutions close to the hunting speed were produced. Resonance cases are commonly classified, with a specific emphasis on studying the circumstance involving two simultaneous external resonances. A damped elastic pendulum, as opposed to an unstretched one, is the subject of additional research on this subject, as described in Ref. 26. A two-scale feed-forward control approach was studied in Ref. 27 using the MSP to lessen undesired residual oscillations in both standard and non-traditional duffing systems. In Ref. 28, the authors applied the method of restricting phase trajectories to explore the dynamics of a parametric pendulum for the stationary and non-stationary scenarios. They suggested a simplified model that might predict highly modulated patterns beyond the usual range of initial conditions. In Ref. 29, it was performed a multi-scale dynamic analysis and established solvability criteria for 1D structures with irregular boundaries, making it easier to examine beams, arches, strings, and cables. In Ref. 30, the presented study is focused on the impact of forced oscillations on a coupled particle to a support by means of two nonlinear springs with viscous dampers. The MSP is used in the time domain in this investigation. However, new algorithms designed specifically to handle differential and algebraic equation problems have resulted in a modified version of the MSP with three scales of the time variable. In Ref. 31, the authors investigated the sinusoidal response of vibrating rods equipped with a variety of nonlinear springs. Response functions are shown over wide frequency ranges in the multi-mode analysis to reveal different resonances.

In Ref. 32, the effect of a piezoelectric device on a 2DOF dynamical system is investigated. LE are applied to get the system’s governing EOM, while the AS was obtained using the MSP, and the NS was derived via the RKM. Secular terms are eliminated to establish solvability criteria, and the various resonance cases are then classified. The ME for two cases are investigated, and the outputs of the energy harvesting device are analyzed. In Ref. 33, the vibrating dynamical motion of a 3DOF auto-parametric system made up of a damped spring pendulum and a damped Duffing oscillator is investigated. The EOM are derived by applying LE, and the MSP are employed to solve this equation. In addition, the study looks into non-linear stability and analyzes resonance cases, time histories, resonance curves, and stability zones. Furthermore, the influence of an electromagnetic harvesting device on another 3DOF dynamical system is examined in Ref. 34. The EOM of the system are solved analytically utilizing the MSP. To determine the areas of instability and stability, the conditions of Routh–Hurwitz are applied. In Ref. 35, the authors examined the influence of three external moments on the movement of a 4DOF nonlinear system. LE and the MSP have been used to get the system’s EOM and to obtain its solutions. The corresponding NS of this system are obtained using RKM, demonstrating high consistency and precision. Graphs representing the motion’s time histories, resonance curves, and the solutions at the steady-state case are used to explore how the physical properties of the system affect the motion under study.

The purpose of this study is to introduce efficient feedback control techniques for reducing vibration in the resonant area. To fulfill that goal, a negative displacement feedback controller (NDFC), a negative velocity feedback controller (NVFC), a negative cubic velocity feedback controller (NCVFC), a NVFC + NCVFC, and composite controllers NDFC + NVFC + NCVFC methods are suggested in this article. Using the MSP, the system’s AS are derived till the second-order approximation. To evaluate the accuracy of the used approach, the NS are calculated using the RKM, graphically represented, and compared to the AS. The solvability criteria and all resonance cases are found when the secular terms are eliminated. Informed of the numerical solutions of the ME’s stability assessments, the temporal histories of the modified phases and amplitudes are displayed. These equations are simplified under steady-state conditions to obtain the frequency response equations, which are subsequently solved together to identify the fixed points. Consequently, the resonance curves at various parameters are used to display the associated domains of stability and instability. In addition, bifurcation sketches, Poincaré maps, and LEs are used to explore the system’s behavior. The transmissibilities of force, displacement, and acceleration are computed and displayed. In domains demanding high precision and stability, such as seismic isolation, precision instruments, and aerospace systems, QZSVI is extensively employed to safeguard delicate equipment from low-frequency vibrations. Their superior isolation capabilities without sacrificing stability make them perfect for situations where minimum external vibration disturbance is required.

The approach and analysis of a QZSVI with negative displacement, velocity, and cubic velocity feedback control techniques to improve the QZSVI’s isolation performance highlight the innovations of this study. Another innovative feature is the use of the MSP to derive the AS up to the second order, which enables a more precise and thorough investigation of the dynamic behavior of the system, especially when nonlinearities are present. By contrasting the AS with the NS derived from the fourth-order RKM, the study confirms the correctness of the AS. This two-pronged strategy guarantees the robustness and dependability of the analytical techniques used. Using force, displacement, and acceleration transmissibility, bifurcation diagrams, Poincaré maps, and Lyapunov exponent spectrums, the study explores complicated dynamic phenomena. Understanding the entire spectrum of system responses requires fresh insights into the system’s stability and possible chaotic behavior, which are provided by this exhaustive examination. Applications such as seismic isolation, aerospace, microelectronics production, and precision instruments use QZSVI to shield delicate equipment from low-frequency vibrations. Their fundamental benefit is that they may achieve almost zero stiffness, which enables very efficient vibration attenuation while preserving system stability. The theoretical results are given a practical dimension by the quantitative proof of the control’s efficacy and the graphical analysis of the amplitude of NDFC, NVFC, NCVFC, NVFC + NCVFC, and composite controllers NDFC + NVFC + NCVFC.

Description of a QZSVI system and the modeling of controller

A negative stiffness corrector (NSC) and two vertical springs constitute the majority of the QZSVI. The components of the NSC include a horizontal spring, connecting rods, hinge axes, and brackets. The connecting rod’s two ends link with the hinge axis and the bracket, respectively, such that every connecting rod has an identical length $L$ . The vertical and horizontal dampers have damping coefficients $C_1$ and $C_2$ . The hinge axis is attached to the edges of the horizontal spring. The lower bracket is mounted to the base plate, whereas the top bracket is connected to the loading support, which is utilized to support a load. Both the base plate and the loading support are affixed to the ends of the vertical spring. The loading support is restricted to vertical motion via vertical guiding rods, as demonstrated in Figure 1(a). When a payload is applied, the loading support undergoes a displacement $x$ from its static equilibrium position under the influence force $f$ , as indicated in Figure 1(b). ³⁶

Figure 1.

(a) The QZSVI system in flat view using two linear dampers; (b) A diagram illustrating the static analysis of the load-bearing support and the scissor-like structure.

Notably, the restoring force in the system will be equal in size but opposite in direction to the applied force. The equation presented below describes how the applied force is related to the resulting displacement

$f\left(x\right)=f_v+mg-f_t,$

(1)where $f_v=k_v\left(x-\mathrm{\Delta }x\right)$ signifies the restoring force provided by the vertical springs, $f_t=f_h\hspace{0.17em}\mathit{tan}\hspace{0.17em}\alpha$ is the produced force by the NSC, and $f_h=k_h\left(d-2L\left(1-\mathit{cos}\hspace{0.17em}\alpha \right)\right)$ refers to the restoring force exerted by the horizontal spring. $\alpha$ represents the angle between the horizontal plane and the connecting rod, along with $\mathit{tan}\hspace{0.17em}\alpha =\frac{x}{\sqrt{4L^2-x^2}}$ . Then, the applied force can be reformulated as ³⁶

$f\left(x\right)=k_{v}x-k_{h}x\left(\frac{d-2L}{\sqrt{4L^2-x^2}}+1\right).$

(2)

The regulating load’s equation under the impact of harmonic force is given as ³⁶

$m\ddot{x}+C_1\dot{x}+f_{d2}\left(x,\dot{x}\right)+f\left(x\right)=F_e\hspace{0.17em}\mathit{cos}\hspace{0.17em}{\Omega }_{e}t.$

(3)here, ${\Omega }_e$ and $F_e$ indicate the frequency and the amplitude of the excitation force, respectively, while the following is an expression for $f_{d2}\left(x,\dot{x}\right)$ , which stands for the applied nonlinear damping force to the loading support

$f_{d2}\left(x,\dot{x}\right)=\frac{C_2{x}^2}{4L^2-x^2}\dot{x}.$

(4)

It is possible to convert equations (2) and (4) into a dimensionless form

$$\begin{gathered} h\left(u\right)=u-\beta u\left(\frac{\delta -2}{\sqrt{4-u^2}}+1\right), \\ {h}_{d2}\left(u,\dot{u}\right)=\frac{c_2{u}^2}{4-u^2}\dot{u}. \end{gathered}$$

(5)

$\text{Here},h=\frac{f}{k_{v}L},h_{d2}=\frac{f_{d2}}{k_{v}L},u=\frac{x}{L},\beta =\frac{k_h}{k_v},\delta =\frac{d}{L}.$

(6)

Using the Taylor series till the third approximation of (5) and the non-dimensional parameters (6), the equation of motion (3) will be

$\ddot{u}+c_1\dot{u}+k_{1}u+k_2{u}^3+k_3{u}^5+k_4{u}^2\dot{u}+k_5{u}^4\dot{u}=f_e\hspace{0.17em}\mathit{cos}\hspace{0.17em}\Omega \tau ,$

(7)where

$k_1=\frac{2-\beta \left(\delta -2\right)}{2}={\omega }^2,k_2=-\frac{\beta \left(\delta -2\right)}{16},k_3=-\frac{3\beta \left(\delta -2\right)}{256},k_4=\frac{C_2}{4m{\omega }_0},k_5=\frac{C_2}{16m{\omega }_0},c_1=\frac{C_1}{m{\omega }_0},f_e=\frac{F_e}{k_{v}L},{\omega }_0^2=\frac{k_v}{m},\tau ={\omega }_{0}t,\Omega =\frac{{\Omega }_e}{{\omega }_0}.$

At this stage, we are going to establish the QZSVI mathematical model, focusing on the impact of NDFC, NVFC, and NCVFC on the response cases for reducing induced vibrations, as shown in Figure 2. Given these considerations, the mathematical model for the QZSVI system will be constructed as follows

$\ddot{u}+c_1\dot{u}+k_{1}u+k_2{u}^3+k_3{u}^5+k_4{u}^2\dot{u}+k_5{u}^4\dot{u}=f_e\hspace{0.17em}\mathit{cos}\hspace{0.17em}\Omega \tau -G_{1}u-G_2\dot{u}-G_3{\dot{u}}^3.$

(8)

Figure 2.

NDVFC, NVFC, and NCVFC of the dynamical model.

The recommended methodology

The asymptotic analytic solutions to equation (8) are obtained in the present section by means of the MSP. Consequently, we direct our study toward the system’s dynamic behavior within a limited region bounded by the static equilibrium position. Following this, the smallness of the parameters and variables was incorporated, in line with

$c_1=\epsilon {\stackrel{\sim }{c}}_1,f_e=\epsilon {\stackrel{\sim }{f}}_e,G_j=\epsilon {\stackrel{\sim }{G}}_j\left(j=1,2,3\right),k_l=\epsilon {\stackrel{\sim }{k}}_l\left(l=2,3,4,5\right)$

(9)where $0<\epsilon \ll 1$ is a small parameter.

As stated by the MSP, the solution $u$ , can be stated as it is illustrated below^18,22

$u={\displaystyle \sum }_{k=0}^1{\epsilon }^k{u}_k\left({\tau }_0,{\tau }_1\right)+O\left({\epsilon }^2\right).$

(10)here ${\tau }_n={\epsilon }^n\tau \left(n=0,1,2\right)$ signify specific time scales in which ${\tau }_0$ is a fast one, while ${\tau }_1$ is the slow one. Now, the derivatives with regard to $\tau$ must be transformed to these scales. Therefore, this objective can be achieved by using the following operators

$$\begin{gathered} \frac{d}{d\tau }=\frac{\partial }{\partial {\tau }_0}+\epsilon \frac{\partial }{\partial {\tau }_1}, \\ \frac{d^2}{d{\tau }^2}=\frac{{\partial }^2}{\partial {\tau }_0^2}+2\epsilon \frac{{\partial }^2}{\partial {\tau }_0\partial {\tau }_1}+O\left({\epsilon }^2\right). \end{gathered}$$

(11)

These operators clearly show that terms $O\left({\epsilon }^2\right)$ and higher are ignored due to their small magnitude. The substitution of (9)–(11) into equation (8) produces the sets of partial DEs listed below, which relate to the distinct powers of $\epsilon$

(a) Order ${\epsilon }^0$

$\frac{{\partial }^2{u}_1}{\partial {\tau }_0^2}+{\omega }^2{u}_1=0,$

(12)

(b) Order of $\epsilon$

$\frac{{\partial }^2{u}_2}{\partial {\tau }_0^2}+{\omega }^2{u}_2=\frac{1}{2}{\tilde{f}}_e{e}^{i\Omega {\tau }_0}-{\tilde{G}}_1{u}_1-{\tilde{k}}_2{u}_1^3-{\tilde{k}}_3{u}_1^5-{\tilde{c}}_1\frac{\partial u_1}{\partial {\tau }_0}-{\tilde{G}}_1\frac{\partial u_1}{\partial {\tau }_0}-{\tilde{k}}_4{u}_1^2\frac{\partial u_1}{\partial {\tau }_0}-{\tilde{k}}_5{u}_1^4\frac{\partial u_1}{\partial {\tau }_0}-{\tilde{G}}_3{\left(\frac{\partial u_1}{\partial {\tau }_0}\right)}^3-2\frac{{\partial }^2{u}_1}{\partial {\tau }_0\partial {\tau }_1}.$

(13)

Therefore, the general solution of (12) can be obtained as follows

$u_1=A{e}^{i\omega {\tau }_0}+\overline{A}{e}^{-i\omega {\tau }_0},$

(14)where $A$ illustrates enigmatic, complex processes operating at slow time scales ${\tau }_0$ and ${\tau }_1$ , while $\overline{A}$ display their complex conjugate. By replacing the previous solution response to (14) with the secular terms, equation (13) can be solved. The following conditions are used to eliminate them

$2i\omega \frac{\partial A}{\partial {\tau }_1}+i\omega \left({\stackrel{\sim }{c}}_1+{\stackrel{\sim }{G}}_2\right)A+{\stackrel{\sim }{G}}_{1}A+\left(3i\omega {\stackrel{\sim }{G}}_3+i\omega {\stackrel{\sim }{k}}_4+3{\stackrel{\sim }{k}}_2\right)A^2\overline{A}+\left(2i\omega {\stackrel{\sim }{k}}_5+10{\stackrel{\sim }{k}}_3\right)A^3{\overline{A}}^2=0.$

(15)

Eventually, we obtained the second-order solution, which is presented below

$$\begin{gathered} u_2=\frac{{\tilde{f}}_e}{2\left({\omega }^2-{\Omega }^2\right)}{e}^{i\Omega {\tau }_0}+\frac{1}{24{\omega }^2}\left({\tilde{k}}_3+i\omega {\tilde{k}}_5\right)A^5{e}^{5i\omega {\tau }_0} \\ +\frac{1}{8{\omega }^2}\left[i\omega \left(3{\tilde{G}}_3+{\tilde{k}}_4+3{\tilde{k}}_{5}A\overline{A}\right)+{\tilde{k}}_2+5{\tilde{k}}_{3}A\overline{A}\right]A^3{e}^{3i\omega {\tau }_0}. \end{gathered}$$

(16)

Resonance classification and modulation equations

This section’s goals are to categorize the different resonance cases, address one of those situations, and obtain the ME. If any of the denominators in the final solution approaches disappear, such conditions could appear. ³² Consequently, opportunities that follow occur

• Primary external resonance case can be found at $\Omega =\omega ,$

• Internal resonance case at $\omega =0.$

If any of the resonance requirements outlined earlier are satisfied, the system will show intricate and complicated behavior. It’s important to keep in mind that even when the vibrations go away from resonance, the solutions obtained are still appropriate. Let’s now investigate the issue more closely and look at the main external resonance that happened at $\Omega \approx \omega$ . Next, the detuning parameters $\sigma$ that help achieve this objective can be taken into consideration; it shows how close $\Omega$ to $\omega$ . Therefore, one can write as follows

$\Omega =\omega +\sigma ,$

(17)where $\sigma =\epsilon \stackrel{\sim }{\sigma }$ . Detuning parameters can be conceptualized as representing the deviation from precise resonance and its corresponding vibrations. ²⁶ By removing secular elements, the solvability criterion may be obtained. Therefore, the following equations indicate the likelihood that the requirements will be met

$2i\omega \frac{\partial A}{\partial {\tau }_1}+i\omega \left({\stackrel{\sim }{c}}_1+{\stackrel{\sim }{G}}_2\right)A+{\stackrel{\sim }{G}}_{1}A+\left(3i\omega {\stackrel{\sim }{G}}_3+i\omega {\stackrel{\sim }{k}}_4+3{\stackrel{\sim }{k}}_2\right)A^2\overline{A}+\left(2i\omega {\stackrel{\sim }{k}}_5+10{\stackrel{\sim }{k}}_3\right)A^3{\overline{A}}^2-\frac{1}{2}{\stackrel{\sim }{f}}_e{e}^{i{\tau }_1{\stackrel{\sim }{\sigma }}_1}=0.$

(18)

Two nonlinear partial DEs with regard to $A$ that rely on ${\tau }_1$ are contained in the established solvability condition (18). Thus, we may express these functions in polar form as follows

$A\left({\tau }_1\right)=\frac{\stackrel{\sim }{q}\left({\tau }_1\right)}{2}{e}^{i\Psi \left({\tau }_1\right)};q=\epsilon \stackrel{\sim }{q}.$

(19)

The operator of the first-order derivative can be written in the following manner because the functions $A$ are independent of the scales ${\tau }_0$

$\frac{\partial A}{\partial {\tau }_0}=0,\frac{\partial A}{\partial \tau }=\epsilon \frac{\partial A}{\partial {\tau }_1}.$

(20)

In dependence on (20), the subsequent modified phase can be utilized to convert the solvability criteria of partial DE (18) into ordinary ones as

$\theta \left({\tau }_1\right)={\tau }_1\stackrel{\sim }{\sigma }-\Psi \left({\tau }_1\right).$

(21)

By substituting (19)–(21) into (18) and dividing the imaginary and real components, the system below can be achieved based on the previous

$$\begin{gathered} q\frac{d\theta }{d\tau }=q\sigma -\frac{5}{16\omega }{k}_3{q}^5-\frac{3}{8\omega }{k}_2{q}^3-\frac{1}{2\omega }{G}_{1}q+\frac{1}{2\omega }{f}_e\hspace{0.17em}\mathit{cos}\hspace{0.17em}\theta , \\ \frac{dq}{d\tau }=-\frac{1}{2}q\left(c_1+G_2\right)-\frac{3}{8}{G}_3{q}^3-\frac{1}{8}{k}_4{q}^3-\frac{1}{16}{k}_5{q}^5+\frac{1}{2\omega }{f}_e\hspace{0.17em}\mathit{sin}\hspace{0.17em}\theta . \end{gathered}$$

(22)

Steady-state approach and stability analysis

The solutions for the dynamical system’s steady-state condition are the main topic of this section. In this scenario, the solutions would show up at the conclusion of the temporary procedures. In this case, we take into consideration the zero value of $\frac{dq}{d\tau }=\frac{d\theta }{d\tau }=0$ . ²³ Thus, equation (22) may be used to generate the following set of algebraic equations

$$\begin{gathered} f_e\hspace{0.17em}cos\hspace{0.17em}\theta +2\omega q\sigma -\frac{5}{8}{k}_3{q}^5-\frac{3}{4}{k}_2{q}^3-G_{1}q=0, \\ {f}_e\hspace{0.17em}sin\hspace{0.17em}\theta -\omega q\left(c_1+G_2\right)-\frac{3}{4}\omega G_3{q}^3-\frac{1}{4}\omega k_4{q}^3-\frac{1}{8}\omega k_5{q}^5=0. \end{gathered}$$

(23)

After the modified phase $\theta$ is removed from the last system (23), an analysis of that system produces the frequency response equations listed below

$f^2=4{\omega }^2\left\{{\left(q\sigma -\frac{5}{16\omega }{k}_3{q}^5-\frac{3}{8\omega }{k}_2{q}^3-\frac{1}{2\omega }{G}_{1}q\right)}^2+{\left(\frac{1}{2}q\left(c_1+G_2\right)+\frac{3}{8}{G}_3{q}^3+\frac{1}{8}{k}_4{q}^3+\frac{1}{16}{k}_5{q}^5\right)}^2\right\}.$

(24)

The stability behavior of these solutions is discussed by linearizing (22) using the Lyapunov first (indirect) approach, which yields the following system ⁸

$\left[\begin{array}{l}\dot{q}\\ \dot{\theta }\end{array}\right]=\left[\begin{array}{l}\frac{\partial \dot{q}}{\partial q}\frac{\partial \dot{q}}{\partial \theta }\\ \frac{\partial \dot{\theta }}{\partial q}\frac{\partial \dot{\theta }}{\partial \theta }\end{array}\right]\left[\begin{array}{l} q\\ \theta \end{array}\right]=\left[\begin{array}{l}{J}_{11}{J}_{12}\\ J_{21}{J}_{22}\end{array}\right]\left[\begin{array}{l} q\\ \theta \end{array}\right],$

(25)where

$$\begin{gathered} J_{11}=-\frac{1}{2}\left(c_1+G_2\right)-\frac{9}{8}{G}_3{q}^2-\frac{3}{8}{k}_4{q}^2-\frac{5}{16}{k}_5{q}^4,J_{12}=\frac{1}{2\omega }{f}_e\hspace{0.17em}\mathit{cos}\hspace{0.17em}\theta , \\ {J}_{21}=\frac{\sigma }{q}-\frac{25}{16\omega }{k}_3{q}^3-\frac{9}{8\omega }{k}_{2}q-\frac{1}{2\omega }\frac{G_1}{q},J_{22}=-\frac{1}{2\omega q}{f}_e\hspace{0.17em}\mathit{sin}\hspace{0.17em}\theta . \end{gathered}$$

(26)

Consequently, the ability to analyze and solve the characteristic equation determines the steady-state solution’s stability. The Jacobian matrix’s eigenvalues may be written in the following manner ⁹

$|\begin{array}{l}{J}_{11}-\lambda \\ J_{21}\end{array}\begin{array}{l}{J}_{12}\\ J_{22}-\lambda \end{array}|=0,$

(27)or

${\lambda }^2+{\Gamma }_1\lambda +{\Gamma }_2=0,$

(28)where

$$\begin{gathered} {\Gamma }_1=-\left(J_{11}+J_{22}\right), \\ {\Gamma }_2=J_{11}{J}_{22}-J_{12}{J}_{21}. \end{gathered}$$

(29)

By using the Routh–Hurwitz criterion, ³³ we can observe that the state’s solution is asymptotically stable if and only if all real portions for the roots of (28) are negative. This condition occurs if the coefficients fulfill

${\Gamma }_1>0\hspace{0.17em}\text{and}\hspace{0.17em}{\Gamma }_2>0,$

(30)which are examined numerically.

Results and discussion

This section illustrates the system’s performance in terms of amplitude and phase, examining both controlled and uncontrolled cases, including the resonance condition. We compare different control systems and analyze how specific parameters influence system amplitude. Additionally, we explore the relationship between system amplitude and the detuning parameter $\sigma$ , as well as conduct bifurcation analysis and calculate LEs to assess system stability and chaotic behavior. Moreover, the RKM may be utilized to numerically solve the system (22) of the above modulation to illustrate the behavior of the function $u$ and its associated phase planes $u\dot{u}$ at resonance, both with and without control. Thus, the following data and initial conditions are taken into account

$\begin{array}{l}\omega =2,c_1=0.2,k_2=0.005,k_3=0.004,k_4=0.001,\\ k_5=0.001,f_e=2,\Omega =\omega ,q\left(0\right)=0.01,\theta \left(0\right)=0.\end{array}$

Figures 3(a) and (b) show the time history of the function $u$ at normal operation (without resonance and without control) for short and long time scales, respectively. In addition to its phase portrait at the same conditions is shown in Figure 3(c). Whereas the curves in Figures 4(a) and (b) represent the time history of the solution $u$ at resonance case without control $\left(G_1=0,G_2=0,G_3=0\right)$ for short and long time intervals, respectively. The phase diagram of this solution is drawn in Figure 4(c) at the same restrictions. We note that the wave’s amplitudes at the basic case and resonance case without control are, respectively, 0.09 and 3.434, which indicates a danger to the system; for this reason, it is essential to use control to restrict and minimize these damaging vibrations in order to keep the system in operational condition. Conversely, the time history of the identical function $u$ at resonance with various types of controllers is shown in Figure 5 for short and long time scales, in which black color refers to NDFC $\left(G_1=2,G_2=0,G_3=0\right)$ with an amplitude of the function $u$ equal to 0.9763; therefore, the amplitude of $u$ is reduced by 71.57%, red color point to NCVFC $\left(G_1=0,G_2=0,G_3=5\right)$ with amplitude of 0.4103, so the amplitude of $u$ is reduced by 88%, green color refers to NVFC + NCVFC $\left(G_1=0,G_2=5,G_3=5\right)$ with amplitude of 0.1772; therefore the amplitude of $u$ is reduced by 94.8%, the blue represent NVFC $\left(G_1=0,G_2=5,G_3=0\right)$ with amplitude of 0.2, so the amplitude of $u$ is reduced by 94.18%, and the pink color point to composite controllers NDFC + NVFC + NCVFC $\left(G_1=3,G_2=5,G_3=5\right)$ with amplitude of $u$ 0.1722, therefore the amplitude of $u$ is reduced by 95%. It is obvious from previous results that the combination of NDFC + NVFC + NCVFC controllers is the most effective as it led to an efficient reduction in the quantity. Although it is 95% bigger than the others, we can see that it is quite similar to the combination of NVFC + NCVFC types. For this reason, the combination of NVFC + NCVFC is the best type since it is more economical and lower cost. The following Table 1 concludes the previous results.

Figure 3.

Time response solution $u\left(\tau \right)$ at various scales of $\tau$ and phase portrait $u\dot{u}$ in basic case (without resonance, without control $\left(G_1=0,G_2=0,G_3=0\right)$ ).

Figure 4.

The time response solution $u\left(\tau \right)$ at different scales of $\tau$ and phase portrait $u\dot{u}$ in resonance case $\left(\Omega =\omega \right)$ without control $\left(G_1=0,G_2=0,G_3=0\right)$ .

Figure 5.

The time response solution $u\left(\tau \right)$ in resonance case $\left(\Omega =\omega \right)$ with various types of controllers: (a, b) NDFC with black color $\left(G_1=2,G_2=0,G_3=0\right),$ NCVFC with red color $\left(G_1=0,G_2=0,G_3=5\right),$ NVFC + NCVFC with green color $\left(G_1=0,G_2=5,G_3=5\right)$ with long and short scales, (c, d) NVFC with blue color $\left(G_1=0,G_2=5,G_3=0\right),$ (e, f) combination of NDFC + NVFC + NCVFC with pink color $\left(G_1=3,G_2=5,G_3=5\right).$

Table 1.

Comparison between different types of controllers.

Controller type	Mathematical representation	Controller gains	Amplitude	Controller effectiveness %
NDFC (black curve)	$-G_{1}u$	$G_1=2$	0.9763	71.57
NCVFC (red curve)	$-G_3{\dot{u}}^3$	$G_3=5$	0.4103	88
NVFC + NCVFC (green curve)	$-G_2\dot{u}-G_3{\dot{u}}^3$	$G_2=G_3=5$	0.1772	94.8
NVFC (blue curve)	$-G_2\dot{u}$	$G_2=5$	0.2	94.18
NDFC + NVFC + NCVFC (pink curve)	$-G_{1}u-G_2\dot{u}-G_3{\dot{u}}^3$	$G_1=3,G_2=5,G_3=5$	0.1722	95

The curves in Figures 6(a) and (b) compare the analytic solution induced by MSP and the numerical solution using RKM at resonance case without control for short and long time intervals, respectively. On the other hand, the comparison of the AS and the NS for the resonance case with control (NVFC + NCVFC) is depicted in Figure 7 in two different time ranges. The comparisons between the analytical and numerical solutions demonstrate considerable agreement.

Figure 6.

The comparison between NS, represented by the red line, and AS, represented by the blue line, in resonance case $\left(\Omega =\omega \right)$ without control $\left(G_1=0,G_2=0,G_3=0\right)$ at various scales of $\tau .$

Figure 7.

The comparison between NS, represented by the red curve, and AS, shown by the blue curve in the resonance case $\left(\Omega =\omega \right)$ with NVFC + NCVFC $\left(G_1=0,G_2=5,G_3=5\right)$ at various scales of $\tau .$

Let us demonstrate the amplitude changes of $u\left(\tau \right)$ when the system parameters, $c_1,G_1,G_2,G_3,$ and $f_e$ are varied. It is observed that when $c_1,G_1,G_2,$ and $G_3$ values grow, the amplitude’s $u\left(\tau \right)$ value falls. Conversely, as $f_e$ grows, increases the amplitude of $u\left(\tau \right)$ as observed in Figure 8.

Figure 8.

System parameter effects $c_1,G_1,G_2,G_3,$ and $f_e$ on system amplitude $u\left(\tau \right).$

The fixed points’ stability regions, explored in Figures 9 and 10 will be presented using the resonant response curves produced from the NS of equation (24). It’s important to keep in mind that solid lines indicate stable fixed points, whereas dashed ones illustrate unstable ones. It was shown that some stability criterion parameters have a substantial impact on stability analysis, such as gains of controllers $G_2,G_3$ , damping coefficient $c_1$ , and frequency $\omega$ . The system amplitude $q$ is shown versus the frequency detuning parameter $\sigma$ in Figures 9 and 10 as the values given for the system parameters change. We observe decreasing of the amplitude with the rise of the value of $c_1,G_2,G_3,$ and $\omega$ , but the amplitude increases with the rise of the value of $f_e.$ The ranges containing the stability areas and instability are $\sigma \le 0$ and $0<\sigma$ , respectively.

Figure 9.

Amplitude’s resonance curve $q$ versus σ: (a) at $G_1=0,G_2=G_3=5,f_e=2,$ (b) at $G_1=0,G_3=5,c_1=0.2,f_e=2,$ (c) at $G_1=0,G_2=5,c_1=0.2,f_e=2$ .

Figure 10.

Amplitude’s resonance curve $q$ versus σ: (a) at $G_1=0,G_2=G_3=5,c_1=0.2,$ (b) at $G_1=0,G_2=G_3=5,c_1=0.2,f_e=2$ .

The frequency response curve of a dynamical system shows how the system’s amplitude response changes with the range of excitation frequency. Figure 11 illustrates the amplitude-frequency response of QZSVI without control $\left(G_1=0,G_2=0,G_3=0\right)$ under different values of $c_1,\omega ,$ and $f_e.$ The amplitude of the system $q$ is decreased with increasing the damping coefficient $c_1$ , as explored in Figure 11(a), while Figure 11(b) presents the largest amplitude at $\Omega =\omega ,$ these results are supported by the time history in Figure 4. As indicated in Figure 11(c), system amplitude increases as the excitation amplitude $f_e$ increases. The amplitude-frequency response with control at various controller gain settings is shown in Figure 12, as the controller gain is increased, the system’s amplitude decreases. The time history in Figure 5(a) with the green curve and amplitude’s resonance curve, as shown in Figure 10, confirm these results. Furthermore, Figure 13 describes the comparison between the amplitude-frequency curve without and with control. It is noted that Figures 4 and 5 provide support for these findings.

Figure 11.

Amplitude–frequency curve $q$ without control $\left(G_1=0,G_2=0,G_3=0\right)$ .

Figure 12.

Amplitude–frequency curve $q$ with different values of controller gains at $G_1=0,c_1=0.2,f_e=2$ .

Figure 13.

Amplitude–frequency curve $q$ without and with control at $G_1=0,c_1=0.2,f_e=2$ .

The system’s behavior, as a parameter varies, is shown graphically by bifurcation curves. They identify transitional phases in the behavior of the system, such as the emergence of new equilibrium points or periodic orbits. Conversely, Lyapunov analysis assesses the stability of the dynamical model through the investigation of the solutions that are in close proximity to it. To ascertain whether little deviations from an equilibrium point result in overtime convergence or divergence, Lyapunov functions are employed. In order to forecast and comprehend complicated phenomena like chaos or stability transitions, bifurcation curves and Lyapunov analysis work together to shed light on the behavior and stability of dynamic systems.^37–39 To display the different forms of motion, bifurcation diagrams of $u\left(\tau \right),$ LEs of ${\lambda }_i\left(i=1,2\right)$ with the damping coefficient $c_1,$ and phase portraits and Poincare´ maps have been executed. The bifurcation diagrams, LEs spectra of $u\left(\tau \right)$ with $c_1,$ and phase portraits and Poincare´ maps at different values of $f_e$ are presented in Figures 14–16 which demonstrate features of various motion types.

Figure 14.

Bifurcation diagram, LEs with the damping coefficient $c_1,$ Poincaré maps, and phase portraits of $u\left(\tau \right)$ (c) at $c_1=0.0009$ and (d) at $c_1=0.0002$ at $f_e=0.45$ .

Figure 15.

Bifurcation diagram, LEs with the damping coefficient $c_1,$ Poincaré maps and phase portraits of $u\left(\tau \right)$ (c) at $c_1=0.0009$ and (d) at $c_1=0.0002$ at $f_e=0.5$ .

Figure 16.

Bifurcation diagram, LEs with the damping coefficient $c_1,$ Poincaré maps, and phase portraits of $u\left(\tau \right)$ (c) at $c_1=0.0009$ and (d) at $c_1=0.0002$ at $f_e=0.7$ .

Diagrams in Figure 14 at $f_e=0.45$ show that the damping coefficient $c_1$ has periods, and each period denotes a different kind of motion for the system. There are positive and negative values in the range at $c_1\in \left[0,0.45*10^{-3}\right)$ in the LE spectrum. The system is moving chaotically as a result, and supported by the bifurcation diagram of $u\left(\tau \right).$ We noticed that the LEs lie below the axis $\lambda =0$ during the period at $c_1\in \left[0.45*10^{-3},1*10^{-3}\right]$ , suggesting that the motion has a periodic pattern. The graphic representation of the bifurcation of $u\left(\tau \right)$ over the same range of $c_1$ supports this conclusion, showing that the system represents a nearly straight line. Graphs in Figure 15 at $f_e=0.5$ demonstrate that $c_1$ has periods, and the system experiences distinct types of motion according to each phase. In the LE spectrum, there are both positive and negative values in the range at $c_1\in \left[0,0.48*10^{-3}\right)$ . As a result, the system is moving chaotically, as seen by the $\left(\tau \right)$ ‘s bifurcation diagram. We observed that during the period at $c_1\in \left[0.48*10^{-3},1*10^{-3}\right]$ , the LEs sit below the axis $\lambda =0$ , indicating that the motion follows a periodic pattern. This result is supported by the graphical representation of $\left(\tau \right)$ ‘s bifurcation across the same range of $c_1$ , which demonstrates that the system resembles a virtually straight line. Plots in Figure 16 at $f_e=0.7$ show that $c_1$ has periods, and the system moves in different ways depending on which phase it is in. The range at $c_1\in \left[0,0.5*10^{-3}\right)$ contains both positive and negative values in the LE spectrum. As a result, the bifurcation diagram of $u\left(\tau \right)$ shows that the system is moving chaotically. We found that the LEs are under the axis $\lambda =0$ throughout its duration, at $c_1\in \left[0.5*10^{-3},1*10^{-3}\right]$ , suggesting that the motion has a periodic pattern. The graphical depiction of $\left(\tau \right)$ ‘s bifurcation throughout the same range as $c_1$ , which shows that the system forms a practically straight line, validates this finding.

The phase portraits and Poincaré maps ¹⁰ are simulated in Figures 14(c) and (d)–16(c) and (d), in which blue curves illustrate the phase portraits while the red dots represent the Poincaré maps. The generated figures were displayed for several values of the damping coefficient $c_1$ in order to demonstrate the system’s various motions. We investigate the value $c_1=0.0009$ in Figures 14(c)–16(c). Poincaré's red-dot pattern maps closely to a single point, supporting the previous conclusion about the system’s periodic motion. As seen in Figures 14(d)–16(d), the red dots are disseminated randomly and diverge due to the chaotic condition that appeared in the value $c_1=0.0002$ .

To emphasize the importance of the presence of feedback control in this system, bifurcation diagrams of $u\left(\tau \right),$ LEs of ${\lambda }_i\left(i=1,2\right)$ with the damping coefficient $c_1,$ and phase portraits and Poincare´ maps were drawn without control $\left(G_1=0,G_2=0,G_3=0\right)$ at $f_e=0.45$ . It became clear that in the absence of control, there are positive and negative values in the range at $c_1\in \left[0,0.59*10^{-3}\right)$ in the LE spectrum. The system is moving chaotically as a result, and supported by the bifurcation diagram of $u\left(\tau \right).$ We noticed that the LEs lie below the axis $\lambda =0$ during the period at $c_1\in \left[0.59*10^{-3},1*10^{-3}\right]$ , indicating that there is a periodic pattern to the motion. The graphic representation of the bifurcation of $u\left(\tau \right)$ over the same range of $c_1$ supports this conclusion, showing that the system represents a nearly straight line as shown in Figure 17. This indicates that the area of chaotic motion is increased, and the area of periodic motion is decreased compared to Figure 14 that drawn at the same parameters with control. In addition to the phase portraits and Poincaré maps are presented in Figures 17(c) and (d). We investigate the value $c_1=0.0009$ in Figure 17(c). The prior conclusion on the periodic motion of the system is supported by Poincaré's red-dot pattern, which maps closely to a single point. As seen in Figure 17(d), The chaotic state that developed causes the red dots to spread erratically and diverge in the value $c_1=0.0002$ .

Figure 17.

Bifurcation diagram, LEs with the damping coefficient $c_1,$ Poincaré maps and phase portraits of $u\left(\tau \right)$ (c) at $c_1=0.0009$ and (d) at $c_1=0.0002$ at $f_e=0.45$ without control $\left(G_1=0,G_2=0,G_3=0\right)$ .

Force, displacement, and acceleration transmissibility

In vibration analysis, force, displacement, and acceleration transmissibility are essential ideas that explain the transmission of vibratory energy throughout a system. ³⁶ The amount of vibrating force that flows through a structure is indicated by its force transmissibility ( $T_f$ ), which is the ratio of transmission force to applied force. In order to determine how much motion is conveyed, displacement transmissibility ( $T_d$ ) calculates the ratio of output displacement to input displacement. Likewise, the ratio of transmitted acceleration to input acceleration is measured by acceleration transmissibility ( $T_a$ ). The force transmitted ( $f_t$ ) toward the load is provided by

$f_t=c_1\dot{u}+k_{1}u+k_2{u}^3+k_3{u}^5+k_4{u}^2\dot{u}+k_5{u}^4\dot{u}+G_{1}u+G_2\dot{u}+G_3{\dot{u}}^3.$

(31)

Assume that

$u=A\hspace{0.17em}\mathit{cos}\hspace{0.17em}\varphi ,\dot{u}=-A\Omega \hspace{0.17em}\mathit{sin}\hspace{0.17em}\varphi .$

(32)here, $\varphi =\Omega \tau +\vartheta ,$ $A$ is the response amplitude and $\vartheta$ is the phase, hence the force, displacement, and acceleration transmissibility are given, respectively, by

$T_f=\sqrt{\frac{\left(1+{\left(2\zeta r\right)}^2\right)}{\left({\left(1-r^2\right)}^2+{\left(2\zeta r\right)}^2\right)}},$

(33)

$T_d=\frac{1}{\sqrt{\left({\left(1-r^2\right)}^2+{\left(2\zeta r\right)}^2\right)}},$

(34)

$T_a=\frac{\sqrt{\left(1+{\left(2\zeta r\right)}^2\right)}}{\sqrt{\left({\left(1-r^2\right)}^2+{\left(2\zeta r\right)}^2\right)}}.$

(35)Here $\zeta$ and $r$ are frequency ratio and damped ratio, respectively, given by

$\zeta =\frac{C_1}{2\sqrt{k_{v}m}},r=\frac{\Omega }{{\omega }_0}.$

(36)

Figures 18–21 show the force, displacement, and acceleration transmissibility without control (blue line) and with control (red line) $\left(G_1=0,G_2=5,G_3=5\right)$ at different values of force excitation $f_e,$ such that Figures 18–21(a, c, d) with log scale in $\Omega$ -axis and linear scale with vertical axis, but Figures 18–21(b) present force, displacement, and acceleration transmissibility with linear scale with two axes. Here is the relationship between force, displacement, acceleration transmissibility, and excitation magnitude. It is evident that peak transmissibility rises with increasing stimulation amplitude. The important role played by control in reducing the transmission of vibratory energy throughout a system, as shown in Figures 18–21, where the force, displacement, and acceleration transmissibility decrease by a very large value with control especially at resonance instance (Table 2).

Figure 18.

Force, displacement, and acceleration transmissibility without control (blue line) and with control (red line) $\left(G_1=0,G_2=5,G_3=5\right)$ at $f_e=0.2$ .

Figure 19.

Force, displacement, and acceleration transmissibility without control (blue line) and with control (red line) $\left(G_1=0,G_2=5,G_3=5\right)$ at $f_e=1$ .

Figure 20.

Force, displacement, and acceleration transmissibility without control (blue line) and with control (red line) $\left(G_1=0,G_2=5,G_3=5\right)$ at $f_e=3$ .

Figure 21.

Force, displacement, and acceleration transmissibility without control (blue line) and with control (red line) $\left(G_1=0,G_2=5,G_3=5\right)$ at $m=10,c_1=0.1,k_1=0.4,k_2=0.04,k_3=0.02,f_e=10$ .

Table 2.

Comparison between the present work and previous ones in Refs. 5 and 36.

No.	Comparison items	The present study	The examined work in Ref. 36	The investigated work in Ref. 5
1	DOF	1DOF	1DOF	1DOF
2	AS	The AS are acquired using MSP up to the second order	The averaging procedure yields the AS	The AS are obtained by the averaging method
3	NS	The NS are obtained RKM from the fourth-order	The numerical simulations are presented	The NS are achieved
4	Stability area	The stability and instability areas are investigated according to Amplitude’s resonance curve $q$ versus $\sigma$	Amplitude’s resonance curves are presented	The stability boundaries are drawn
5	Bifurcation diagrams	Graphs and discussions of the bifurcation diagrams have been conducted	Not obtained	Not achieved
6	Lyapunov exponent spectrums	The Lyapunov exponent spectrums have been obtained and examined	Not achieved	Not examined
7	Poincaré maps	Diagramming and analysis are performed on the Poincaré maps	Not drawn	Not plotted
8	Parameters effect	The parameters effect with amplitudes are plotted	Not mapped out	Not shown
9	Feedback control	Multi-control methods are used (NDFC, NCVFC, NVFC, NVFC + NCVFC, NDFC + NVFC + NCVFC)	Not exist	Displacementvelocity feedback controller is used
10	Diagram of the time history of a solution and phase plan	The time history and phase plan are drawn in all cases of control	Not presented	Not graphed
11	Controller effectiveness	The controller effectiveness is investigated	Not exist	Not investigated
12	Force transmissibility	The force transmissibility is plotted	The force transmissibility is shown	The force transmissibility is investigated
13	Displacement transmissibility	The displacement transmissibility is presented	The displacement transmissibility is graphed	Not drawn
14	Acceleration transmissibility	The acceleration transmissibility is simulated	Not obtained	Not shown

Conclusion

The isolation performance of a QZSVI system can be improved by using the NDFC, NVFC, NCVFC, NVFC + NCVFC, and composite controllers NDFC + NVFC + NCVFC approaches. Any potentially dangerous vibrations are reduced by using these controls. The AS of the system is obtained using a MSP up to the second order. It is then verified numerically by applying the RKM. The ME are acquired, and every case of established external resonance is assessed. The bifurcation plots, Poincaré maps, and LEs show and analyze two different motions of the system: periodic and chaotic motion. The resonance response curves show the locations of the system’s stable and unstable fixed points. It was discovered that as a result of the NDFC, NVFC, NCVFC, NVFC + NCVFC, and composite controllers NDFC + NVFC + NCVFC, the amplitude of $u$ decreased by 71.57%, 94.18%, 88%, 94.8%, and 95%, respectively. The force, displacement, and acceleration transmissibility are calculated and presented; it was found that the force, displacement, and acceleration transmissibility are inversely proportional to the controller gains signal, which gives an essential benefit for the used controller at the resonance instance. QZSVI are utilized in industries such as seismic isolation, aerospace, precise instrumentation, and microelectronics manufacture to shield delicate equipment from low-frequency vibrations. Their capacity to achieve almost zero stiffness, which permits extremely efficient vibration absorption while preserving system stability, is their fundamental benefit, in addition to providing excellent vibration isolation.