Tuning Mechanism and Parameter Optimization of a Dynamic Vibration Absorber with Inerter and Negative Stiffness Under Delayed FOPID

Abstract

The dynamic vibration absorber (DVA) based on delayed fractional PID (DFOPID) can achieve a more superior vibration suppression effect. However, the strong nonlinear characteristics of the system and the computational burden resulting from its high dimensionality make solving and optimizing more challenging. This paper presents a coupled model of DFOPID and DVA, exploring its parameter tuning mechanism and optimization problem. First, using the averaging method and Lyapunov stability theory, the amplitude-frequency equation and the stability condition of the steady-state solution of the primary system are derived. Numerical simulations validate the accuracy of the analytical result. Next, based on the mechanics of vibration, the approximate expressions of the controller under different differential conditions are calculated, and their equivalent action mechanisms are analyzed. Finally, by minimizing the maximum amplitude of the primary system as the objective function, the Particle Swarm Optimization (PSO) algorithm is applied to optimize the parameters of the passive DVA and the DVA models controlled by PID, FOPID, and DFOPID, successfully addressing the parameter optimization challenges posed by traditional fixed-point theory. The vibration reduction performance is compared across different loading environments. The results demonstrate that the model presented in this paper performs the best, exhibiting excellent vibration suppression and robustness.

1. Introduction

In recent years, substantial progress has been made in researching nonlinear problems, although the majority of these studies continue to focus on integer-order systems [1,2,3,4,5]. With the ongoing advancement of industrialization, increasingly complex and dynamic application environments will inevitably necessitate more sophisticated requirements for system modeling and control. Fractional-order systems [6,7,8,9,10], recognized for their distinctive “hereditary” and “memory” effects, are increasingly capturing the interest of scholars.

With the rapid rise of fractional calculus, fractional control theory has also experienced a vigorous development trend. In particular, the fractional PID controller ($P{I}^{\lambda }{D}^{\mu }$ controller) proposed by Podlubny is typical [11,12]. By introducing “order” degrees of freedom in the differential and integral terms of the traditional PID controller, the number of adjustable parameters is increased from 3 to 5, thereby providing the actual system with superior robustness and stability, as well as more flexible, precise, and efficient control performance. It is one of the most powerful tools for vibration suppression. Aboelela et al. [13] applied the fractional PID controller to the six-degree-of-freedom flying object model. The parameters of the controller were optimized through the Particle Swarm Optimization (PSO) algorithm to improve the control accuracy and dynamic response. Sharma et al. [14] investigated the trajectory tracking and disturbance suppression of a two-link manipulator under the fractional fuzzy PID controller. The research results showed that the controller, after being optimized by the cuckoo search algorithm, demonstrated excellent control performance and strong adaptability. Niu et al. [15] considered the free vibration of a Duffing oscillator under a class of displacement feedback fractional PID controllers. The parameter tuning process of the fractional PID was revealed by means of the averaging method. Liu [16] further designed an adaptive PSO algorithm to solve the parameter optimization problem in fractional PID controllers. Liu et al. [17] obtained the approximate analytical solution of the gear system under the speed feedback fractional PID controller by using the incremental harmonic balance method. The resonance frequency and resonance amplitude of the system were effectively adjusted by designing the controller parameters. For the dual-mass wind turbine system, Frikh et al. [18] proposed a fractional PID controller design method by combining phase margin and unit gain crossover frequency. Research showed that this method can effectively improve the speed control performance of the turbine system. Liu et al. [19] designed a robust fractional PID controller, providing a new solution for the yaw control of autonomous underwater vehicles. Ansarian and Mahmoodabadi [20] proposed a fuzzy adaptive robust fractional PID controller for the stable control of unmanned aerial vehicle systems. Shafiee et al. [21] combined a fractional PID controller with a fuzzy logic system to enable adaptive adjustment of control parameters in response to low-frequency oscillations in power systems. To enhance the robustness and anti-interference capability of quadrotor robots. In addition, Sui et al. [22] established a tuned inerter damper with a delayed fractional PID controller. Based on the averaging method and Lyapunov stability theory, the amplitude-frequency equation and stability condition of the primary system were obtained, and the parameters were optimized by combining the grey wolf algorithm.

The dynamic vibration absorber (DVA), also known as the tuned mass damper, is a commonly used vibration reduction device. Its working principle is to absorb the energy of the primary system through resonance opposition, thereby achieving the ultimate goal of vibration reduction in the primary system. Since Frahm proposed the concept of DVA in 1909, many scholars have made significant contributions to its development. Shen et al. [23] considered a type of negative stiffness DVA and obtained the optimal frequency ratio and optimal damping ratio of the system based on the fixed-point theory. Similarly, Shen et al. [24] further studied the vibration reduction performance of the DVA with grounded stiffness and an amplification mechanism. Gao et al. [25] designed a new type of DVA model with inerter and negative stiffness, and combined traditional theory with intelligent algorithm to study its parameter optimization and vibration suppression effect. Li et al. [26] proposed a type of DVA model with negative stiffness and an amplification mechanism, and used fixed-point theory and perturbation method to derive analytical results. They also obtained the optimal parameters of the DVA by means of the PSO algorithm and Newton’s iterative method. On the other hand, with the increasing complexity of actual working conditions, nonlinear vibration absorbers have developed rapidly and gradually matured. On the other hand, with the increasing complexity of actual working conditions, nonlinear dynamic DVAs have developed rapidly and gradually matured. Zang et al. [27] used the generalized transfer coefficient to evaluate the vibration reduction effect of a nonlinear DVA and further revealed the relationship between quasi-periodic motion and the optimal transfer rate. Chen et al. [28] designed a nonlinear energy sink to eliminate multi-modal resonances in composite plate structures. Bian and Jing [29] designed a tunable nonlinear DVA with an X-shaped structure based on bionic concepts. Compared with traditional nonlinear DVAs, it can significantly improve the system’s robustness and expand the vibration suppression bandwidth.

Although passive DVAs perform well in many fields, they cannot provide the best vibration suppression effect in complex and variable actual working conditions due to their strong frequency dependence and lack of adaptability [30,31]. The DVA model based on the delayed fractional PID controller can perceive external excitation or a structural response in real time and dynamically adjust its control force, flexibly responding to different vibration modes and system requirements [22]. Correspondingly, the tuning and optimization problems under multiple parameters will also become more difficult.

Motivated by the above analysis, this paper presents a DVA model that integrates an inerter, grounded negative stiffness, and a delayed fractional-order PID (FOPID) controller. The tuning mechanism and parameter optimization of the system are thoroughly investigated. The rest of the paper is organized as follows: In Section 2, based on the averaging method and Lyapunov stability theory, the amplitude-frequency equation and stability criterion for the steady-state solution of the primary system are derived. In Section 3, a numerical simulation is conducted to validate the accuracy of the aforementioned result. In Section 4, the equivalent control mechanism of the DFOPID is analyzed based on the mechanics of vibration. Furthermore, an optimization framework is established using the PSO algorithm, with the objective of minimizing the peak amplitude of the primary system. The vibration attenuation performance of the proposed absorber is then compared under various loading conditions with that of a passive absorber, as well as absorbers controlled by conventional PID and FOPID controllers. The conclusions are given in Section 5.

In this study, the primary objective and contribution of this study lie in applying the controller to a novel DVA model and integrating it with intelligent algorithms to enhance its performance. The contributions of this work are as follows:

• We propose a novel DVA model that incorporates both an inerter and grounded negative stiffness. This configuration improves the dynamic characteristics and energy dissipation capability of the system.
• A delayed FOPID controller is applied to this new DVA model to enhance its adaptability and control performance in complex dynamic environments. The use of delay and fractional-order features makes the controller more flexible in tuning the damping behavior.
• The mechanism of the controller within the coupled DVA system is thoroughly analyzed, and PSO is employed for parameter optimization. This not only ensures efficient tuning but also broadens the application of fractional-order control in engineering.

2. Model Description and Steady-State Response Analysis

Figure 1 shows the DVA model incorporating an inerter, grounded negative stiffness, and a delayed FOPID controller (DFOPID). m is the mass of the primary system. b and c are the inerter and damping coefficients of the DVA. $K_1$ and $K_2$ represent the stiffness coefficients of the primary system and the DVA, respectively, while $x_1$ and $x_2$ denote the displacements of the primary system and the DVA, respectively. K is the negative stiffness coefficient. The differences between the model presented in this paper and the DVA models in References [25,26] are primarily as follows: (1) although all three models incorporate inerter elements, the vibration absorber system in this study replaces the conventional mass element with an inerter [32,33]; (2) this paper integrates a DFOPID controller into the model, harnessing the benefits of active control to overcome limitations such as the poor adaptability of the original vibration absorber system. The controller $\delta (t-\tau )$ is defined as follows:

$$\begin{array}{c}\hfill \delta (t-\tau )={\delta }_P{x}_1(t-\tau )+{\delta }_I{D}^{-\lambda }{x}_1(t-\tau )+{\delta }_D{D}^{\mu }{x}_1(t-\tau ),\end{array}$$

(1)

where ${\delta }_P$, ${\delta }_I$, and ${\delta }_D$ represent the coefficients of the proportional, integral, and derivative terms of the controller, respectively, while $\lambda$ and $\mu$ denote the orders of the fractional integral and derivative terms, respectively.

According to Newton’s second law, when the primary system is subjected to a harmonic excitation $Fcos\left(\omega t\right)$, the corresponding dynamic equation can be expressed as

$$\left\{\begin{array}{c}m{\ddot{x}}_1+K_1{x}_1+K_2(x_1-x_2)=-\delta (t-\tau )+F\cos \left(\omega t\right),\hfill \\ b{\ddot{x}}_2+K{x}_2+K_2(x_2-x_1)+c{\dot{x}}_2=0.\hfill \end{array}\right.$$

(2)

Introduce the parameter transformation

$$\begin{array}{c}\hfill ϵ={\displaystyle \frac{b}{m}},{\omega }_0=\sqrt{{\displaystyle \frac{K_1}{m}}},ϵk_2={\displaystyle \frac{K_2}{m}},ϵk={\displaystyle \frac{K}{m}},ϵ\xi ={\displaystyle \frac{c}{m}},ϵ{\delta }_p={\displaystyle \frac{{\delta }_P}{m}},ϵ{\delta }_i={\displaystyle \frac{{\delta }_I}{m}},ϵ{\delta }_d={\displaystyle \frac{{\delta }_D}{m}},ϵf={\displaystyle \frac{F}{m}},\end{array}$$

system (2) is transformed into

$$\left\{\begin{array}{cc}\hfill {\ddot{x}}_1& +{\omega }_0^2{x}_1+ϵk_2(x_1-x_2)+ϵ{\delta }_p{x}_1(t-\tau )+ϵ{\delta }_i{D}^{-\lambda }{x}_1(t-\tau )\hfill \\ \hfill & +ϵ{\delta }_d{D}^{\mu }{x}_1(t-\tau )=ϵf\cos \left(\omega t\right),\hfill \\ \hfill {\ddot{x}}_2& +k{x}_2+k_2(x_2-x_1)+\xi {\dot{x}}_2=0.\hfill \end{array}\right.$$

(3)

For nonlinear systems, frequency matching either within the system or with external excitation may lead to amplified responses concentrated at certain frequencies, such as internal resonance or primary resonance. For example, Liu et al. [34] designed and experimentally validated a broadband dual-functional device leveraging 1:2 internal resonance, capable of simultaneously achieving vibration mitigation and energy harvesting within an integrated structure. In this paper, primary resonance is analyzed by introducing a detuning parameter $\sigma$ via ${\omega }^2={\omega }_0^2+ϵ\sigma ,$ where $\sigma$ (units: ${\mathrm{rad}}^2/{\mathrm{s}}^2$) quantifies the scaled squared-frequency deviation. Physically, this corresponds to $\sigma \approx 2{\omega }_0(\omega -{\omega }_0)/ϵ$, where $\omega -{\omega }_0$ (units: rad/s) is the linear detuning frequency. This normalization simplifies the perturbation analysis while retaining the physical essence of detuning. Then system (3) can be transformed into

$$\left\{\begin{array}{cc}\hfill {\ddot{x}}_1& +{\omega }^2{x}_1=ϵ[f\cos \left(\omega t\right)+\sigma x_1-k_2(x_1-x_2)-{\delta }_p{x}_1(t-\tau )\hfill \\ \hfill & -{\delta }_i{D}^{-\lambda }{x}_1(t-\tau )-{\delta }_d{D}^{\mu }{x}_1(t-\tau )],\hfill \\ \hfill {\ddot{x}}_2& +k{x}_2+k_2(x_2-x_1)+\xi {\dot{x}}_2=0.\hfill \end{array}\right.$$

(4)

Using the averaging method, assume that the solution of system (4) satisfies

$$x_1=r_1\cos (\omega t+{\vartheta }_1),x_2=r_2\cos (\omega t+{\vartheta }_2).$$

(5)

Substituting Equation (5) into system (4), we obtain

$$\left\{\begin{array}{c}{\dot{r}}_1\cos {\varphi }_1-r_1{\dot{\vartheta }}_1\sin {\varphi }_1=0,\hfill \\ {\dot{r}}_1\sin {\varphi }_1+r_1{\dot{\vartheta }}_1\cos {\varphi }_1={\displaystyle \frac{ϵ}{\omega }}(U_2-U_1),\hfill \end{array}\right.$$

(6a)

$$\left\{\begin{array}{c}{\dot{r}}_2\cos {\varphi }_2-r_2{\dot{\vartheta }}_2\sin {\varphi }_2=0,\hfill \\ {\dot{r}}_2\sin {\varphi }_2+r_2{\dot{\vartheta }}_2\cos {\varphi }_2=-{\displaystyle \frac{1}{\omega }}{U}_3,\hfill \end{array}\right.$$

(6b)

where

• ${\varphi }_1=\omega t+{\vartheta }_1$, ${\varphi }_2=\omega t+{\vartheta }_2$,
• $U_1=f\cos \left(\omega t\right)+\sigma r_1\cos {\varphi }_1-k_2(r_1\cos {\varphi }_1-r_2\cos {\varphi }_2)-{\delta }_p{r}_1\cos ({\varphi }_1-\omega \tau ),$
• $U_2={\delta }_i{D}^{-\lambda }\left[r_1\cos ({\varphi }_1-\omega \tau )\right]+{\delta }_d{D}^{\mu }\left[r_1\cos ({\varphi }_1-\omega \tau )\right],$
• $U_3=({\omega }^2-k)r_2\cos {\varphi }_2-k_2(r_2\cos {\varphi }_2-r_1\cos {\varphi }_1)+r_2\xi \omega \sin {\varphi }_2.$

Solving Equation (6) yields

$$\left\{\begin{array}{c}{\dot{r}}_1={\displaystyle \frac{ϵ}{\omega }}(U_2-U_1)\sin {\varphi }_1,\hfill \\ r_1{\dot{\vartheta }}_1={\displaystyle \frac{ϵ}{\omega }}(U_2-U_1)\cos {\varphi }_1,\hfill \end{array}\right.$$

(7a)

$$\left\{\begin{array}{c}{\dot{r}}_2=-{\displaystyle \frac{1}{\omega }}{U}_3\sin {\varphi }_2,\hfill \\ r_2{\dot{\vartheta }}_2=-{\displaystyle \frac{1}{\omega }}{U}_3\cos {\varphi }_2.\hfill \end{array}\right.$$

(7b)

Based on the averaging method, by integrating Equation (7) over the interval $[0,T]$, we obtain the approximate expressions for the amplitudes $r_1$ and $r_2$, as well as the phases ${\vartheta }_1$ and ${\vartheta }_2$, namely,

$$\left\{\begin{array}{c}{\dot{r}}_1={\displaystyle \frac{ϵ}{\omega T}}{\int }_0^T(U_2-U_1)\sin {\varphi }_1\mathrm{d}{\varphi }_1,\hfill \\ r_1{\dot{\vartheta }}_1={\displaystyle \frac{ϵ}{\omega T}}{\int }_0^T(U_2-U_1)\cos {\varphi }_1\mathrm{d}{\varphi }_1,\hfill \end{array}\right.$$

(8a)

$$\left\{\begin{array}{c}{\dot{r}}_2=-{\displaystyle \frac{1}{\omega T}}{\int }_0^T{U}_3\sin {\varphi }_2\mathrm{d}{\varphi }_2,\hfill \\ r_2{\dot{\vartheta }}_2=-{\displaystyle \frac{1}{\omega T}}{\int }_0^T{U}_3\cos {\varphi }_2\mathrm{d}{\varphi }_2.\hfill \end{array}\right.$$

(8b)

If the integrand is a periodic function and the upper limit of integration T is taken as $2\pi$, then we have

$$\begin{array}{cc}\hfill V_{11}& =-{\displaystyle \frac{ϵ}{2\pi \omega }}{\int }_0^{2\pi }{U}_1\sin {\varphi }_1\mathrm{d}{\varphi }_1\hfill \\ \hfill & =-{\displaystyle \frac{ϵ}{2\omega }}\left[f\sin {\vartheta }_1+k_2{r}_2\sin ({\vartheta }_1-{\vartheta }_2)-{\delta }_p{r}_1\sin \left(\omega \tau \right)\right],\hfill \end{array}$$

(9a)

$$\begin{array}{cc}\hfill V_{21}& =-{\displaystyle \frac{ϵ}{2\pi \omega }}{\int }_0^{2\pi }{U}_1\cos {\varphi }_1\mathrm{d}{\varphi }_1\hfill \\ \hfill & =-{\displaystyle \frac{ϵ}{2\omega }}\left[f\cos {\vartheta }_1+\sigma r_1-k_2{r}_1+k_2{r}_2\cos ({\vartheta }_1-{\vartheta }_2)-{\delta }_p{r}_1\cos \left(\omega \tau \right)\right],\hfill \end{array}$$

(9b)

$$\begin{array}{cc}\hfill V_{31}& =-{\displaystyle \frac{1}{2\pi \omega }}{\int }_0^{2\pi }{U}_3\sin {\varphi }_2\mathrm{d}{\varphi }_2\hfill \\ \hfill & ={\displaystyle \frac{1}{2\omega }}\left[k_2{r}_1\sin ({\vartheta }_1-{\vartheta }_2)-r_2\xi \omega \right],\hfill \end{array}$$

(9c)

$$\begin{array}{cc}\hfill V_{41}& =-{\displaystyle \frac{1}{2\pi \omega }}{\int }_0^{2\pi }{U}_3\cos {\varphi }_2\mathrm{d}{\varphi }_2\hfill \\ \hfill & ={\displaystyle \frac{1}{2\omega }}\left[(k+k_2-{\omega }^2)r_2-k_2{r}_1\cos ({\vartheta }_1-{\vartheta }_2)\right].\hfill \end{array}$$

(9d)

If the integrand is a non-periodic function, the upper limit of integration T is taken as $+\infty$. With reference to [10], we obtain

$$\begin{array}{cc}\hfill V_{12}& ={\displaystyle \frac{ϵ}{\omega T}}{\int }_0^T{U}_2\sin {\varphi }_1\mathrm{d}{\varphi }_1\hfill \\ \hfill & ={\displaystyle \frac{ϵr_1}{2}}\left[{\delta }_i{\omega }^{-\lambda -1}\sin ({\displaystyle \frac{\lambda \pi }{2}}+\omega \tau )-{\delta }_d{\omega }^{\mu -1}\sin ({\displaystyle \frac{\mu \pi }{2}}-\omega \tau )\right],\hfill \end{array}$$

(10a)

$$\begin{array}{cc}\hfill V_{22}& ={\displaystyle \frac{ϵ}{\omega T}}{\int }_0^T{U}_2\cos {\varphi }_1\mathrm{d}{\varphi }_1\hfill \\ \hfill & ={\displaystyle \frac{ϵr_1}{2}}\left[{\delta }_i{\omega }^{-\lambda -1}\cos ({\displaystyle \frac{\lambda \pi }{2}}+\omega \tau )+{\delta }_d{\omega }^{\mu -1}\cos ({\displaystyle \frac{\mu \pi }{2}}-\omega \tau )\right].\hfill \end{array}$$

(10b)

By associating Equations (9) and (10), it can be inferred that

$$\left\{\begin{array}{c}{\dot{r}}_1={\displaystyle \frac{ϵr_1}{2}}\left[{\delta }_i{\omega }^{-\lambda -1}\sin ({\displaystyle \frac{\lambda \pi }{2}}+\omega \tau )-{\delta }_d{\omega }^{\mu -1}\sin ({\displaystyle \frac{\mu \pi }{2}}-\omega \tau )\right]\hfill \\ -{\displaystyle \frac{ϵ}{2\omega }}\left[f\sin {\vartheta }_1+k_2{r}_2\sin ({\vartheta }_1-{\vartheta }_2)-{\delta }_p{r}_1\sin \left(\omega \tau \right)\right],\hfill \\ r_1{\dot{\vartheta }}_1={\displaystyle \frac{ϵr_1}{2}}\left[{\delta }_i{\omega }^{-\lambda -1}\cos ({\displaystyle \frac{\lambda \pi }{2}}+\omega \tau )+{\delta }_d{\omega }^{\mu -1}\cos ({\displaystyle \frac{\mu \pi }{2}}-\omega \tau )\right]\hfill \\ -{\displaystyle \frac{ϵ}{2\omega }}\left[f\cos {\vartheta }_1+\sigma r_1-k_2{r}_1+k_2{r}_2\cos ({\vartheta }_1-{\vartheta }_2)-{\delta }_p{r}_1\cos \left(\omega \tau \right)\right],\hfill \end{array}\right.$$

(11a)

$$\left\{\begin{array}{c}{\dot{r}}_2={\displaystyle \frac{1}{2\omega }}\left[k_2{r}_1\sin ({\vartheta }_1-{\vartheta }_2)-r_2\xi \omega \right],\hfill \\ r_2{\dot{\vartheta }}_2={\displaystyle \frac{1}{2\omega }}\left[(k+k_2-{\omega }^2)r_2-k_2{r}_1\cos ({\vartheta }_1-{\vartheta }_2)\right].\hfill \end{array}\right.$$

(11b)

By simplifying Equation (11b), we obtain

$$\left\{\begin{array}{c}\sin {\vartheta }_2={\displaystyle \frac{\left[(k+k_2-{\omega }^2)\sin {\vartheta }_1-\xi \omega \cos {\vartheta }_1\right]r_2}{k_2{r}_1}},\hfill \\ \cos {\vartheta }_2={\displaystyle \frac{\left[(k+k_2-{\omega }^2)\cos {\vartheta }_1+\xi \omega \sin {\vartheta }_1\right]r_2}{k_2{r}_1}}.\hfill \end{array}\right.$$

(12)

By eliminating ${\vartheta }_2$, the amplitude-frequency equation for system (3) derived as

$$\begin{array}{c}\hfill r_2^2={\displaystyle \frac{k_2^2{r}_1^2}{{\xi }^2{\omega }^2+{(k+k_2-{\omega }^2)}^2}}.\end{array}$$

(13)

By substituting Equation (11a), we get

$$\left\{\begin{array}{cc}\hfill \sin {\vartheta }_1=& {\displaystyle \frac{r_1}{f}}[{\delta }_i{\omega }^{-\lambda }\sin ({\displaystyle \frac{\lambda \pi }{2}}+\omega \tau )-{\delta }_d{\omega }^{\mu }\cos ({\displaystyle \frac{\mu \pi }{2}}-\omega \tau )\hfill \\ \hfill & +{\delta }_p\sin \left(\omega \tau \right)-{\displaystyle \frac{\xi \omega k_2^2}{{\xi }^2{\omega }^2+{(k+k_2-{\omega }^2)}^2}}],\hfill \\ \hfill \cos {\vartheta }_1=& {\displaystyle \frac{r_1}{f}}[{\delta }_i{\omega }^{-\lambda }\cos ({\displaystyle \frac{\lambda \pi }{2}}+\omega \tau )+{\delta }_d{\omega }^{\mu }\cos ({\displaystyle \frac{\mu \pi }{2}}-\omega \tau )+k_2\hfill \\ \hfill & -\sigma +{\delta }_p\cos \left(\omega \tau \right)-{\displaystyle \frac{(k+k_2-{\omega }^2)k_2^2}{{\xi }^2{\omega }^2+{(k+k_2-{\omega }^2)}^2}}].\hfill \end{array}\right.$$

(14)

Using the relation ${\sin }^2{\vartheta }_1+{\cos }^2{\vartheta }_1=1$, the term ${\vartheta }_1$ in Equation (14) can be eliminated. Substituting Equation (13) into the resulting expression leads to the amplitude-frequency equation of the primary system

$$\begin{array}{cc}\hfill r_1=& f/\left\{\right[{\delta }_i{\omega }^{-\lambda }\sin ({\displaystyle \frac{\lambda \pi }{2}}+\omega \tau )-{\delta }_d{\omega }^{\mu }\cos ({\displaystyle \frac{\mu \pi }{2}}-\omega \tau )+{\delta }_p\sin \left(\omega \tau \right)\hfill \\ \hfill & -{\displaystyle \frac{\xi \omega k_2^2}{{\xi }^2{\omega }^2+{(k+k_2-{\omega }^2)}^2}}{]}^2+[{\delta }_i{\omega }^{-\lambda }\cos ({\displaystyle \frac{\lambda \pi }{2}}+\omega \tau )\hfill \\ \hfill & +{\delta }_d{\omega }^{\mu }\cos ({\displaystyle \frac{\mu \pi }{2}}-\omega \tau )+k_2-\sigma +{\delta }_p\cos \left(\omega \tau \right)\hfill \\ \hfill & -{\displaystyle \frac{(k+k_2-{\omega }^2)k_2^2}{{\xi }^2{\omega }^2+{(k+k_2-{\omega }^2)}^2}}{]}^2{\}}^{{\displaystyle \frac{1}{2}}}.\hfill \end{array}$$

(15)

The stability of the system’s periodic solution is analyzed using the Lyapunov stability theory. The non-zero constant particular solutions ${\overline{r}}_1,{\overline{\vartheta }}_1$ and ${\overline{r}}_2,{\overline{\vartheta }}_2$ in Equation (11) correspond to the steady-state periodic motion of the system. By setting ${\dot{r}}_1=0,{\dot{\vartheta }}_1=0,$ ${\dot{r}}_2=0,$ ${\dot{\vartheta }}_2=0$ and introducing the perturbation variables

$$\begin{array}{c}\hfill \Delta r_1=r_1-{\overline{r}}_1,\Delta {\vartheta }_1={\vartheta }_1-{\overline{\vartheta }}_1,\Delta r_2=r_2-{\overline{r}}_2,,\Delta {\vartheta }_2={\vartheta }_2-{\overline{\vartheta }}_2,\end{array}$$

the linear perturbation equations for the steady-state solution $({\overline{r}}_1,{\overline{\vartheta }}_1,{\overline{r}}_2,{\overline{\vartheta }}_2)$ be obtained, that is,

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left[\begin{array}{c}\Delta r_1\\ \Delta {\vartheta }_1\\ \Delta r_2\\ \Delta {\vartheta }_2\end{array}\right]=\mathbf{J}\left[\begin{array}{c}\Delta r_1\\ \Delta {\vartheta }_1\\ \Delta r_2\\ \Delta {\vartheta }_2\end{array}\right]=\left[\begin{array}{cccc}{H}_{11}& H_{12}& H_{13}& H_{14}\\ H_{21}& H_{22}& H_{23}& H_{24}\\ H_{31}& H_{32}& H_{33}& H_{34}\\ H_{41}& H_{42}& H_{43}& H_{44}\end{array}\right]\left[\begin{array}{c}\Delta r_1\\ \Delta {\vartheta }_1\\ \Delta r_2\\ \Delta {\vartheta }_2\end{array}\right],$$

(16)

where the elements of $\mathbf{J}$ are described in Appendix A. The characteristic equation of $\mathbf{J}$ is given by

$$\begin{array}{c}\hfill {\chi }^4+a_1{\chi }^3+a_2{\chi }^2+a_3\chi +a_4=0.\end{array}$$

According to the Routh–Hurwitz criterion, a sufficient condition for the steady-state solution $({\overline{r}}_1,{\overline{\vartheta }}_1,{\overline{r}}_2,{\overline{\vartheta }}_2)$ to be asymptotically stable is as follows:

$$\begin{array}{c}\hfill a_1>0,a_4>0,a_1{a}_2>a_3,(a_1{a}_2-a_3)a_3>a_4{a}_1^2.\end{array}$$

(17)

The expressions of $a_1,a_2,a_3,a_4$ in the above equation are given in Appendix B.

3. Numerical Validation

System (3) is simulated using a numerical calculation method to validate the accuracy and precision of the obtained approximate analytical solution. When $\mu \in (0,1)$, the discretized iteration scheme for system (3) is given by

$$\begin{array}{cc}\hfill & x_1\left(t_m\right)=x_1\left(t_{m-1}\right)+h{y}_1\left(t_{m-1}\right),\hfill \\ \hfill & x_2\left(t_m\right)=x_2\left(t_{m-1}\right)+h{y}_2\left(t_{m-1}\right),\hfill \\ \hfill & y_1\left(t_m\right)=y_1\left(t_{m-1}\right)+h[ϵf\cos \left(\omega t_m\right)-{\omega }_0^2{x}_1\left(t_m\right)-ϵk_2(x_1\left(t_m\right)-x_2\left(t_m\right))\hfill \\ \hfill & -ϵ{\delta }_p{x}_1\left(t_{m-l}\right)-ϵ{\delta }_i{z}_1\left(t_{m-1-l}\right)-ϵ{\delta }_d{z}_2\left(t_{m-1-l}\right)],\hfill \\ \hfill & y_2\left(t_m\right)=y_2\left(t_{m-1}\right)+h\left[k_2(x_1\left(t_m\right)-x_2\left(t_m\right))-k{x}_2\left(t_m\right)-\xi y_2\left(t_{m-1}\right)\right],\hfill \\ \hfill & z_1\left(t_m\right)=x_1\left(t_m\right)h^{\lambda }-\sum _{i=1}^m{C}_m^{\lambda }{z}_1\left(t_{m-i}\right),\hfill \\ \hfill & z_2\left(t_m\right)=y_1\left(t_m\right)h^{1-\mu }-\sum _{i=1}^m{C}_m^{1-\mu }{z}_2\left(t_{m-i}\right),\hfill \end{array}$$

(18)

where $t_m=mh$ is the time sampling point, and $h=0.01$ is the time step size. $l={\displaystyle \frac{\tau }{h}}$, with $l\in \mathbb{N}$. $C_n^q$ is the fractional binomial coefficient, which satisfies

$$\begin{array}{c}\hfill C_0^q=1,C_j^q=(1-{\displaystyle \frac{1+q}{j}})C_{j-1}^q.\end{array}$$

When $\mu \in (1,2)$, $z_2\left(t_m\right)$ in Equation (18) should be replaced with

$$\begin{array}{cc}\hfill z_2\left(t_m\right)=& h^{2-\mu }[ϵf\cos \left(\omega t_m\right)-{\omega }_0^2{x}_1\left(t_m\right)-ϵk_2(x_1\left(t_m\right)-x_2\left(t_m\right))-ϵ{\delta }_p{x}_1\left(t_{m-l}\right)\hfill \\ \hfill & -ϵ{\delta }_i{z}_1\left(t_{m-1-l}\right)-ϵ{\delta }_d{z}_2\left(t_{m-1-l}\right)]-\sum _{i=1}^m{C}_m^{2-\mu }{z}_2\left(t_{m-i}\right).\hfill \end{array}$$

Let the static displacement be $A_{st}={\displaystyle \frac{F}{K_1}}$. The displacement amplitude transfer function of the primary system is then given by

$$\begin{array}{c}\hfill T_A={\displaystyle \frac{r_1}{A_{st}}}.\end{array}$$

(19)

Let $\mathsf{\Omega }={\displaystyle \frac{\omega }{{\omega }_0}}$. Figure 2a shows the numerical solution and the approximate analytical solution of system (3) when $\mathsf{\Omega }=1$. Furthermore, by taking the maximum steady-state response amplitude for different values of $\mathsf{\Omega }$ and combining this with Equation (19), the amplitude-frequency curve of the primary system is plotted, as shown in Figure 2b. From the figure, it can be observed that the numerical solution and the approximate analytical solution agree well, indicating the reliability and validity of the above theoretical analysis.

4. Parameter Analysis and Optimization

Next, we analyze the impact of the integral term order on the amplitude-frequency curve (related results for the derivative term can be found in [10]) and further explore the parameter tuning mechanism of the DFOPID. Matlab codes for reproducing the results in this work are available at https://github.com/jlli-hust/math_code (accessed on 22 May 2025).

4.1. Parameter Analysis

Keeping all other parameters constant, the integral term order $\lambda$ is set to 0.1, 0.3, 0.5, 0.7, and 0.9, respectively. The corresponding amplitude-frequency curves of the primary system are illustrated in Figure 3a. It can be seen that as $\lambda$ increases, the second peak of the amplitude-frequency curve shifts downward vertically. However, the larger the value of $\lambda$, the smaller the change in amplitude. Hence, increasing parameter $\lambda$ contributes to improved robustness of the primary system. Figure 3b shows the amplitude-frequency curves of the primary system for $\lambda \in (1,2)$. As illustrated, the overall trend of the curve is similar to that in Figure 3a. Furthermore, by observing Figure 3, it is evident that when $\lambda \in (0,1)$, the damping efficiency of the controller outperforms that when $\lambda \in (1,2)$. Therefore, in the subsequent study, we will focus solely on the parameter optimization problem under the condition $\lambda \in (0,1)$.

Figure 3c illustrates how variations in parameter $\tau$ influence the amplitude-frequency response of the primary system. As illustrated in the figure, as parameter $\tau$ increases toward 1, the second peak of the amplitude-frequency curve decreases significantly. However, when $\tau$ exceeds 1, the peak starts to increase, indicating that a moderate increase in the time-delayed term can reduce the amplitude of the primary system, thereby enhancing the robustness of the system.

By combining Equations (9) and (10), the approximate analytical expression for the DFOPID is derived

$$\begin{array}{cc}\hfill \delta (t-\tau )=& r_1[{\delta }_P\cos ({\varphi }_1-\omega \tau )+{\delta }_I{\omega }^{-\lambda }\cos ({\varphi }_1-\omega \tau -{\displaystyle \frac{\lambda \pi }{2}})\hfill \\ \hfill & +{\delta }_D{\omega }^{\mu }\cos ({\varphi }_1-\omega \tau +{\displaystyle \frac{\mu \pi }{2}})].\hfill \end{array}$$

(20)

In addition, based on the mechanics of vibration, the elastic, damping, and inertial forces of the primary system are presented in the following forms:

$$\begin{array}{c}\hfill {\overline{F}}_k=\overline{k}{x}_1=\overline{k}{r}_1\cos {\varphi }_1,{\overline{F}}_c=\overline{c}{\dot{x}}_1=-\overline{c}\omega r_1\sin {\varphi }_1,{\overline{F}}_b=\overline{b}{\ddot{x}}_1=-\overline{b}{\omega }^2{r}_1\cos {\varphi }_1,\end{array}$$

(21)

where $\overline{k}$, $\overline{c}$, and $\overline{b}$ are the elastic, damping, and inertia coefficients, respectively.

When $\mu \in (0,1)$, Equation (20) is expanded using Equation (21), yielding

$$\begin{array}{c}\hfill \delta (t-\tau )={\overline{k}}_1{x}_1+{\overline{c}}_1{\dot{x}}_1,\end{array}$$

(22)

where

$$\begin{array}{cc}\hfill & {\overline{k}}_1={\delta }_P\cos \left(\omega \tau \right)+{\omega }^{-\lambda }{\delta }_I\cos ({\displaystyle \frac{\lambda \pi }{2}}+\omega \tau )+{\omega }^{\mu }{\delta }_D\cos ({\displaystyle \frac{\mu \pi }{2}}-\omega \tau ),\hfill \\ \hfill & {\overline{c}}_1={\omega }^{-\lambda -1}\left[{\omega }^{\lambda +\mu }{\delta }_D\sin ({\displaystyle \frac{\mu \pi }{2}}-\omega \tau )-{\omega }^{\lambda }{\delta }_P\sin \left(\omega \tau \right)-{\delta }_I\sin ({\displaystyle \frac{\lambda \pi }{2}}+\omega \tau )\right].\hfill \end{array}$$

Similarly, when $\mu \in (1,2)$, we have

$$\begin{array}{c}\hfill \delta (t-\tau )={\overline{k}}_2{x}_1+{\overline{c}}_2{\dot{x}}_1+{\overline{b}}_2{\ddot{x}}_1,\end{array}$$

(23)

where

$$\begin{array}{cc}\hfill & {\overline{k}}_2={\delta }_P\cos \left(\omega \tau \right)+{\omega }^{-\lambda }{\delta }_I\cos ({\displaystyle \frac{\lambda \pi }{2}}+\omega \tau ),\hfill \\ \hfill & {\overline{c}}_2={\omega }^{-\lambda -1}\left[{\omega }^{\lambda +\mu }{\delta }_D\sin ({\displaystyle \frac{\mu \pi }{2}}-\omega \tau )-{\omega }^{\lambda }{\delta }_P\sin \left(\omega \tau \right)-{\delta }_I\sin ({\displaystyle \frac{\lambda \pi }{2}}+\omega \tau )\right],\hfill \\ \hfill & {\overline{b}}_2=-{\omega }^{\mu -2}{\delta }_D\cos ({\displaystyle \frac{\mu \pi }{2}}-\omega \tau ).\hfill \end{array}$$

By examining Equations (22) and (23), it can be seen that the parameters of DFOPID (time delay $\tau$; proportional-integral-derivative coefficients ${\delta }_P,{\delta }_I,{\delta }_D$; and orders $\lambda ,\mu$) adjust the damping performance of the primary system through elastic, damping, and inertial forces. However, due to the large number of parameters and the complex coupling relationships, traditional fixed-point theory optimization methods are no longer applicable. Therefore, this section will employ the PSO algorithm for parameter optimization.

4.2. Parameter Optimization

The Particle Swarm Optimization (PSO) algorithm is an optimization technique rooted in swarm intelligence, inspired by the foraging behavior of bird flocks. It seeks the optimal solution by simulating the movement of a group of particles within the search space. The fundamental principles of PSO can be summarized in the following steps:

1. Particle Swarm Initialization: Initially, a set of particles is randomly generated in the solution space, with each particle representing a potential solution. Each particle has two key attributes: position and velocity. The position corresponds to the specific value of the solution, while the velocity determines the direction and step size of the particle’s movement within the solution space.

2. Particle Fitness Evaluation: For each particle, the fitness value (objective function value) corresponding to its current position is computed. A higher fitness value indicates that the solution is closer to the optimal one.

3. Velocity and Position Update: In each iteration, the particles update both their velocity and position. The velocity update formula is

$$\begin{array}{c}\hfill {\upsilon }_i(t+1)=w{\upsilon }_i\left(t\right)+c_1{s}_1(p_{best}-x_i\left(t\right))+c_2{s}_2(g_{best}-x_i\left(t\right)),\end{array}$$

(24)

where ${\upsilon }_i\left(t\right)$ and $x_i\left(t\right)$ represent the velocity and position of particle i at time t. $p_{best}$ is the best position found by particle i. $g_{best}$ is the best position found by all particles. $s_1$ and $s_2$ are random numbers in the interval $[0,1]$. w is the inertia weight, which controls the degree to which the particle retains its current velocity. $c_1$ and $c_2$ are learning factors that control the particle’s ability to learn from both its own best position and the global best position. Generally, $c_1,c_2\in [0,2]$.

Additionally, the position update formula can be expressed as

$$\begin{array}{c}\hfill x_i(t+1)=x_i\left(t\right)+{\upsilon }_i(t+1).\end{array}$$

(25)

After the particle’s velocity is updated, it moves to a new position based on the updated velocity.

4. Individual and Global Best Update: Each particle compares its current fitness value with its historical best fitness value, $p_{best}$. If the current fitness is better, the individual best position is updated. Meanwhile, the position corresponding to the particle with the best fitness among all particles is set as the global best position, $g_{best}$.

5. Termination Condition: The Particle Swarm Optimization algorithm terminates when a specified condition is met, such as reaching the maximum number of iterations or when the fitness value exceeds a defined threshold.

The goal is to minimize the maximum amplitude of the primary system, with parameter optimization carried out using the Particle Swarm Optimization algorithm. The corresponding objective function can be defined as

$$\begin{array}{c}\hfill A=\underset{p\in D}{min}\left[\underset{0\ll \mathsf{\Omega }\le 3}{max}{T}_A\left(p\right)\right],\end{array}$$

(26)

where p represents the parameters to be optimized, and D is the parameter optimization range, satisfying $k_2\in [0,200]$, $k\in [-100,0]$, $\xi \in [0,8]$, ${\delta }_p,{\delta }_i,{\delta }_d\in [0,80]$, $\lambda \in [0,1]$, $\mu \in [0,2]$, and $\tau \in [0.01,1]$. Note that the above parameter ranges are established based on the variability of complex application environments and the diversity of materials used in the model. The maximum inertia weight is $w_{max}=0.8$, the minimum inertia weight is $w_{min}=0.2$, and the learning factors are $c_1=c_2=1.5$. The optimization results of the passive vibration absorber as well as the PID, FOPID, and DFOPID (See Figure 1) controllers are summarized in

Table 1. The optimal parameter obtained using the PSO algorithm.

Parameter	Passive	PID	FOPID	DFOPID	Units
$k_2$	178.07	178.07	178.07	178.07	N/m
k	−63.31	−63.31	−63.31	−63.31	N/m
$\xi$	6.19	6.19	6.19	6.19	$\mathrm{N}·\mathrm{s}/\mathrm{m}$
${\delta }_p$	—	80	73.99	76.76	N/m
${\delta }_i$	—	80	69.81	65.60	$\mathrm{N}·{\mathrm{s}}^{-\lambda }/\mathrm{m}$
${\delta }_d$	—	80	75.48	78.02	$\mathrm{N}·{\mathrm{s}}^{\mu }/\mathrm{m}$
$\lambda$	—	1	0.0012	0.39	—
$\mu$	—	1	0.96	1.16	—
$\tau$	—	—	—	0.07	s
A	2.62	1.18	1.14	1.09	—

, while the corresponding iteration curves are illustrated in Figure 4a.

The termination condition is 800 iterations, with a total of 80 particles. As observed, the amplitude of the primary system rapidly decreases during the initial iterations and gradually stabilizes as the number of iterations increases. Ultimately, the optimal $T_A$ values are $2.62$, $1.18$, $1.14$, and $1.09$. Substituting the optimized parameters into Equation (19), the corresponding amplitude-frequency curves are shown in Figure 4b. A comparison reveals a good match for the maximum amplitudes, confirming the accuracy of the intelligent algorithm. Additionally, as seen in Figure 4b, the DFOPID demonstrates superior vibration reduction performance compared to the other models, with the amplitude-frequency curve exhibiting nearly equal double peaks, thus validating the effectiveness of the model presented in this paper.

4.3. Comparison of Models Under Stochastic and Impulsive Excitation

Many engineering systems (such as mechanical equipment, building structures, vehicles, etc.) are not always subjected to ideal harmonic excitation but also need to account for the effects of complex random and impulsive excitations. Random excitation refers to signals that are uncertain and unpredictable, with parameters such as amplitude, frequency, and duration that cannot be precisely forecasted. These signals typically follow a statistical distribution or stochastic process. Impulsive excitation, on the other hand, is a high-energy signal that occurs over a very short duration, concentrated in time but containing a broad range of frequency components. Assume the primary system is subjected to white noise excitation with a zero mean and a power spectral density of $S\left(\omega \right)=S_0$. The power spectral density (PSD) function of the primary system’s displacement response is then given by

$$\begin{array}{c}\hfill S\left(\omega \right)=|X_{T_A}{|}^2{S}_0.\end{array}$$

Additionally, the simulation duration is set to 100 s. The results are presented in Figure 5.

Figure 5 visually illustrates the impact of using a vibration absorber on the primary system’s vibration reduction performance under white noise excitation. As observed, with the addition of the passive vibration absorber, as well as PID, FOPID, and DFOPID controllers, the displacement of the primary system is significantly reduced, demonstrating clear damping advantages. To further quantify the analysis, the displacement variance statistics and attenuation ratios of the primary system shown in Figure 5 are calculated, as presented in

Table 2. Variances of the primary system displacement and its decrease ratios.

Models	Variances	Decrease Ratios (%)
Uncontrol	$1.4043\times {10}^{-3}$	—
Passive	$5.6464\times {10}^{-5}$	95.9792
PID	$4.0474\times {10}^{-5}$	97.1178
FOPID	$3.2025\times {10}^{-5}$	97.7195
DFOPID	$6.4240\times {10}^{-6}$	99.5426

. It can be observed that, compared to the uncontrolled case, the primary system displacement after the DFOPID-absorber (See Figure 1) is reduced by three orders of magnitude, with an attenuation rate reaching 99.54%, making it the most effective among all controllers. This highlights the excellent vibration suppression and robustness of the model presented in this paper under random excitation (Gaussian white noise).

The primary system is typically subjected to impulsive excitation as well. In the following, displacement impulsive excitation is used as an example to compare the vibration reduction performance of different models. To simplify the analysis, during the simulation process, the excitation signal is converted into the system’s initial conditions, that is,

$$\begin{array}{c}\hfill x_1\left(0\right)=\rho ,{\dot{x}}_1\left(0\right)=x_2\left(0\right)={\dot{x}}_2\left(0\right)=0,\end{array}$$

(27)

where $\rho$ represents the magnitude of the impact of the impulsive excitation on the primary system.

Figure 6 presents the time histories of the primary system under displacement impulsive excitation for different models. As observed, all the absorbers are able to effectively absorb the vibration energy of the primary system within a short period of time, with the corresponding minimum peak values of the attenuation curves being $x_1=-0.88,-0.41,-0.40,-0.28$. The results indicate that, under the selected impact load amplitude, the DFOPID-absorber (See Figure 1) significantly reduces the instantaneous vibration peaks of the primary system, demonstrating the best shock resistance performance. The absorber model proposed in this paper offers new perspectives and references for the design of vibration absorption systems under complex operating conditions.

5. Conclusions

In practical engineering, DFOPID controllers (as referenced in [15,17,35,36]) and linear vibration absorbers (such as those used in Taipei 101, Shanghai Tower, and car seats) have been widely used and have demonstrated excellent control performance. However, due to the inherent limitations of linear absorbers, such as strong frequency dependence leading to a lack of adaptability, they struggle to cope with complex and dynamic engineering environments. To address this, the present study explores the integration of a DFOPID controller with a linear vibration absorber, leveraging the controller’s active control capabilities to enhance adaptability. This paper further analyzes the system’s dynamic characteristics and vibration mitigation performance. Specifically, based on fractional control theory, we systematically discuss the tuning mechanism and parameter setting issues of a coupled DVA model involving inerter elements, a grounded negative stiffness structure, and DFOPID. The results show that for different orders of the differential terms, the DFOPID adjusts the vibration reduction performance of the primary system through elastic, damping, and inertial forces. Furthermore, by minimizing the maximum amplitude of the primary system as the objective function, the model parameters are optimized using the PSO algorithm. The optimized model is compared with passive DVA and active DVA models under PID, FOPID, and DFOPID in various loading environments (harmonic excitation, random excitation, and impulsive excitation). The results demonstrate that the DFOPID-DVA system performs the best, demonstrating excellent vibration suppression and robustness. The models and methods presented in this paper can provide valuable insights for the design of wide-frequency vibration reduction systems in practical engineering applications, offering a solid theoretical foundation for future real-world implementations.

In the future, we will consider applying the coupled model to practical engineering applications. Moreover, the current design of the DFOPID-DVA system is limited to coupling models with linear vibration absorbers. For more widely used nonlinear energy sinks and non-smooth DVA models, the parameter tuning mechanisms and optimization issues still require further exploration and development.