Research on Target Energy Transfer and Energy Dissipation of Coupled Fractional-Order Inerter-Based Nonlinear Energy Sinks Vibration System

Abstract

This study investigates the critical role of resonance capture dynamics in determining the energy dissipation performance of nonlinear energy sinks (NES). A fluid inerter combining mass amplification and damping characteristics is proposed as a core component, based on which two configurations of fractional-order NES (configured in series and parallel) are systematically constructed. The applicability of the complex averaging method in fractional-order systems has been addressed through fractional calculus (such as Leibniz properties), enabling it to be analyzed like integer-order systems. Employing the multi-scale perturbation method, the energy transfer mechanism between the primary oscillator and the NES is derived, leading to the analytical determination of optimal cubic stiffness and maximum energy transfer efficiency. Comparative simulation shows that the parameters of the inerter directly affect the magnitude of critical damping. The optimal cubic stiffness design method is more reliable than traditional methods and can ensure effective target energy transfer triggering. Further analysis of dissipation time shows that the performance of fractional-order NES is superior to integer-order NES; notably, the dissipation time of series fractional-order NES is significantly shorter than that of parallel and traditional NES. In summary, this study provides theoretical guidance for the design of lightweight and high-performance NES and will also promote the application of fractional calculus theory in the field of engineering vibration reduction.

1. Introduction

The advantages of nonlinear energy sinks have been widely recognized [1,2,3,4]: they can achieve energy transfer through target energy transfer (abbreviated as TET) and dissipate most of the energy through NES damping. Under the context of TET, undesirable energy can be spatially transferred from a main structure to an attached NES in an irreversible way and efficiently dissipated. This irreversible energy transfer constitutes the core principle of vibration suppression in NES structures [5]. The preceding research has paved the road for the final goal of optimization design for engineering applications. References [6,7] proposed an optimization method for the cubic stiffness of a cubic NES single-degree-of-freedom vibration system. The TET can be triggered when the initial energy value is higher than the optimization value. However, the author found during verification that the optimal stiffness designed in the literature cannot fully guarantee the effectiveness of all positions. Therefore, based on its optimization, this paper supplemented it to obtain a more reliable optimization method. To trigger TET, the initial energy needs to be greater than a fixed value, and NES needs to have a certain mass. Therefore, coupling into the main structure requires the introduction of a large additional mass, and the damping of the main structure is still the main factor in dissipating energy, which limits its practical application.

To overcome the problem of large additional mass, scholars have proposed the application of new structures, such as multiple NES parallel connections [7,8,9,10,11] and lever-type NES [12], which can effectively increase the vibration-reduction effect and possibly appropriately reduce the mass of NES, but obviously their structure is complex and increases the cost. With the emergence of inerters, research on the vibration suppression of integer-order inertial NES has begun to enter the field of view [13,14,15]. In most of the related literature, the study still focuses on the effect of mechanical-type inerters [16,17] under ideal conditions, ignoring damping and nonlinear factors, which is equivalent to only considering the effect of inertia. A significant advancement was achieved by Swift et al. [18], who pioneered a novel fluid inerter design utilizing a spiral tube as its core component. This architecture offers distinct advantages over mechanical counterparts, including structural simplicity, compact dimensions, high inertance, the elimination of backlash and eccentricity issues, and enhanced compatibility with diverse passive network layouts. While the dynamics of mechanical inerters are well-characterized (typically neglecting damping effects [18,19]), fluid inerters inherently exhibit damping behavior due to liquid viscosity and intrinsic structural properties. Consequently, research focus has shifted toward quantifying these damping mechanisms and even exploiting them for control purposes l [20,21,22].

Existing models for fluid inerters predominantly rely on empirical formulations derived from classical experiments, often amalgamated through fitting procedures. Although simulations generally align with experimental data, discrepancies persist—particularly in low- versus high-frequency regimes [23,24]. This suggests limitations in fully capturing the underlying fluid dynamics. The apparent direct correspondence between components and flow behavior is an oversimplification; internally, fluid motion manifests complex phenomena. Whether laminar, turbulent, or transitional, localized flow structures exhibit fractal characteristics [25], challenging conventional modeling paradigms. Therefore, after studying typical models of fluid inerters with different hypotheses [23,24], the author proposed two fractional-order inerter models and verified their effectiveness through comparison with classical experimental data and existing models. Using fractional-order models to describe fluid inerters also solves the theoretical research problem of traditional nonlinear models [26]. This model only has one fractional derivative term that can simultaneously represent the inertia and damping characteristics of the fluid inerter. The model is more concise, and the parameter meanings are relatively clear. The fluid inerter expressed by the fractional-order model is called a fractional-order inerter. The author then applied fractional-order inerters to the study of critical damping in vibration isolation systems [27] and the study of vibration characteristics in nonlinear energy sinks [22].

This article will investigate the target energy transfer and energy dissipation mechanisms of NES with a fractional-order inerter, explore the influence of fractional-order inerter parameters, and compare the performance of different combinations of fractional-order inerter-based NES (abbreviated as FOIB-NES). The specific tasks of this article are as follows:

Section 2: Firstly, the single-degree-of-freedom vibration system models of coupled parallel and series FOIB-NES are presented. Using fractional-order properties to address the difficulties in applying the complex averaging method [28,29,30,31] to fractional-order nonlinear vibration systems, the slow-varying dynamic equations of the system are obtained, and the multi-scale method [32,33,34,35] is used to transform and further obtain the slow-varying manifold of the system. Section 3: Firstly, based on the conclusions drawn in Section 2, we derived the energy transfer relationship between the main oscillator and the NES oscillator. Then, we provided scenarios for triggering TET, optimal design criterion, and the influence of NES parameters on energy transfer, and summarized the optimal conditions for generating effective TET. Section 4: In order to verify the reliability of the TET conditions proposed in Section 3, (1) a comparison and verification of the energy transfer relationship with actual simulation results were provided for different initial energies and cubic stiffnesses; (2) and the corresponding energy dissipation curve of the main oscillator was provided to verify whether the dissipation capacity is stronger when TET is satisfied. (3) At the same time, different fractional-order dissipation curves were compared to verify that the fractional order has better dissipation ability. (4) To study the robustness or parameter margin of NES parameters, the dissipation time versus cubic stiffness curves were plotted for different inerter parameters and the robustness strength under different conditions was also discussed. Section 5: The conclusion of the entire text is presented.

2. Asymptotic Expansion of the Linear Oscillator Attach to a Single FOIB-NES

2.1. Modeling

The analytical development of a system comprising a single FOIB-NES and a linear oscillator was carried out, as presented in Figure 1, where Scheme A (abbreviated as SA) is a parallel FOIB-NES, and Scheme B (abbreviated as SB) is a series FOIB-NES, where one end of the inerter needs to be grounded. In the figure, m₁, k₁, and c₁ are the main structural mass, stiffness, and damping coefficient, respectively; m₂, k_n, and c_n represent the mass, ideal nonlinear cubic stiffness, and damping coefficient of the mass block on NES, respectively; b_n and μ are the inertance and fractional derivative order of the fractional-order inerter, respectively.

The dynamic Equations of two schemes can be obtained from Newton’s second law, as shown below:

$$\mathrm{Scheme} \mathrm{A}: \left\{\begin{array}{l}{m}_1{\ddot{z}}_1+m_2{\ddot{z}}_2+c_1{\dot{z}}_1+k_1{z}_1=f\cos \omega t\\ m_2{\ddot{z}}_2-\left[b_{\mathrm{n}}{D}^{\mu }\left(z_1-z_2\right)+c_{\mathrm{n}}\left({\dot{z}}_1-{\dot{z}}_2\right)+k_{\mathrm{n}}{\left(z_1-z_2\right)}^3\right]=0\end{array}\right.$$

(1)

$$\mathrm{Scheme} \mathrm{B}: \left\{\begin{array}{l}{m}_1{\ddot{z}}_1+m_2{\ddot{z}}_2+b_{\mathrm{n}}{D}^{\mu }{z}_2+c_1{\dot{z}}_1+k_1{z}_1=f\cos \omega t\\ m_2{\ddot{z}}_2+b_{\mathrm{n}}{D}^{\mu }{z}_2-c_{\mathrm{n}}\left({\dot{z}}_1-{\dot{z}}_2\right)-k_{\mathrm{n}}{\left(z_1-z_2\right)}^3=0\end{array}\right.$$

(2)

Obviously, when b_n = 0, the two models are the same, which is the dynamic model coupled with traditional NES. The model of the fluid inerter in Figure 1 adopts the fractional-order model in Reference [26]. The reference also mentioned that Coulomb friction force accounts for a small proportion of the total force under other excitation frequencies outside the ultra-low-frequency region (such as 0.3 Hz, 0.5 Hz). However, the main vibration-reduction device proposed in this article is mainly used in future applications such as vehicles, spacecraft, machine tools, etc. Its normal and stable operation is not prone to ultra-low-frequency operation, so this article also ignores this. The D^α appearing in all equations represents the fractional differential operator defined by Caputo, and the specific definition of Caputo derivative is as follows [36]:

$$D^{\alpha }={}_a{}^{C}D_t^{\alpha }f(t)=\left\{\begin{array}{l}{\displaystyle \frac{1}{\Gamma (n-\alpha )}}{\displaystyle {\int }_0^t\frac{f^n(\tau )d\tau }{{(t-\tau )}^{\alpha +1-n}},n-1<\alpha <n}\\ f^n(t),\alpha =n\end{array}\right.\alpha \in ℝ,n\in \mathbb{N}$$

(3)

where $\Gamma \left(n-\alpha \right)={\displaystyle {\int }_0^{+\infty }{t}^{n-\alpha -1}}{e}^{-x}dt$, the domain of f(t) is (a, b), and n is the smallest integer greater than or equal to α.

Fractional-order differentiation also has similar properties to integer-order differentiation. Several properties required for this article are presented below [36]:

(1) Set $a,b\in ℝ$, and $0<\alpha ,\beta <1$, then there are

$$\left\{\begin{array}{l}{D}^{\alpha }\left[ax(t)+bg(t)\right]=a{D}^{\alpha }x(t)+b{D}^{\alpha }g(t)\\ D^{\alpha }\left[D^{\beta }x(t)\right]=D^{\beta }\left[D^{\alpha }x(t)\right]=D^{\alpha +\beta }x(t)\Rightarrow D^{\alpha +\beta -1}\left[\dot{x}(t)\right]\end{array}\right.$$

(4)

Here, functions x(t) and g(t) and their derivatives are continuous on the interval $[t_0,t]$.

(2) If $t=\omega \tau$ and $x\left(t\right)=X\left(\tau \right)$, then there is

$${\displaystyle \frac{D^{\alpha }x\left(t\right)}{D{t}^{\alpha }}}={\omega }^{\alpha }{\displaystyle \frac{D^{\alpha }X\left(\tau \right)}{D{\tau }^{\alpha }}}$$

(5)

(3) If $x\left(t\right)=a\sin \left(\omega t+\phi \right)$ or $x\left(t\right)=a\cos \left(\omega t+\phi \right)$ and 0 < α <1, then there are

$$\left\{\begin{array}{l}{D}^{\alpha }\left[a\sin \left(\omega t+\phi \right)\right]=a{\omega }^{\alpha }\sin \left(\omega t+\phi +{\displaystyle \frac{\alpha \pi }{2}}\right)\\ D^{\alpha }\left[a\cos \left(\omega t+\phi \right)\right]=a{\omega }^{\alpha }\cos \left(\omega t+\phi +{\displaystyle \frac{\alpha \pi }{2}}\right)\end{array}\right.$$

(6)

According to Reference [37], under generalized harmonic excitation, the fractional derivative terms in Equations (1) and (2) can be equivalently treated to obtain [26]

$$D^{\mu }{z}_2=B{\ddot{z}}_2+C{\dot{z}}_2,D^{\mu }\left(z_1-z_2\right)=B\left({\ddot{z}}_1-{\ddot{z}}_2\right)+C\left({\dot{z}}_1-{\dot{z}}_2\right)$$

(7)

Here, B and C are equivalent masses and equivalent damping coefficients, where the first term on the right-hand side of the two Equations acts as an inertial force and the second term acts as a damping force.

Introducing a new time scale $\tau ={\omega }_{0}t,{\omega }_0=\sqrt{k_1/m_1}$ and nondimensionalizing Equations (1) and (2) to obtain

$$\mathrm{Scheme} \mathrm{A}: \left\{\begin{array}{l}{z^{″}}_1+\epsilon {z^{″}}_2+\epsilon {\xi }_1{z^{\prime }}_1+z_1=f_{\mathrm{e}}\cos \Omega \mathit{\tau }\\ \epsilon {z^{″}}_2-\left[\epsilon \lambda {\omega }_0^{\mu -2}\cdot D^{\mu }\left(z_1-z_2\right)+\epsilon {\xi }_2\left({z^{\prime }}_1-{z^{\prime }}_2\right)+\kappa {\left(z_1-z_2\right)}^3\right]=0\end{array}\right.$$

(8)

$$\mathrm{Scheme} \mathrm{B}: \left\{\begin{array}{l}{z^{″}}_1+\epsilon {z^{″}}_2+\epsilon \lambda {\omega }_0^{\mu -2}\cdot D^{\mu }{z}_2+\epsilon {\xi }_1{z^{\prime }}_1+z_1=f_{\mathrm{e}}\cos \Omega \mathit{\tau }\\ \epsilon {z^{″}}_2+\epsilon \lambda {\omega }_0^{\mu -2}\cdot D^{\mu }{z}_2-\epsilon {\xi }_2\left({z^{\prime }}_1-{z^{\prime }}_2\right)-\kappa {\left(z_1-z_2\right)}^3=0\end{array}\right.$$

(9)

where $\begin{array}{l}\Omega ={\displaystyle \frac{\omega }{{\omega }_0}},\epsilon ={\displaystyle \frac{m_2}{m_1}},{\epsilon }_1={\displaystyle \frac{b_{\mathrm{n}}}{m_1}},\lambda ={\displaystyle \frac{{\epsilon }_1}{\epsilon }},\kappa ={\displaystyle \frac{k_{\mathrm{n}}}{k_1}},\\ \epsilon {\xi }_1={\xi }_{\mathrm{s}}={\displaystyle \frac{c_1}{\sqrt{m_1{k}_1}}},\epsilon {\xi }_2={\xi }_{\mathrm{n}}={\displaystyle \frac{c_{\mathrm{n}}}{\sqrt{m_1{k}_1}}},f_{\mathrm{e}}={\displaystyle \frac{f}{k_1}}\end{array}$.

2.2. Slow-Varying Dynamic Equations and Slow-Varying Manifolds of the System

Target energy transfer refers to the process of the unidirectional transfer of vibration energy from the main structure to the NES: energy is transferred to the NES and dissipated through the damping effect of the NES, and finally only a small, or even no, vibration energy returns to the main structure. TET is mainly achieved through resonance capture, which is the main principle of NES to suppress structural free vibration [5]. The following focuses on the impact of introducing fractional-order inerters to TET. To reduce the repetition of the derivation process of the two schemes, Scheme A is given as the main and complete step, while Scheme B provides the conclusions that need to be used later.

2.2.1. Complexification

This article mainly discusses the characteristics of free vibration, i.e., f_e = 0. To better analyze the relationship between energy transfer and dissipation, a variable replacement is introduced: $y_1=z_1/\sqrt{\epsilon },y_2=\left(z_2-z_1\right)/\sqrt{\epsilon }$. Equations (8) and (9) can be transformed into

$$\mathrm{Scheme} \mathrm{A}: \left\{\begin{array}{l}{\ddot{y}}_1+\epsilon \left({\ddot{y}}_1+{\ddot{y}}_2\right)+\epsilon {\xi }_1{\dot{y}}_1+y_1=0\\ {\ddot{y}}_1+{\ddot{y}}_2+\lambda {\omega }_0^{\mu -2}\cdot D^{\mu }{y}_2+{\xi }_2{\dot{y}}_2+\kappa y_2^3=0\end{array}\right.$$

(10)

$$\mathrm{Scheme} \mathrm{B}: \left\{\begin{array}{l}{\ddot{y}}_1+\epsilon \left({\ddot{y}}_1+{\ddot{y}}_2\right)+\epsilon \lambda {\omega }_0^{\mu -2}\cdot D^{\mu }\left(y_1+y_2\right)+\epsilon {\xi }_1{\dot{y}}_1+y_1=0\\ \left({\ddot{y}}_1+{\ddot{y}}_2\right)+\lambda {\omega }_0^{\mu -2}\cdot D^{\mu }\left(y_1+y_2\right)+{\xi }_2{\dot{y}}_2+\kappa y_2^3=0\end{array}\right.$$

(11)

Internal resonance refers to the strong coupling phenomenon between different modes in a system due to frequency ratios satisfying specific integer ratio relationships (such as 1:1, 2:1, 3:1, etc.). The ability of the system to undergo internal resonance is one of the prerequisites for achieving the target energy transfer. This article mainly studies the dynamics and characteristics of 1:1 internal resonance, where ω = ω₀, i.e., Ω = 1. We introduce complex variables of Manevitch as follows [29]:

$${\phi }_j{\mathrm{e}}^{\mathrm{i}\tau }={\dot{y}}_j+\mathrm{i}{y}_j,{\phi }_j^{*}{\mathrm{e}}^{-\mathrm{i}\tau }={\dot{y}}_j-\mathrm{i}{y}_j,(j=1,2)$$

(12)

Here, φ₁ and φ₂, respectively, represent the variables of the amplitude of the displacement of the main oscillator and the relative displacement of the two oscillators as a function of time, where ${\phi }_j^{*}$ is the corresponding conjugate complex variable and i is the imaginary unit. Therefore, the following relationship exists:

$$y_j={\displaystyle \frac{{\phi }_j{\mathrm{e}}^{\mathrm{i}\tau }-{\phi }_j^{*}{\mathrm{e}}^{-\mathrm{i}\tau }}{2\mathrm{i}}},{\dot{y}}_j={\displaystyle \frac{{\phi }_j{\mathrm{e}}^{\mathrm{i}\tau }+{\phi }_j^{*}{\mathrm{e}}^{-\mathrm{i}\tau }}{2}},{\ddot{y}}_j={\dot{\phi }}_j{\mathrm{e}}^{i\tau }+\mathrm{i}{\displaystyle \frac{{\phi }_j{\mathrm{e}}^{\mathrm{i}\tau }-{\phi }_j^{*}{\mathrm{e}}^{-\mathrm{i}\tau }}{2}}$$

(13)

Due to the presence of fractional derivative terms, the derivation process of the complex averaging method encounters certain difficulties, and the following key steps need to be added. Firstly, Equation (13) can be further transformed to obtain

$${\dot{\phi }}_j{\mathrm{e}}^{\mathrm{i}\tau }-{\dot{\phi }}_j^{*}{\mathrm{e}}^{-\mathrm{i}\tau }=0,{\dot{\phi }}_j{\mathrm{e}}^{\mathrm{i}\tau }+{\dot{\phi }}_j^{*}{\mathrm{e}}^{-\mathrm{i}\tau }=2{\dot{\phi }}_j{\mathrm{e}}^{\mathrm{i}\tau }$$

(14)

Then, according to the properties of fractional calculus [36], $D^{\mu }{y}_j=D^{\alpha }{\dot{y}}_j$, α = μ − 1 (0 < α < 1) are obtained, and the Leibniz property of fractional calculus [36] is introduced. If $f\left(t\right)$ is continuous in [t₀, t] and $g\left(t\right)$ as n + 1 continuous derivatives in [t₀, t], then the fractional derivative of the product $f\left(t\right)g\left(t\right)$ is given by

$${}_{t_0}D_t^{\alpha }\left[f(t)g(t)\right]={\displaystyle \sum _{k=0}^n\left(\begin{array}{c}\alpha \\ k\end{array}\right)f^{\left(k\right)}\left(t\right)g^{\left(\alpha -k\right)}\left(t\right)}-R_{\mathrm{n}}^{\alpha }\left(t\right),n\ge \alpha +1$$

(15)

This rule is a generalization of the Leibniz rule for integer-order differentiation. Here, $\left(\begin{array}{c}\alpha \\ k\end{array}\right)={\displaystyle \frac{\Gamma \left(\alpha +1\right)}{k!\Gamma \left(\alpha -k+1\right)}}$, $R_{\mathrm{n}}^{\alpha }\left(t\right)={\displaystyle \frac{1}{n!\Gamma \left(-\alpha \right)}}{\displaystyle \underset{t_0}{\overset{t}{\int }}{\left(t-\tau \right)}^{-\alpha -1}g\left(\tau \right)d\tau }{\displaystyle \underset{\tau }{\overset{t}{\int }}{f}^{n+1}\left(\xi \right)}{\left(t-\xi \right)}^{n}d\xi$.

To find the limit for τ = t, we can obtain $\underset{\tau \to t}{\lim }{R}_n^{\alpha }\left(t\right)=g\left(\tau \right){\left(t-\tau \right)}^{-\alpha -1}{\displaystyle \underset{\tau }{\overset{t}{\int }}{f}^{n+1}\left(\xi \right)}{\left(t-\xi \right)}^{n}d\xi$ by using the L’Hospital rule. Reference [36] points out that if m < α ≤ m + 1, obviously, k cannot be greater than m + 2. We can make n ≥ m + 2, at which point $\underset{\tau \to t}{\lim }{R}_n^{\alpha }\left(t\right)$ will tend toward 0 for τ → t. Since 0 < α < 1, n = 2 and k < 2, $R_n^{\alpha }\left(t\right)$ can be ignored in [t₀, t]. Therefore, $D^{\mu }{y}_j$ can be simplified based on Equations (12)–(15), the fractional derivative properties of trigonometric functions, and Euler’s formula, as shown below:

$$\begin{array}{l}{D}^{\mu }{y}_j=D^{\alpha }{\dot{y}}_j=D^{\alpha }\left({\displaystyle \frac{{\phi }_j{\mathrm{e}}^{\mathrm{i}\tau }+{\phi }_j^{*}{\mathrm{e}}^{-\mathrm{i}\tau }}{2}}\right)\\ ={\displaystyle \frac{1}{2}}\left[\left({\phi }_j{\mathrm{e}}^{\mathrm{i}\tau }+{\phi }_j^{*}{\mathrm{e}}^{-\mathrm{i}\tau }\right)\cos {\displaystyle \frac{\alpha \mathsf{\pi }}{2}}+\left({\phi }_j{\mathrm{e}}^{\mathrm{i}\tau }-{\phi }_j^{*}{e}^{-\mathrm{i}\tau }\right)\sin {\displaystyle \frac{\alpha \mathsf{\pi }}{2}}\mathrm{i}\right]+\alpha {\dot{\phi }}_j{\mathrm{e}}^{\mathrm{i}\tau }\sin {\displaystyle \frac{\alpha \mathsf{\pi }}{2}}\end{array}$$

(16)

After substituting Equations (12)–(16) into Equations (10) and (11) and performing averaging, we obtain the system’s slowly varying dynamic equations for the two schemes.

$$\mathrm{Scheme} \mathrm{A}: \left\{\begin{array}{l}{\dot{\phi }}_1+\epsilon \left({\dot{\phi }}_1+{\dot{\phi }}_2\right)+\epsilon {\displaystyle \frac{{\phi }_1+{\phi }_2}{2}}\mathrm{i}+\epsilon {\xi }_1{\displaystyle \frac{{\phi }_1}{2}}=0\\ \left({\dot{\phi }}_1+\rho {\dot{\phi }}_2\right)+{\displaystyle \frac{{\phi }_1}{2}}\mathrm{i}+\left[{\displaystyle \frac{{\xi }_{\mathrm{NES}}}{2}}+\left({\displaystyle \frac{\overline{B}}{2}}-\kappa {\displaystyle \frac{3{\left|{\phi }_2\right|}^2}{8}}\right)\mathrm{i}\right]{\phi }_2=0\end{array}\right.$$

(17)

$$\mathrm{Scheme} \mathrm{B}: \left\{\begin{array}{l}{\dot{\phi }}_1+\epsilon \rho \left({\dot{\phi }}_1+{\dot{\phi }}_2\right)+\epsilon \left(\overline{c}+\overline{B}\mathrm{i}\right){\displaystyle \frac{{\phi }_1+{\phi }_2}{2}}+\epsilon {\xi }_1{\displaystyle \frac{{\phi }_1}{2}}=0\\ \rho \left({\dot{\phi }}_1+{\dot{\phi }}_2\right)+\left(\overline{c}+\overline{B}\mathrm{i}\right){\displaystyle \frac{{\phi }_1}{2}}+\left[{\displaystyle \frac{{\xi }_{\mathrm{NES}}}{2}}+\left({\displaystyle \frac{\overline{B}}{2}}-\kappa {\displaystyle \frac{3{\left|{\phi }_2\right|}^2}{8}}\right)\mathrm{i}\right]{\phi }_2=0\end{array}\right.$$

(18)

Among them, $b_0={\omega }_0^{\mu -2}\sin {\displaystyle \frac{\alpha \mathsf{\pi }}{2}},c_0={\omega }_0^{\mu -2}\cos {\displaystyle \frac{\alpha \mathsf{\pi }}{2}},\overline{c}=\lambda c_0,\overline{b}=\lambda b_0,\rho =1+\alpha \overline{b},\overline{B}=1+\overline{b},{\xi }_{\mathrm{NES}}=\overline{c}+{\xi }_2$, and all b₀ and c₀ in the following text express the same meaning.

2.2.2. Multiple Scales Expansion

Multi-scale methods [32,33,34,35,38] have been widely applied and can ensure accuracy in analysis, making them particularly suitable for the fast–slow approximation analysis discussed in this section. Therefore, the following uses multi-scale transformation to obtain the approximate fast–slow manifold of the system and obtain the energy transfer relationship. Firstly, briefly introduce the steps of multi-scale methods [38], introduce a set of gradually slower time scales T_n = εⁿT₀, n= 1, 2,…, and consider these time scales as independent variables.

The idea of seeking approximate solutions of different orders using different time scales [38] is formed by representing the solution of the dynamic equation as

$$u=u_0\left(T_0,T_1,\dots \right)+\epsilon u_1\left(T_0,T_1,\dots \right)+{\epsilon }^2{u}_2\left(T_0,T_1,\dots \right)+\dots +{\epsilon }^n{u}_n\left(T_0,T_1,\dots T_n\right)$$

(19)

Then, a partial derivative operator can be defined as representing the derivative operator.

$$\begin{array}{l}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}T}}={\displaystyle \sum _{n=0}^{+\infty }{\epsilon }^n\frac{\partial }{\partial T_n}}\stackrel{def}{=}{\displaystyle \sum _{n=0}^{+\infty }{\epsilon }^n{D}_n}\\ {\displaystyle \frac{{\mathrm{d}}^2}{\mathrm{d}{T}^2}}={\displaystyle \sum _{n=0}^{+\infty }{\epsilon }^n{D}_n\left({\displaystyle \sum _{s=0}^{+\infty }{\epsilon }^s{D}_s}\right)}=D_0^2+2\epsilon D_0{D}_1+{\epsilon }^2\left(D_1^2+2D_0{D}_2\right)+\cdots \end{array}$$

(20)

Substitute Equations (19) and (20) into the dynamic equation to be solved and compare the coefficients of ε to the same power to obtain a series of linear partial differential equations, which can be solved sequentially.

Referring to the above steps, further approximate the slowly varying dynamic Equation (17) of Scheme A in the previous section and introduce the following multi-scale transformation [32]:

$$\begin{array}{l}{\phi }_j={\phi }_{j0}+\epsilon {\phi }_{j1}+{\epsilon }^2{\phi }_{j2}+\cdots +{\epsilon }^n{\phi }_{jn},j=1,2\\ \tau ={\tau }_0+{\tau }_1+{\tau }_3+\cdots {\tau }_n,{\tau }_n={\epsilon }^n{\tau }_0,\\ {\displaystyle \frac{d}{d\tau }}={\displaystyle \frac{\partial }{\partial {\tau }_0}}+\epsilon {\displaystyle \frac{\partial }{\partial {\tau }_1}}+\cdots +{\epsilon }^n{\displaystyle \frac{\partial }{\partial {\tau }_n}}\end{array}$$

(21)

Here, only the first two orders are taken. For Scheme A, substituting expansions (21) into Equation (17) and setting the coefficients of powers of ε as equal to zero, one derives the following hierarchy of problems at successive orders of approximation:

$$O\left({\epsilon }^0\right):{\displaystyle \frac{\partial {\phi }_{10}}{\partial {\tau }_0}}+\rho {\displaystyle \frac{\partial {\phi }_{20}}{\partial {\tau }_0}}+\mathrm{i}{\displaystyle \frac{{\phi }_{10}}{2}}+\left[{\displaystyle \frac{{\xi }_{\mathrm{NES}}}{2}}+\mathrm{i}\left({\displaystyle \frac{\overline{B}}{2}}-{\displaystyle \frac{3\kappa }{8}}{\left|{\phi }_{20}\right|}^2\right)\right]{\phi }_{20}=0$$

(22)

$$O\left({\epsilon }^1\right):{\displaystyle \frac{\partial {\phi }_{10}}{\partial {\tau }_1}}+{\displaystyle \frac{\partial {\phi }_{11}}{\partial {\tau }_0}}+{\displaystyle \frac{\partial {\phi }_{10}}{\partial {\tau }_0}}+{\displaystyle \frac{\partial {\phi }_{20}}{\partial {\tau }_0}}+\mathrm{i}{\displaystyle \frac{{\phi }_{10}+{\phi }_{20}}{2}}+{\xi }_1{\displaystyle \frac{{\phi }_{10}}{2}}=0$$

(23)

When resonance occurs at a 1:1 ratio, the variables in the above two equations are independent of the rapidly changing time scale τ₀. Therefore, as one approaches the equilibrium state, the following relationship exists:

$$O\left({\epsilon }^0\right):{\displaystyle \frac{{\phi }_{10}}{2}}\mathrm{i}+\left[{\xi }_{\mathrm{NES}}+\left(\overline{B}-3/4\kappa {\left|{\phi }_{20}\right|}^2\right)\mathrm{i}\right]{\displaystyle \frac{{\phi }_{20}}{2}}=0$$

(24)

$$O\left({\epsilon }^1\right):{\displaystyle \frac{\partial {\phi }_{10}}{\partial {\tau }_1}}+\mathrm{i}{\displaystyle \frac{{\phi }_{10}+{\phi }_{20}}{2}}+{\xi }_1{\displaystyle \frac{{\phi }_{10}}{2}}=0$$

(25)

Equations (22) and (23) are the approximate slowly varying manifolds of Scheme A. For Scheme B, Equation (18) adopts the same steps as above to obtain its approximate slowly varying manifold

$$\left\{\begin{array}{l}O\left({\epsilon }^0\right):\left(\overline{c}+\overline{B}\mathrm{i}\right){\displaystyle \frac{{\phi }_{10}}{2}}+\left[{\xi }_{\mathrm{NES}}+\left(\overline{B}-3/4\kappa {\left|{\phi }_{20}\right|}^2\right)\mathrm{i}\right]{\displaystyle \frac{{\phi }_{20}}{2}}=0\\ O\left({\epsilon }^1\right):{\displaystyle \frac{\partial {\phi }_{10}}{\partial {\tau }_1}}+\left(\overline{c}+\overline{B}\mathrm{i}\right){\displaystyle \frac{{\phi }_{10}+{\phi }_{20}}{2}}+{\xi }_1{\displaystyle \frac{{\phi }_{10}}{2}}=0\end{array}\right.$$

(26)

3. Analysis and Parameter Optimization of Target Energy Transfer

3.1. Deduction of Energy Transfer Relationship

To study the changes in the amplitude and phase trajectories of the oscillator in the equilibrium state of Scheme A, complex variables can be represented by modulus and phase angle [7]:

$${\phi }_{10}=R_1{\mathrm{e}}^{\mathrm{i}{\theta }_1},{\phi }_{20}=R_2{\mathrm{e}}^{\mathrm{i}{\theta }_2}$$

(27)

Substitute Equation (27) into Equations (22) and (23) and make its real and imaginary parts zero to obtain the intermediate relationship equation. Then, simplify and organize it to obtain

$$\left\{\begin{array}{l}{R}_1{\displaystyle \frac{\partial R_1}{\partial {\tau }_1}}=-{\displaystyle \frac{{\xi }_1{R}_1^2+{\xi }_{\mathrm{NES}}{R}_2^2}{2}}\\ {R_1}^2={\left(\overline{B}-3/4\kappa R_2^2\right)}^2{R}_2^2+{\xi }_{\mathrm{NES}}^2{R}_2^2\end{array}\right.$$

(28)

For the convenience of analysis, further simplification is needed by introducing a dimensionless variable: $H_i=\kappa E_{i0}=\kappa R_i^2\left(i=1,2\right)$. From the definition of complex variables, we can note that

$$\left\{\begin{array}{l}{E}_{10}={\left|{\phi }_{10}\right|}^2={\dot{y}}_1^2+y_1^2\\ E_{20}={\left|{\phi }_{20}\right|}^2=\underset{{\tau }_0\to \infty }{\lim }\left({\dot{y}}_2^2+y_2^2\right)\end{array}\right.$$

(29)

where E_i0 is a conservative system with a dimensionless energy form like that of the main structure energy E₁, differing only by a multiple of 1/2. Therefore, Equation (28) can be transformed into

$$\left\{\begin{array}{l}{H}_1=\left[{\xi }_{\mathrm{NES}}^2+{\left(\overline{B}-3/4H_2\right)}^2\right]H_2\\ {\displaystyle \frac{\partial H_1}{\partial {\tau }_1}}=-\left({\xi }_1{H}_1+{\xi }_{\mathrm{NES}}{H}_2\right)\end{array}\right.$$

(30)

By following the same derivation steps as above, the energy relationship formula for Scheme B can be obtained

$$\left\{\begin{array}{l}{H}_1={\displaystyle \frac{1}{{\overline{c}}^2+{\overline{B}}^2}}\cdot \left[{\xi }_{\mathrm{NES}}^2+{\left(\overline{B}-3/4H_2\right)}^2\right]H_2\\ {\displaystyle \frac{\partial H_1}{\partial {\tau }_1}}=-H_1\left(\overline{c}+{\xi }_1\right)+{\displaystyle \frac{2\overline{c}\overline{B}\left(\overline{B}-3/4H_2\right)+\left({\overline{c}}^2-{\overline{B}}^2\right){\xi }_{\mathrm{NES}}}{{\overline{c}}^2+{\overline{B}}^2}}\cdot H_2\end{array}\right.$$

(31)

In the equation, variable H₁ represents the energy of the primary oscillator, which approximates the energy of the linear main structure at multiple scales, and H₂ represents a first-order approximation of the energy of the interaction between NES and the main structure at multiple scales. Obviously, H₁ and H₂ can be expressed as invariant manifolds of nonlinear modes at the time scale T₀ → ∞, i.e., oscillator energy.

Observing the second Equations of (30) and (31), the energy dissipation rate relationship of the main structure of system A is relatively clear. When ξ₁ = ξ₂ = 0 and b_n = 0 or ξ₁ = ξ₂ = 0 and μ = 2, ∂H₁/∂T₁ = 0, this is a conservative system, and the loss of non-existent energy must have a value of 0. When the factors related to dissipation are individually zero, the dissipation of energy is only related to non-zero damping. The right-hand end of the equation remains less than 0, indicating that the energy is constantly decreasing.

The energy dissipation rate relationship of the main structure of Scheme B system is relatively complex. When ξ₁ = ξ₂ = 0 and b_n = 0 or ξ₁ = ξ₂ = 0 and μ = 2, ∂H₁/∂T₁ = 0 is consistent with Scheme A and is a conservative system. However, the influence of various damping needs further analysis.

3.2. The Generation Conditions of TET

Special note: During the simulation analysis below, m₁ = 10, k₁ = 0.6 remain unchanged, while the other parameters are suitable for discussion. Observing the first Equations of (30) and (31), both equations are very close to being cubic functions of H₁ with respect to H₂, and Scheme B has an additional coefficient of 1/($\stackrel{\mathrm{-}}{c}$² + $\overline{B}$²) compared to Scheme A. Since $\stackrel{\mathrm{-}}{B}$ = 1 + λ${\mathit{\omega }}_0^{\mathit{\mu }-1}$sin (μ − 1) π/2, with μ − 1 ∈ (0, 1), $\stackrel{\mathrm{-}}{B}$ > 1, it can be predicted that under the same parameter conditions, the H₁ amplitude of Scheme B will be smaller. A relationship diagram between H₁ and H₂ was drawn based on these two equations, as shown in Figure 2, with parameters m₂ = 0.3, μ = 1.9, c₁ = 0.003, c_n = 0.002, and b_n = 0.4.

When there are three solutions for H₂ in Figure 2, the line between two extreme points (black circles) and the peak valley (red circles) is unstable and easy to obtain. Due to the damping effect, the energy of the main structure is continuously dissipated, and the main energy is always in a decreasing state. Its flow must follow the direction of the black arrow route 1 or the blue arrow route 2 in the figure, rather than an unstable route, resulting in a unique jumping phenomenon in nonlinearity. This is also the cause of TET in the system. When the initial positions are different, the energy takes different routes [6,8].

By taking the derivative of H₂ in the first equation of Equations (30) and (31) and making it equal to zero, the expression for the extremum coordinates can be obtained as follows:

$$\mathrm{Scheme} \mathrm{A}:\begin{array}{l}\left(H_2^{+},H_{1a}^{+}\right)=\left({\displaystyle \frac{4}{9}}\left(2\overline{B}-\sqrt{{\Delta }_{\mathrm{p}}}\right),{\displaystyle \frac{8}{81}}\left[\overline{B}\left({\overline{B}}^2+9{\xi }_{\mathrm{NES}}^2\right)+{\Delta }_{\mathrm{p}}\sqrt{{\Delta }_{\mathrm{p}}}\right]\right),\hfill \\ \left(H_2^{-},H_{1a}^{-}\right)=\left({\displaystyle \frac{4}{9}}\left(2\overline{B}+\sqrt{{\Delta }_{\mathrm{p}}}\right),{\displaystyle \frac{8}{81}}\left[\overline{B}\left({\overline{B}}^2+9{\xi }_{\mathrm{NES}}^2\right)-{\Delta }_{\mathrm{p}}\sqrt{{\Delta }_{\mathrm{p}}}\right]\right)\hfill \end{array}$$

(32)

$$\mathrm{Scheme} \mathrm{B}:\begin{array}{l}\left(H_2^{+},H_{1b}^{+}\right)=\left({\displaystyle \frac{4}{9}}\left(2\overline{B}-\sqrt{{\Delta }_{\mathrm{p}}}\right),{\displaystyle \frac{1}{{\overline{c}}^2+{\overline{B}}^2}}{H}_{1a}^{+}\right), \hfill \\ \left(H_2^{-},H_{1b}^{-}\right)=\left({\displaystyle \frac{4}{9}}\left(2\overline{B}+\sqrt{{\Delta }_{\mathrm{p}}}\right),{\displaystyle \frac{1}{{\overline{c}}^2+{\overline{B}}^2}}{H}_{1a}^{-}\right)\hfill \end{array}$$

(33)

where

$${\Delta }_{\mathrm{p}}={\overline{B}}^2-3{\xi }_{\mathrm{NES}}^2>0$$

(34)

To study the relationship between damping and energy dissipation time in NES structures, it is necessary to determine all parameters except for the damping of the main structure. Therefore, the primary task is to design the structural parameters of NES. In addition to satisfying the condition of Equation (34), the parameters must be optimized. From Figure 2 and Reference [6], to achieve the optimal energy transfer method for NES, the following conditions must be met:

$$H_1\left(0\right)=\kappa E_{10}\left(0\right)>H_1^{+}\Rightarrow \kappa >{\displaystyle \frac{H_1^{+}}{E_{10}\left(0\right)}}={\kappa }_{\mathrm{opt}}$$

(35)

Among them, E₁₀ is the initial energy of the system’s main oscillator, calculated by Equation (29). So, a preliminary design can be carried out for its nonlinear stiffness κ, with schemes a and b having

$$\begin{array}{l}\mathrm{Scheme} \mathrm{A}: {\kappa }_{\mathrm{opt}}=4{\displaystyle \frac{\overline{B}\left({\overline{B}}^2+9{\xi }_{\mathrm{NES}}^2\right)+{\Delta }_{\mathrm{p}}\sqrt{{\Delta }_{\mathrm{p}}}}{81E_{10}}}\\ \mathrm{Scheme} \mathrm{B}: {\kappa }_{\mathrm{opt}}=4{\displaystyle \frac{\overline{B}\left({\overline{B}}^2+9{\xi }_{\mathrm{NES}}^2\right)+{\Delta }_{\mathrm{p}}\sqrt{{\Delta }_{\mathrm{p}}}}{81E_{10}\left({\overline{c}}^2+{\overline{B}}^2\right)}}\end{array}$$

(36)

Although there are two routes in Figure 2, the energy transfer of TET is a prerequisite for rapid energy dissipation. Obviously, route 1 is better, while route 2 is prone to monotonic decreases without jumping phenomenon. Therefore, only satisfying H₁(0) = κH₁₀(0) > $H_1^{+}$ is problematic. When H₂(0) approaches route 2, the response is easily dropped into route 2. Therefore, to improve reliability, this article proposes, for the first time, a supplementary condition that the initial H₁(0) and H₂(0) should be above the maximum value and located to the right of the extreme value. Obviously, the two schemes have the same relationship equation

$$H_2\left(0\right)=\kappa E_{20}\left(0\right)>H_2^{+}\Rightarrow \kappa >{\kappa }_{\mathrm{opt}}={\displaystyle \frac{H_2^{+}}{E_2\left(0\right)}}={\displaystyle \frac{4}{9E_{20}}}\left(2\overline{B}-\sqrt{{\Delta }_{\mathrm{p}}}\right)$$

(37)

So far, the optimal design can be completed when the cubic stiffness meets these two equations, which can ensure the triggering of TET, but cannot guarantee the optimal energy dissipation performance. Therefore, other parameters of NES also need to be designed.

When the damping of the main structure ξ_s = 0 and the NES damping ratio ξ₂ and the inerter parameters ($\overline{B}, \stackrel{\mathrm{-}}{c}$) jointly affect the energy transfer efficiency of the main structure (denoted as H_transfer)

$$\begin{array}{l}\begin{array}{l}\mathrm{Scheme} \mathrm{A}:\hfill \end{array} H_{\mathrm{transfer}}=1-{\displaystyle \frac{H_{1\mathrm{a}}^{-}}{H_{1\mathrm{init}}}}=1-{\displaystyle \frac{8}{81H_{1\mathrm{init}}}}\left[\overline{B}\left({\overline{B}}^2+9{\xi }_{\mathrm{NES}}^2\right)-{\Delta }_{\mathrm{p}}\sqrt{{\Delta }_{\mathrm{p}}}\right]\\ \begin{array}{l}\mathrm{Scheme} \mathrm{B}:\hfill \end{array} H_{\mathrm{transfer}}=1-{\displaystyle \frac{H_{1b}^{-}}{H_{1\mathrm{init}}}}=1-{\displaystyle \frac{H_{1a}^{-}}{\left({\overline{B}}^2+{\overline{c}}^2\right)H_{1\mathrm{init}}}},H_{1\mathrm{init}}=\kappa E_1\left(0\right)\end{array}$$

(38)

When resonance energy capture is at its optimal state, where all initial energy is transferred, there is

$$H_{\mathrm{opt}}=1-{\displaystyle \frac{H_1^{-}}{H_1^{+}}}$$

(39)

Deduction reveals that both schemes have the same H_opt

$$H_{\mathrm{opt}}={\displaystyle \frac{2{\Delta }_{\mathrm{p}}\sqrt{{\Delta }_{\mathrm{p}}}}{\overline{B}\left({\overline{B}}^2+9{\xi }_{\mathrm{NES}}^2\right)+{\Delta }_{\mathrm{p}}\sqrt{{\Delta }_{\mathrm{p}}}}}$$

(40)

3.3. Influences of NES Parameters on Energy Transfer

The expressions for H₁ and H₂ in schemes A and B are similar. Therefore, the influence of parameters on H₁ and H₂ is also consistent, except that H₁ in scheme B is ${\displaystyle \frac{1}{{\overline{B}}^2+{\overline{c}}^2}}$ that of scheme A. Only Scheme A is shown below: Figure 3 shows the relationship between H₁ and H₂ for different inertia coefficients bn (μ = 1.9) and different fractional orders μ (b_n = 0.4), where the circle in the figure is the extreme value of H₁ The remaining parameters are m₂ = 0.3, k_n = 0.12, and c_n = 0.002.

Overall, b_n has a significant impact on the amplitudes of H₁ and H₂, with μ having a greater effect on H₁ and a relatively smaller effect on H₂. Under the same parameters, as b_n increases, the amplitudes of H₁ and H₂ become larger, and the height difference between the two extremes becomes greater. Combining Figure 3a and Equation (35), the optimal cubic stiffness of the two schemes is significantly higher than that of the non-inerter NES (b_n = 0). Conversely, as μ increases, the effect is the opposite. In addition, it was found that when using fractional-order inerters, the maximum value is greater than b_n = 0 (non-inerter NES), but the distance $H_2^{+}-H_2^{-}$ between the two extreme values increases.

According to Equation (34), ∆_p must be greater than or equal to 0. If all the parameters of the inerter are 0, we can obtain ${\xi }_2<\sqrt{3}/3$. Otherwise, we can simplify the inequality relationship: ${\xi }_2<\sqrt{3}/3-2\sqrt{3}/3{\omega }_0^{\mu -2}{b}_{\mathrm{n}}/m_2\cos \left(\left(3\mu -2\right)\mathsf{\pi }/6\right)$.

When μ ∈ (5/3, 2), the size range of the cosine term on the right side of the inequality is cosπ/2~cos2π/3, which is obviously less than 0. Therefore, the critical damping ($\sqrt{3}/3$) of FIOB-NES is larger than that of traditional NES, while the opposite is true when μ∈ (1, 5/3). Based on the expression of ∆_p, the relationship curve between parameters and ∆_p can be drawn as shown in Figure 4 (other parameters: (a) c_n = 0.002; (b) and (c) c_n = 0.002 and m₂ = 0.3).

From Figure 4a, as m₂ increases, ∆_p also increases. Figure 4b shows that as μ increases, ∆_p has a maximum value between 1 and 2 at μ, and the location where ∆_p is greater than 0 is around μ = 2. Furthermore, as b_n increases, ∆_p becomes relatively larger. Figure 4c shows that μ determines the influence of b_n; for example, when μ = 1.5, μ = 1.6, b_n cannot make ∆_p greater than 0 regardless of the value taken. When μ = 1.7, 1.8, 1.9, and 2, the larger b_n, the larger ∆_p, indicating that the value of μ determines the trend of ∆_p variation. The addition of fractional-order inerters can significantly alter the critical conditions of the original NES system.

It is obvious that ∆_p = 0 and H_transfer = 0, and there is no energy transfer. Currently, the corresponding damping is called the critical value of energy transfer. Traditional NES (when b_n = 0, $\stackrel{\mathrm{-}}{c}$= 0, and $\stackrel{\mathrm{-}}{b }$= 0, then the critical value is ξ_NES = ξ₂ < $\sqrt{3}/3$ is a constant). When introducing a fractional-order inerter, the energy transfer efficiency is not only related to NES damping ξ₂, but also to bn and μ. The relationship curve between transfer efficiency and NES damping ξ₂ at different derivative orders μ can be obtained from Equation (40) and ${\xi }_{\mathrm{NES}}=\overline{c}+{\xi }_2$, as shown in Figure 5a b_n = 0.1 and Figure 5b b_n = 0.4. The critical damping value is located when the transfer efficiency is equal to 0 in the figure, and the value on the right side is meaningless and not plotted.

Figure 5 shows that the larger the μ, the higher the efficiency for the same ξ₂, indicating that the damping attached to the fractional-order inerter is detrimental to energy transfer efficiency. When the order μ is the same, the larger b_n, the greater the damping at which the transfer efficiency equals 0—that is, the greater the critical damping. However, with the same ξ₂, efficiency is significantly reduced. In summary, firstly, the damping ξ₂ of NES should be far away from the critical value, and the fractional order should be as close to 2 as possible. Finally, under the premise of meeting the requirements, choose a smaller b_n. When ξ₂ approaches or exceeds the critical damping, the energy transfer efficiency is almost zero, and other parameters cannot achieve TET. Here, we focus on the case of c_n = 0.002 (i.e., ξ₂ = 0.027).

Based on the above analysis and discussion, the optimization conditions for generating effective TET are as follows: (1) ∆_p >0; (2) NES damping cannot exceed the critical damping (the NES damping corresponding to a transfer efficiency of 0); (3) there exists an optimal minimum value κ_opt (i.e., threshold for triggering TET) for cubic stiffness (satisfying both Equations (36) and (37)); (4) while satisfying condition 2, try to choose the parameters that maximize the optimal energy transfer.

4. Discussion

Below, we will provide multiple typical simulation examples to verify the validity of the above conclusions. When the initial energy is concentrated in the main structure (i.e., z₁ ≠ 0 or $\dot{z}$₁ ≠ 0, z₂ = 0, $\dot{z}$₂ = 0), due to y₁ = z₁/$\sqrt{\mathit{\epsilon }}$, y₂ = (z₂ − z₁)/$\sqrt{\mathit{\epsilon }}$, it can be seen from Equation (29) that H₁(0) and H₂(0) are not equal to 0.

All the original differential equations of the system are solved using the Runge–Kutta method, where the fractional-order terms are approximated by the Oustaloup approximation algorithm [39]. The simulation time step is 0.1 and the duration is 1000 s. After obtaining the responses, they are converted into H₁(τ₁), H₂(τ₁), and E₁(τ₁) in the flow figures. When the analytical relationship between H₁ and H₂ (red thick solid line) and the comparison of numerical simulations with different initial energies H₁(0) and H₂(0) (initial conditions ẏ₁(0) = 0.4, ẏ₂(0) = −0.4) are given below, the conditions for the occurrence of jumps are verified. The cubic stiffness κ (calculated solely based on Equation (36)) corresponding to the four initial energies is shown in Figure 6 and Figure 7. At this point, the relationship between H₁(0) and H₂(0) is combined to satisfy the following inequality relationship.

For Scheme A:

(1)

$H_1\left(0\right)<H_{1\mathrm{a}}^{+}, H_2\left(0\right)<H_2^{+}$ (solid line).

(2)

$H_1\left(0\right)=H_{1\mathrm{a}}^{+}, H_2\left(0\right)>H_2^{+}$ (dashed line).

(3)

$H_1\left(0\right)>H_{1\mathrm{a}}^{+}, H_2\left(0\right)>H_2^{+}$ (dotted line).

(4)

$H_1\left(0\right)>H_{1\mathrm{a}}^{+}, H_2\left(0\right)>H_2^{-}$ (dash dot line)

For Scheme B:

(1)

$H_1\left(0\right)<H_{1\mathrm{b}}^{+}, H_2\left(0\right)<H_2^{+}$ (solid line).

(2)

$H_1\left(0\right)=H_{1\mathrm{b}}^{+}, H_2\left(0\right)<H_2^{+}$ (dashed line).

(3)

$H_1\left(0\right)>H_{1\mathrm{b}}^{+}, H_2\left(0\right)<H_2^{+}$ (dotted line).

(4)

$H_1\left(0\right)>H_{1\mathrm{b}}^{+}, H_2\left(0\right)>H_2^{+}$ (dash dot line).

The remaining parameters in Figure 6 and Figure 7 are μ = 1.9, m₂ = 0.3, b_n = 0.3, c_s = 0, and c_n = 0.002. The circle in Figure 6 clearly indicates the starting point of the energy response curve, and the following circles in Figure 8 and Figure 10 are the same.

From Figure 6, when Equation (36) is satisfied and the conditions of Equation (37) are also satisfied, both can go towards route 1. However, the dotted line in Figure 6b, which satisfies the calculation of κ_opt via Equation (36) but does not meet the conditions of Equation (37), requires route 2 is followed at this stage. If Equation (36) is not satisfied, we can only follow route 2 (as shown by the dashed line in Figure 6).

Figure 7 shows the time-domain response of the main structure energy (E₁ = 1/2E₁₀) for different H₁(0) and H₂(0): (a) Scheme A: κ_opt = 12.3 and (b) Scheme B: κ_opt = 2.68; all other parameters are the same as Figure 7.

Figure 7 shows that there is a jump in the energy of the main oscillator along route 1, and the energy dissipation is fast; when following route 2, the energy dissipation of the main oscillator is very slow.

The second comparison is given below, with the same parameters of m₂ = 0.3, b_n = 0.3, c_s = 0, and c_n = 0.002. The initial conditions are changed to ẏ₁ (0) = 0.2, ẏ₂ (0) = −0.2, H₁ (0), and H₂ (0) initial positions (dots) and cubic stiffness (as shown in the figures), and the remaining parameters in Figure 8 and Figure 9 are μ = 1.9; Figure 10 and Figure 11 show μ = 2.

Figure 8 and Figure 9 show the case where a fractional-order inerter is used, where the blue dashed line in Figure 8b only satisfies condition (36), resulting in traveling along route 2, while the others satisfy both inequalities simultaneously. Therefore, the energy of the main oscillator can only travel along route 1. The case where the cubic stiffness is much greater than the optimal value is also given. It can be seen from the figure that energy transfer and jumping can also occur when the initial energy is particularly large, and Scheme A has obvious energy transfer, but slower energy dissipation compared to Scheme B.

Figure 10 and Figure 11 show the use of integer-order inerters, which significantly improve the transfer efficiency. The energy between the two oscillators is fully transferred, and jumps can also occur. However, the energy dissipation time of the main structure is obviously much longer than that of fractional-order inerters.

From the previous analysis, Scheme A, like traditional NES, basically needs to satisfy Equation (36) to achieve effective energy transfer and jump in most cases, while Scheme B needs to satisfy two conditional Equations (36) and (37) simultaneously to achieve it.

Figure 12 shows the comparison of the third group under the initial conditions of ẏ₁(0) = 0.4, ẏ₂(0) = −0.4. Two FOIB-NES schemes (b_n = 0.3) are compared with traditional NES (b_n = 0). When both SA and SB satisfy Equations (36) and (37) simultaneously, their k values are taken as the optimal κ_opt (with κ_opt values of 12.3 and 6, respectively). The simulation results are shown in Figure 12a,b. The traditional NES only satisfies Equation (36) (with κ_opt =1.2), as shown in Figure 12b, and κ = κ_opt +1.6 in Figure 12a. The other parameters are m₂ = 0.3, c_s = 0, and c_n = 0.002.

From Figure 12, Scheme B has the lowest number of energy transfers, but the fastest energy consumption. Scheme A shows a certain improvement compared to the non-inerter NES, but not significantly. Moreover, under the same initial energy, its cubic stiffness is the highest, and the non-inerter NES scheme has a particularly high transfer efficiency, but its dissipation capacity is clearly insufficient. Since the optimal stiffness of the non-inerter NES is only 1.2, it is prone to the influence of other nonlinear disturbances, as shown in Figure 12a. When κ = 2.8, its dissipation rate significantly decreases.

To study the robustness or parameter margin of the optimal cubic stiffness, a discriminant index is proposed based on the above analysis: dissipation time t_d, which corresponds to the time when the initial energy E₁(0) dissipation of the main structure is 95% (η(τ_d) = 95%). The formula for the energy dissipation rate η(τ₁) of the main structure is η(τ₁) = [E₁(0) − E₁(τ₁)]/E₁(0) × 100%.

The optimal cubic stiffness value k_opt is selected as the starting point for simulation research, and then the cubic stiffness is increased by a step size of 1 (SB) and 0.5 (SB) to output the dissipation time curves of two different schemes with different orders and η(τ_d) = 95% as shown in Figure 13 (the parameters, except for k and μ, are consistent with Figure 12). Special note: The simulation time for the following example is 30 s. In the area where τ_d reaches 30 s in the figure, the actual τ_d is already greater than 30 s, and TET is basically ineffective. Therefore, to save computational costs, τ_d is uniformly taken as 30 s.

Similarly, we also plotted the dissipation time curves (μ = 1.9) for different inertance b_n, as shown in Figure 14.

Figure 13 and Figure 14 show that the dissipation time and robustness of the inerter of fractional order are significantly better than those with μ = 2, and scheme B is better than scheme A. In addition, the smaller the μ, the better the robustness, while the larger the b_n, the better the robustness.

From the above simulation comparison and discussion, the following results can be drawn:

(1)

When Equation (36) is mostly satisfied, it is effective, especially for scheme A and traditional NES without inerters, while Scheme B will follow route 2, as shown in Figure 6b and Figure 8b.

(2)

Figure 7 and Figure 9 show that the energy of the main structure decreases very slowly when traveling along route 2, without jumping, monotonically decreasing. However, when traveling along route 1, both schemes can quickly decrease and then jump to a lower position towards 0.

(3)

After meeting the TET conditions, the higher the transmission efficiency, the lower the dissipation capacity, which is only a prerequisite for effectively suppressing vibration. Further research is needed on the speed of dissipation. When using an integer-order inerter, as shown in Figure 10, both schemes follow route 1 and have sufficient transmission. However, from Figure 11, the energy drop rate is much slower than in Figure 9.

(4)

At the same initial velocity, the optimal cubic stiffness κ_opt, required for TET generation in Scheme A, is much larger than that in Scheme B, while the κ_opt is minimized in integer-order inertial NES. In addition, from Figure 13 and Figure 14, scheme B has better parameter robustness than scheme A, and both schemes are more robust than integer-order when using fractional-order inerters.

In summary, the conditions for generating TET are as follows: satisfy the inequality of Equation (34), ensuring that the equation has multiple roots; simultaneously satisfy the inequalities of Equations (36) and (37), and try to take values near the lower limit to ensure the occurrence of jumps; keep NES damping away from the critical damping to ensure a certain transmission efficiency; considering the energy dissipation capacity, it is necessary to select smaller values of ξ₂ and a larger values of b_n, and to select a value of μ that is less than but close to 2.

5. Conclusions

This article is based on the fractional-order complex averaging method and uses an FOIB-NES structure design as the core carrier to comprehensively reveal the key role of fractional-order inerters in target energy transfer and dissipation mechanisms. The main conclusions are as follows:

(1)

By utilizing the properties of fractional calculus (such as fractional-order Leibniz and trigonometric functions), the difficulties in mathematical processing of fractional-order complex averaging methods have been solved, allowing for the use of the same steps as integer-order systems to obtain slowly varying dynamic equations and manifolds.

(2)

Supplementing traditional TET conditions makes the selection of optimal cubic stiffness (i.e., threshold for triggering TET) more reliable. In addition, when the TET condition is met, the optimal cubic stiffness of series FOIB-NES is much lower than that of parallel FOIB-NES, but greater than that of traditional integer-order inertial NES.

(3)

An effective NES damping range for energy dissipation should be ensured: FOIB-NES has a larger damping limit than non-inerter NES, and the lower limit of NES damping for series FOIB-NES is smaller than that for parallel FOIB-NES. The dissipation speed of series FOIB-NES is greater than that of parallel FOIB-NES and greater than that of non-inerter NES. When the critical damping value or over the critical damping value is selected for the main structure, regardless of the design parameters, NES almost does not dissipate energy—that is, NES is ineffective.

(4)

The transfer efficiency of traditional non-inerter NES and integer-order inertial NES is higher than that of FOIB-NES, and the transfer efficiency of parallel FOIB-NES is higher than that of the series type. When both meet the TET conditions, the series FOIB-NES is more prone to energy jumps, resulting in a faster decrease in the energy of the main structure.

Although this study achieved certain results in analyzing the energy transfer mechanism and energy dissipation of FOIB-NES, there are still several theoretical and engineering limitations that urgently need further breakthroughs in subsequent research. Although current research has verified that cubic stiffness can achieve efficient targeted energy transfer within a specific range, it has not yet answered a fundamental question: is there a physically achievable upper or lower bound that leads to a sharp decrease or even failure of TET efficiency beyond this range? This study shows that when FOIB-NES is connected in series, TET can be triggered under low initial energy conditions, exhibiting significant inertia substitution ability and a better dissipation effect. This provides a new idea for reducing or even removing traditional additional mass blocks and achieving massless shock absorbers. However, the current conclusions are mainly based on numerical simulations and lack physical prototype verification. The method proposed in this article is mainly aimed at the case of coupling FOIB-NES with a single-degree-of-freedom main system. However, actual engineering structures (beams, plates, shells, etc.) have an essentially infinite degree of freedom continuum, and their coupling with fractional-order NES will lead to a series of new problems: the embedding of fractional-order differential operators in partial differential equations makes system decoupling more difficult. Therefore, it is unclear whether the approximate method proposed in this article would still be applicable, and the efficiency and stability of numerical calculations need to be verified. Future work could aim to construct a reduced-order model through modal truncation and perturbation analysis, and use nonlinear orthogonal decomposition to identify the dominant energy channel, providing theoretical support for vibration suppression in large structures.