Studies on Vibration and Synchronization Characteristics of an Anti-Resonance System Driven by Triple-Frequency Excitation

Abstract

In the continuous drilling process of oil wells, to achieve the efficient screening of drilling fluids by the vibrating screen while ensuring the safety of the screening operation, an anti-resonance system driven by two exciters with triple-frequency (denoted as 3:1 frequency ratio) is proposed. Initially, differential motion equations are formulated utilizing Lagrange’s equation, followed by the definition of vibration isolation coefficients adopting ratios. Triple-frequency synchronization and stability criterion between two eccentric blocks are subsequently elucidated via the asymptotic method and Routh–Hurwitz criterion. Concurrently, the effects of structural parameters on vibration isolation capacity, steady-state trajectory, and the triple-frequency synchronization phase are investigated through numerical computation. Ultimately, the reliability of the theoretical study is corroborated by simulation analysis. Results indicate that under the allowable system parameters for the practical project, the amplitude of the vibration body can exceed three times that of the isolation body; the two solutions of the stable phase difference (SPD) are different by π, one of which is stable and the other is unstable, and the stability of phase difference is determined by the sign of the stability coefficient. This work is useful for developing new vibrating screens and other multi-frequency vibration machines.

1. Introduction

Vibrating screens are widely applied to recycle and separate drilling fluid from wellhole mud in the drilling process [1,2,3]. Since the operation performance of a vibrating screen is determined by dynamic characteristics, the synchronization phenomenon between unbalanced rotors of vibrating screens is crucial for the dynamic behavior of the vibrating system. The so-called synchronization phenomenon is usually considered as the rhythm adjustment of vibrating objects due to their internal weak coupling [4], which also can be found everywhere in nature. For example, pendulums with diverse speed in the same beam, flashing bugs with inconsistent frequencies in the wild, and finally limit-cycle oscillators with different working frequencies in their network exhibit consistent frequency operation [5]. In a mechanical system, Blekhman explained the synchronization phenomenon of double eccentric rotors, and the concept and theory of synchronization related to rotors operated in planes was defined [6,7,8]. The theory was further extended by Paz from the special case of parallel-axis rotors in a space system, and the stability condition of phase difference was revealed by adopting the Hamiltonian principle [9,10]. Moreover, Zhang et al. studied the synchronous transmission of a dry friction cylindrical roller in a vibrating mechanical system excited with two exciters. By means of the average method and Routh–Hurwitz criterion, the condition of synchronization and the criterion of stability were obtained [11,12,13]. The vibrators of the vibration system analyzed above are mainly fixed in the oscillating body. Fang proposed a vibration system where the vibrators wobbled relative to the oscillating body, and the correctness of synchronization behavior in a rotor-pendula system was verified by simulations [14,15,16]. In addition, to investigate the self-synchronization of the sub-resonance region and resonance region, the interaction between unbalanced DC motors and the flexible structural frame was analyzed by Balthazar through several numerical simulations [17,18]. To ensure the linear vibration of the system driven by four exciters, the influence of structural parameters on the composite synchronization of four induction motors was discussed by Kong et al. [19,20]; Kong’s results show that the zero-phase synchronization of four different exciters can be obtained at different frequencies of power supply. In order to improve the screening efficiency of the vibration system and prevent screen clogging during the rapid drilling process, Zou studied the double-frequency synchronization of the dual-vibration motor system and proposed a frequency doubling control method [21], and the behavior of double/triple-frequency synchronization when a system is driven by two, three, and four motors was investigated by Zhang using asymptotic and average methods [22,23,24]. However, in engineering practice, the adverse effects caused by vibration (vibration fatigue, vibration noise, etc.) are obvious, especially for vibration systems under multi-frequency excitation. Consequently, a nonlinear dynamic model reflecting the actual working state of anti-resonance machinery was discussed by Li and Tao [25]. On this basis, the double-frequency eccentric rotor excitation vibration system was applied to the anti-resonance system by Sun [26,27], whereas the anti-resonance and synchronization mechanism of the triple-frequency rotor excitation anti-resonance system is less revealed. In high-frequency ratio systems, the particles on the screen surface are subjected to more intense periodic variable stress and exhibit a larger acceleration gradient. This characteristic renders such systems highly suitable for screening wet, cohesive, and multi-sized mixed materials. Therefore, a dual-body anti-resonance system driven by two motors with triple-frequency excitation is discussed in the presented work; the synchronization and stability are obtained by the asymptotic method and Routh–Hurwitz criterion.

The innovations and contributions of the triple-frequency anti-resonance system in engineering are mainly reflected in the following aspects: (1) In the screening process, the exciting forces of different amplitudes and frequencies are worked alternately, with the alternation of the excitation force occurring twice within one vibration cycle. This configuration is suitable for screening material particles with a relatively wide range of intrinsic properties. (2) The stable trajectory of the triple-frequency vibration system presents as a superposition of three circles, and it belongs to an unconventional trajectory vibration system. In contrast to the same-frequency and double-frequency systems, the direction of the acceleration vector of its vibration trajectory changes more rapidly. In engineering applications, for the screening process, this rapid change means that materials on the vibrating screen are subjected to more complex and frequent force changes. For example, in the screening of granular materials, it can make materials tumble and move more fully, promoting the separation of particles of different sizes. Compared with the single-frequency excitation which has a relatively simple and regular trajectory and less force change on materials, and the double-frequency excitation whose trajectory change frequency is lower than that of triple-frequency, the triple-frequency excitation can improve the screening efficiency and effect, which is more advantageous for the screening engineering operation. (3) The alternating excitation force generated by the vibrator is prone to induce fatigue damage in the base, while in an anti-resonance system, the vibrations of the mass connected to the foundation are significantly less than those of the working mass. Therefore, the application of anti-resonance theory allows for a rational distribution of the excitation force, thereby reducing harmful vibrations and enhancing beneficial vibrations. (4) Both vibrators are mounted within the isolation body; during operation, the vibration amplitude of the isolation body is relatively small compared to the main vibration body. This configuration provides protection for the vibrators and mitigates the risk of loosening and damage.

The structure of the paper is planned as follows: In Section 2, the dynamic equation of the system is derived. In Section 3, the steady-state responses of the anti-resonance system are obtained by Laplace transform. In Section 4, the system modes are derived, and the coefficient of the vibration absorption capacity is determined. In Section 5, the conditions of synchronization and stability of the system are given. In Section 6, the vibration characteristics and stable self-synchronization conditions of the system are discussed in combination with theory and practice. In Section 7, the reliability of the theory is verified by two sets of simulation. In Section 8, several important conclusions from the analysis are summarized.

2. Dynamic Equation

The simplified dynamical model of an anti-resonance vibrating screen driven by two eccentric blocks (EBs) with triple-frequency excitation is established in Figure 1a. The mass of the vibration body (VB) and isolation body (IB) is expressed in terms of m₁ and m₂. VB is linked to IB through springs with a stiffness coefficient k_u (u = x₁, y₁) and damping coefficient fu (u = x₁, y₁), and IB is connected to the ground through springs with a stiffness coefficient k_v (v = x₂, y₂) and damping coefficient f_v (v = x₂, y₂). Meanwhile, the instantaneous swing angles of two bodies around their centroid are denoted as ψ₁ and ψ₂; the rotational stiffness and damping coefficient of the two bodies are denoted as k_w and f_w (w = ψ₁, ψ₂). Additionally, two EBs of different frequencies with masses m_o1 and m_o2 are symmetrically mounted in the IB. The installation position of the vibrators is determined by the installation angle β_i (i = 1, 2) and installation distance l_i (i = 1, 2); the phase angle of two EBs are represented by φ_i (i = 1, 2), and the radius of two EBs is set as r.

In Figure 1b, C₁ and C₂ are the origin of VB and IB centroids when the system is stationary; $C_i^{\prime }$ (i = 1, 2) and $C_i^{″}$ (i = 1, 2) are the mass center of the VB and IB when the system is vibrating, and the axes of two vibrators are deemed as o_i (i = 1, 2). Combined with Figure 1a, $C_1{x}_1{y}_1$ and $C_2{x}_2{y}_2$ are fixed reference frames of the VB and IB; $C_1^{\prime }{x}_1^{\prime }{y}_1^{\prime }$ and $C_2^{\prime }{x}_2^{\prime }{y}_2^{\prime }$ correspond to the translational reference frames of the VB and IB. Owing to the instantaneous swing angles of rigid bodies that are denoted by ψ_i (i = 1, 2), $C_1^{″}{x}_1^{″}{y}_1^{″}$ and $C_2^{″}{x}_2^{″}{y}_2^{″}$ are considered the motion reference frames of the VB and IB, respectively. In addition, when the system is working, the relative position of vibrator shafts and the IB is unchanged, and $o_i{x}_i^{‴}{y}_i^{‴}$ is regarded as the fixed reference frame in $c_2^{″}{x}_2^{″}{y}_2^{″}$.

Therefore, the centroids of two EBs in the reference frame $o_i{x}_i^{‴}{y}_i^{‴}$ (i = 1, 2) are expressed as follows:

$${{\Phi }_1}^{‴}={\left[r\cos {\phi }_1,r\sin {\phi }_1\right]}^T \begin{array}{c}{{\Phi }_2}^{‴}={\left[r\cos {\phi }_2,r\sin {\phi }_2\right]}^T\end{array}$$

(1)

Point o_i (i = 1, 2) in the reference frame $c_2^{″}{x}_2^{″}{y}_2^{″}$ is denoted as follows:

$${{\Phi }_1}^{″}={\left[l\cos {\beta }_1,l\sin {\beta }_1\right]}^T \begin{array}{c}{{\Phi }_2}^{″}={\left[l\cos {\beta }_2,l\sin {\beta }_2\right]}^T\end{array}$$

(2)

The centroids of the VB and IB in the reference frames $C_1{x}_1{y}_1$ and $C_2{x}_2{y}_2$ can be expressed as follows:

$${\Phi }_1^{\prime }={\left[x_1,y_1\right]}^T \begin{array}{c}{\Phi }_2^{\prime }={\left[x_2,y_2\right]}^T\end{array}$$

(3)

Owing to the swing of VB and IB, the centroids of the EBs are expressed in $C_2{x}_2{y}_2$ as follows:

$${\Phi }_1={\Phi }_2^{\prime }+{\Gamma }_2({\Phi }_1^{″}+{{\Phi }_1}^{‴}) {\Phi }_2={\Phi }_2^{\prime }+{\Gamma }_2({\Phi }_2^{″}+{{\Phi }_2}^{‴})$$

(4)

Here, Φ₁ and Φ₂ can be understood as the coordinates of the center of mass of the eccentric block in the reference frame $C_2{x}_2{y}_2$. Since neither of the two motors is installed in the upper body, ${\Phi }_1^{\prime }$ is not involved in the calculation of Φ₁ and Φ₂ coordinates, but the center of mass of the lower body ${\Phi }_2^{\prime }$ is adopted instead. Equation (4) is essentially a coordinate transformation relationship. The coordinates of the center of mass of the eccentric block in the local/intermediate coordinate system are “mapped” to the coordinate system used for overall analysis through the rotation matrix Γ, achieving the unification of coordinates among different moving parts (the EB associated with the VB and IB), and preparing for subsequent energy modeling. Γ₁ and Γ₂ are rotation matrices, describing the coordinate rotation mapping caused by the structural swing angles ψ₁ and ψ₂. cosψ and sinψ are the standard constructions of rotation matrices, used to convert the coordinates in one coordinate system to the coordinate system after rotating ψ around the axis perpendicular to the paper plane. The explanations of Γ₁ and Γ₂ are provided in Appendix A.

Based on the energy method in Analytical Mechanics and combined with the basic principles of Multibody System Dynamics, the kinetic energy T, potential energy V, and dissipation energy D of the system are expressed as follows:

$$\begin{array}{l}T={\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^2{m}_i({\dot{x}}_i^2+{\dot{y}}_i^2)}+{\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^2{j}_{0i}{\dot{\phi }}_i^2}+{\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^2{J}_i{\dot{\psi }}_i^2}+{\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^2{m}_{oi}}{\dot{\Phi }}_i^T{\dot{\Phi }}_i\\ V={\displaystyle \frac{1}{2}}{k}_{x2}{x}_2^2+{\displaystyle \frac{1}{2}}{k}_{y2}{y}_2^2+{\displaystyle \frac{1}{2}}{k}_{\psi 2}{\psi }_2^2+{\displaystyle \frac{1}{2}}{k}_{x1}{(x_1-x_2)}^2+{\displaystyle \frac{1}{2}}{k}_{y1}{(y_1-y_2)}^2+{\displaystyle \frac{1}{2}}{k}_{\psi 1}{({\psi }_1-{\psi }_2)}^2\\ D={\displaystyle \frac{1}{2}}{f}_{x2}{\dot{x}}_2^2+{\displaystyle \frac{1}{2}}{f}_{y2}{\dot{y}}_2^2+{\displaystyle \frac{1}{2}}{f}_{\psi 2}{\dot{\psi }}_2^2+{\displaystyle \frac{1}{2}}{f}_{x1}{({\dot{x}}_1-{\dot{x}}_2)}^2+{\displaystyle \frac{1}{2}}{f}_{y1}{({\dot{y}}_1-{\dot{y}}_2)}^2+{\displaystyle \frac{1}{2}}{f}_{\psi 1}{({\dot{\psi }}_1-{\dot{\psi }}_2)}^2+{\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^2{f}_i{\dot{\phi }}_i^2}\end{array}$$

(5)

The derivation process of the energy equation is shown in Appendix B. Since the vibrating system is solely powered by the electromagnetic output torque and the gravity of EBs, the generalized coordinates and forces are described as follows:

$$[Q_{x1} Q_{x2} Q_{y1} Q_{y2} Q_{\psi 1} Q_{\psi 2} Q_{\phi 1} Q_{\phi 2}]=[0 0 0 0 0 0 T_{e1}+m_{o1}rgsin{\phi }_1 T_{e2}+m_{o1}rgsin{\phi }_1]$$

where $T_{ei}(i=1,2)$ is the effective electromagnetic output torque of the two vibrators, according to the Lagrange equation:

$${\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{\partial T}{\partial \dot{q}}}\right)\partial {\displaystyle \frac{\partial \left(T-V\right)}{\partial q}}+{\displaystyle \frac{\partial D}{\partial \dot{q}}}=Q$$

(6)

where q is the vector of generalized coordinates, including independent coordinates x₁, x₂, y₁, y₂, ψ₁, ψ₂, φ₁, and φ₂ that describe the configuration of the system. The dynamic equation of the system is obtained as follows:

$$\begin{array}{l}{M}_1{\ddot{x}}_1+f_{x1}({\dot{x}}_1-{\dot{x}}_2)+k_{x1}(x_1-x_2)=0\\ M_2{\ddot{x}}_2+f_{x2}{\dot{x}}_2+f_{x1}({\dot{x}}_2-{\dot{x}}_1)+k_{x2}{x}_2+k_{x1}(x_2-x_1)={\displaystyle \sum _{i=1}^2{m}_{oi}r({\dot{\phi }}_i^2\sin {\phi }_i-{\ddot{\phi }}_i\cos {\phi }_i)}\\ M_1{\ddot{y}}_1+f_{y1}({\dot{y}}_1-{\dot{y}}_2)+k_{y1}(y_1-y_2)=0\\ M_2{\ddot{y}}_2+f_{y2}{\dot{y}}_2+f_{y1}({\dot{y}}_2-{\dot{y}}_1)+k_{y2}{y}_2+k_{y1}(y_2-y_1)=-{\displaystyle \sum _{i=1}^2{m}_{oi}r({\dot{\phi }}_i^2\cos {\phi }_i+{\ddot{\phi }}_i\sin {\phi }_i)}\\ J_1{\ddot{\psi }}_1+f_{\psi 1}({\dot{\psi }}_1-{\dot{\psi }}_2)+k_{\psi 1}({\psi }_1-{\psi }_2)=0\\ J_2{\ddot{\psi }}_2+f_{\psi 2}{\dot{\psi }}_2+f_{\psi 1}({\dot{\psi }}_2-{\dot{\psi }}_1)+k_{\psi 2}{\psi }_2+k_{\psi 1}({\psi }_2-{\psi }_1)\\ =-m_{o1}rl[{{\dot{\phi }}_1}^2\cos ({\phi }_1-{\beta }_1)+{\ddot{\phi }}_1\sin ({\phi }_1-{\beta }_1)]-m_{o2}rl[{{\dot{\phi }}_2}^2\cos ({\phi }_2-{\beta }_2)+{\ddot{\phi }}_2\sin ({\phi }_2-{\beta }_2)]\\ J_{o1}{\ddot{\phi }}_1+f_1{\dot{\phi }}_1=T_{e1}+m_{o1}rgsin{\phi }_1-m_{o1}r[{\ddot{x}}_2\cos {\phi }_1+{\ddot{y}}_2\sin {\phi }_1]-m_{o1}lr[{\ddot{\psi }}_2\sin ({\phi }_1-{\beta }_1)-{\dot{\psi }}_2^2\cos ({\phi }_1-{\beta }_1)]\\ J_{o2}{\ddot{\phi }}_2+f_2{\dot{\phi }}_2=T_{e2}+m_{o2}rgsin{\phi }_2-m_{o2}r[{\ddot{x}}_2\cos {\phi }_2+{\ddot{y}}_2\sin {\phi }_2]-m_{o2}lr[{\ddot{\psi }}_2\sin ({\phi }_2-{\beta }_2)-{\dot{\psi }}_2^2\cos ({\phi }_2-{\beta }_2)]\end{array}$$

(7)

where M_j (j = 1, 2) denotes the mass of body j and its components, and we have $M_1=m_1$, $M_2=m_2+{m_o}_1+m_{o2}$. f_i (i = 1, 2) stands for the friction coefficient between motor i and motor shaft i; $J_{oi}=j_{oi}+m_{oi}{r}^2$; J_oi and j_oi are the moment of inertia of EBs and their axes, respectively, in which j_oi can be ignored for its small value. In addition, J₁ and J₂ are the rotational inertia of VB and IB around their centroids, which are expressed as $J_1=M_1{l_{e1}}^2, J_2=M_2{l_{e2}}^2$.

3. Vibration Responses

In the simplified dynamical model shown in Figure 1a, the instantaneous phase of EBs is noted as φ_i (i = 1, 2). Since the period of φ₂ is one-third of φ₁, the phase of the EBs can be expressed as follows:

$${\phi }_1=\tau +{\Delta }_1 \begin{array}{c}{\phi }_2=3\tau +{\Delta }_2.\end{array} \tau ={\omega }_{m}t$$

(8)

where Δ_i (i = 1, 2) is the relative phase angle. During the process of achieving stable synchronization, Δ_i is slowly changed relative to φ_i. It should be noted that when the system is stable, the dynamic balance between the load torque and the output torque of the vibrator is achieved, but the rapid change of Δ_i is caused by the imbalance between torques; thus, ${\dot{\Delta }}_i$ (i = 1, 2) can be ignored. Through Differentiating Equation (8), the velocities of the EBs at steady-state are obtained as follows:

$${\dot{\phi }}_1={\omega }_m \begin{array}{c}{\dot{\phi }}_2=3{\omega }_m\end{array}$$

(9)

Substituting Equation (9) into Equation (7), the damping-related term can be ignored by considering the proposed system as a weak damping system. This simplification is justified because the damping coefficient of the system, primarily derived from spring collisions and air damping, is approximately 20 N·s·m/rad. Such a value is significantly smaller than the range (0–10,000 N·s·m/rad) where the swing-direction damping affects the phase difference solution by less than 0.1 rad, as shown in the literature [28]. Thus, the steady-state transient response approximate solutions of the system are derived as follows:

$$\begin{array}{l}{x}_1={\mu }_{x11}{\eta }_{1}r\sin {\phi }_1+{\mu }_{x12}{a}_{21}{\eta }_{1}r{n}^2\sin {\phi }_2\\ x_2={\mu }_{x21}{\eta }_{1}r\sin {\phi }_1+{\mu }_{x22}{a}_{21}{\eta }_{1}r{n}^2\sin {\phi }_2\\ y_1=-{\mu }_{y11}{\eta }_{1}r\cos {\phi }_1-{\mu }_{y12}{a}_{21}{\eta }_{1}r{n}^2\cos {\phi }_2\\ y_2=-{\mu }_{y21}{\eta }_{1}r\cos {\phi }_1-{\mu }_{y22}{a}_{21}{\eta }_{1}r{n}^2\cos {\phi }_2\\ {\psi }_1=-{\mu }_{\psi 11}{\eta }_1{A}_{22}rl\cos ({\phi }_1-{\beta }_1)-{\mu }_{\psi 12}{a}_{21}{\eta }_1{A}_{22}rl{n}^2\cos ({\phi }_2-{\beta }_2)\\ {\psi }_2=-{\mu }_{\psi 21}{\eta }_1{A}_{22}rl\cos ({\phi }_1-{\beta }_1)-{\mu }_{\psi 22}{a}_{21}{\eta }_1{A}_{22}rl{n}^2\cos ({\phi }_2-{\beta }_2)\end{array}$$

(10)

where μ_xa, μ_ya, and μ_ψa (a = 11, 12, 21, 22) are the amplitude amplification factors of steady-state responses in the system; the parameter n (n = 3) represents the multiple of the high-frequency excitation force frequency relative to the fundamental frequency. Additionally, the detailed descriptions of the symbols in Equation (10) are as follows:

$$\begin{array}{l}{\omega }_{x1}=\sqrt{{\displaystyle \frac{k_{x1}}{M_1}}},n_{x1}={\displaystyle \frac{{\omega }_{x1}}{{\omega }_m}},{\omega }_{x2}=\sqrt{{\displaystyle \frac{k_{x2}}{M_2}}},n_{x2}={\displaystyle \frac{{\omega }_{x2}}{{\omega }_m}},a_{21}={\displaystyle \frac{m_{o2}}{m_{o1}}},{\omega }_{y1}=\sqrt{{\displaystyle \frac{k_{y1}}{M_1}}},\\ n_{y1}={\displaystyle \frac{{\omega }_{y1}}{{\omega }_m}},{\omega }_{y2}=\sqrt{{\displaystyle \frac{k_{y2}}{M_2}}},n_{y2}={\displaystyle \frac{{\omega }_{y2}}{{\omega }_m}},{\eta }_M={\displaystyle \frac{M_1}{M_2}},{\omega }_{\psi 1}=\sqrt{{\displaystyle \frac{k_{\psi 1}}{J_1}}},n_{\psi 1}={\displaystyle \frac{{\omega }_{\psi 1}}{{\omega }_m}},\\ {\omega }_{\psi 2}=\sqrt{{\displaystyle \frac{k_{\psi 2}}{J_2}}},n_{\psi 2}={\displaystyle \frac{{\omega }_{\psi 2}}{{\omega }_m}},{\eta }_1={\displaystyle \frac{m_{o1}}{M_2}},{\eta }_M={\displaystyle \frac{M_1}{M_2}},A_{22}={\displaystyle \frac{M_2}{J_2}}.\end{array}$$

$$\begin{array}{l}{\mu }_{p11}={\displaystyle \frac{n_{p1}^2}{1-n_{p2}^2-n_{p1}^2{\eta }_M-n_{p1}^2+n_{x1}^2{n}_{x2}^2}},{\mu }_{p12}={\displaystyle \frac{n_{p1}^2}{n^4-n^2{n}_{p2}^2-n^2{n}_{p1}^2{\eta }_M-n^2{n}_{p1}^2+n_{p1}^2{n}_{p2}^2}}\\ {\mu }_{p21}={\displaystyle \frac{-1+n_{x1}^2}{1-n_{p2}^2-n_{p1}^2{\eta }_M-n_{p1}^2+n_{p1}^2{n}_{p2}^2}},{\mu }_{p22}={\displaystyle \frac{-n^2+n_{x1}^2}{n^4-n^2{n}_{p2}^2-n^2{n}_{p1}^2{\eta }_M-n^2{n}_{p1}^2+n_{p1}^2{n}_{p2}^2}},p=(x,y,\psi )\end{array}$$

4. Vibration Characters

4.1. Natural Frequency

To investigate the vibration absorption characteristics of the system, a comprehensive analysis of natural frequency is of necessity. From Equation (7), the mass-coupling matrix M and stiffness-coupling matrix K in a vertical direction can be deduced as follows:

$$M=\left[\begin{array}{cc}{M}_1& 0\\ 0& M_2\end{array}\right]\hspace{1em}K=\left[\begin{array}{cc}{k}_{y1}& -k_{y1}\\ -k_{y1}& k_{y1}+k_{y2}\end{array}\right]$$

(11)

The system frequency equation can be derived from M and K as follows:

$$\Delta ({\omega }^2)=\left|\begin{array}{cc}{k}_{y1}-M_1{\omega }_n^2& -k_{y1}\\ -k_{y1}& k_{y1}+k_{y2}-M_2{\omega }_n^2\end{array}\right|=0$$

(12)

The frequency equation is expanded and rearranged as follows:

$$\begin{array}{c}a{({\omega }_n^2)}^2+b{\omega }_n^2+c=0,\\ a=M_1{M}_2,b=-M_1(k_{y1}+k_{y2})-M_2{k}_{y1},c=k_1(k_{y1}+k_{y2})-k_{y1}^2\end{array}$$

(13)

Then, the natural frequency of the vibration system is obtained as follows:

$${\omega }_{n1,2}^2={\displaystyle \frac{-b\mp \sqrt{b^2-4ac}}{2a}}$$

(14)

Eliminating the negative values of ${\omega }_{n1,2}^2$, the intrinsic frequency of the system in the y-direction is represented by a pair of meaningful roots (ω_n1 and ω_n2). After arranging the roots according to ω_n1 ≤ ω_n2, ω_n1 is the first natural frequency, and ω_n2 is the second natural frequency of the system. In addition, utilizing the aforementioned algorithm for y-direction natural frequencies, natural frequencies in x- and ψ-directions can also be derived.

4.2. Vibration Absorption Coefficient

When the system is operated in a steady-state, the instantaneous positions of two bodies can be obtained by Equation (10). Thus, Y₁ and Y₂ are employed as the maximum displacements of the VB and IB in the y-direction, and X₁ and X₂ are adopted as the maximum displacements in the x-direction. To reduce the number of resonance frequencies and simplify system design, the structural parameters (such as spring constants and mass distributions) in the x- and y-directions are identical. This symmetry ensures that the dynamic behaviors in the x- and y-directions are fundamentally equivalent (i.e., X₁ = Y₁ and X₂ = Y₂). Consequently, the vibration absorption coefficient ζ can be expressed as the ratio of the amplitudes of the VB and IB in the y- direction, i.e., as follows:

$$\zeta ={\displaystyle \frac{Y_1}{Y_2}}$$

(15)

When ζ > 1, the vibration absorption performance of the system is observed in the y-direction.

5. Triple-Frequency Synchronization Condition and Stability Criterion

5.1. Self-Synchronization Condition

The displacement solutions in previous work are obtained under the condition when the system is stable. Since the synchronization characteristics of EBs are crucial to the study on the stability of the vibrating system, the asymptotic method is introduced to derive the synchronization phase difference of EBs. To obtain the phase equations for EBs, the solutions for every degree of freedom are brought into the last two equations of Equation (7). After simplifying the obtained equations, the instantaneous acceleration expressions of EBs are written in the following form:

$$\begin{array}{l}{\ddot{\phi }}_1=\epsilon \{T_1^{(1)}-2F_1^{(1)}{\dot{\phi }}_1+k_{1}sin{\phi }_1-{\mu }_{x21}{\ddot{\phi }}_1-{\displaystyle \frac{{\mu }_{\psi 21}{A}_{22}{l}^2}{2}}[{\ddot{\phi }}_1-{\ddot{\phi }}_1\cos {\mathrm{\Phi }}_{\beta 11}^{+}+{\dot{\phi }}_1^2\sin {\mathrm{\Phi }}_{\beta 11}^{+}]\\ -{\mu }_{x22}{a}_{21}{n}^2[{\ddot{\phi }}_2\cos {\mathrm{\Phi }}_{12}^{-}+{\dot{\phi }}_2^2\sin {\mathrm{\Phi }}_{12}^{-}]-{\displaystyle \frac{1}{2}}{\mu }_{\psi 22}{a}_{21}{A}_{22}{n}^2{l}^2[{\ddot{\phi }}_2\cos {\mathrm{\Phi }}_{\beta 12}^{-}\\ -{\ddot{\phi }}_2\cos {\mathrm{\Phi }}_{\beta 21}^{+}+{\dot{\phi }}_2^2\sin {\mathrm{\Phi }}_{\beta 21}^{+}+{\dot{\phi }}_2^2\sin {\mathrm{\Phi }}_{\beta 12}^{-}]\}+{\epsilon }^2{T}_1^{(2)}-2{\epsilon }^2{F}_1^{(2)}{\dot{\phi }}_1\\ {\ddot{\phi }}_2=\epsilon \{T_2^{(1)}-2F_2^{(1)}{\dot{\phi }}_2+k_{2}sin{\phi }_2-{\mu }_{x21}[{\ddot{\phi }}_1\cos {\mathrm{\Phi }}_{12}^{-}-{\dot{\phi }}_1^2\sin {\mathrm{\Phi }}_{12}^{-}]\\ +{\displaystyle \frac{{\mu }_{\psi 21}{A}_{22}{l}^2}{2}}[-{\dot{\phi }}_1^2\sin {\mathrm{\Phi }}_{\beta 21}^{+}+{\dot{\phi }}_1^2\sin {\mathrm{\Phi }}_{\beta 12}^{-}+{\ddot{\phi }}_1\cos {\mathrm{\Phi }}_{\beta 21}^{+}-{\ddot{\phi }}_1\cos {\mathrm{\Phi }}_{\beta 12}^{-}]\\ -{\mu }_{x22}{a}_{21}{n}^2{\ddot{\phi }}_2+{\displaystyle \frac{{\mu }_{\psi 22}{a}_{21}{A}_{22}{n}^2{l}^2}{2}}[-{\dot{\phi }}_2^2\sin {\mathrm{\Phi }}_{\beta 22}^{+}+{\ddot{\phi }}_2\cos {\mathrm{\Phi }}_{\beta 22}^{+}-{\ddot{\phi }}_2]\}\\ +{\epsilon }^2\{T_2^{(2)}-2F_2^{(2)}{\dot{\phi }}_2\}+{\epsilon }^3\dots \end{array}$$

(16)

with

$$\begin{array}{l}{\mathrm{\Phi }}_{11}^{+}=2{\phi }_1,{\mathrm{\Phi }}_{\beta 11}^{+}=2{\phi }_1-2{\beta }_1,{\mathrm{\Phi }}_{\beta 12}^{-}=-{\mathrm{\Phi }}_{\beta 21}^{-}={\phi }_1-{\phi }_2+{\beta }_2-{\beta }_1,{\mathrm{\Phi }}_{21}^{+}={\phi }_2+{\phi }_1,\\ {\mathrm{\Phi }}_{22}^{+}=2{\phi }_2,{\mathrm{\Phi }}_{\beta 22}^{+}=2{\phi }_2-2{\beta }_2,{\mathrm{\Phi }}_{12}^{-}=-{\mathrm{\Phi }}_{21}^{-}={\phi }_1-{\phi }_2,{\mathrm{\Phi }}_{\beta 21}^{+}={\phi }_2+{\phi }_1-{\beta }_2-{\beta }_1\end{array}$$

$$\begin{array}{l}{\displaystyle \frac{T_{e1}}{J_{o1}{\omega }_m^2}}=\epsilon T_1^{(1)}+{\epsilon }^2{T}_1^{(2)},{\displaystyle \frac{T_{e2}}{J_{o2}{\omega }_m^2}}=\epsilon T_2^{(1)}+{\epsilon }^2{T}_2^{(2)},{\displaystyle \frac{f_1}{J_{o1}{\omega }_m}}=2\epsilon F_1^{(1)}+2{\epsilon }^2{F}_1^{(2)},\\ {\displaystyle \frac{f_2}{J_{o2}{\omega }_m}}=2\epsilon F_2^{(1)}+2{\epsilon }^2{F}_2^{(2)}, k_1={\displaystyle \frac{M_{2}g}{m_{o1}r{\omega }_m^2}},k_2={\displaystyle \frac{M_{2}g}{m_{o2}r{\omega }_m^2}}\end{array}$$

In Equation (16), parameter η₁ denoted as the mass ratio of m_o1 to M₁ is treated as a small parameter ε. Carrying Equation (8) into Equation (16), the acceleration equation with respect to the phase angle can be expressed as follows:

$$\begin{array}{l}{\ddot{\Delta }}_1=\epsilon \{T_1^{(1)}-2F_1^{(1)}(1+{\dot{\Delta }}_1)+k_{1}sin(\tau +{\Delta }_1)-{\mu }_{x21}{\ddot{\Delta }}_1-{\displaystyle \frac{{\mu }_{\psi 21}{A}_{22}{l}^2}{2}}[(1+2{\dot{\Delta }}_1+{\dot{\Delta }}_1^2)\sin {\mathrm{\Phi }}_{\beta 11}^{+}\\ +{\ddot{\Delta }}_1-{\ddot{\Delta }}_1\cos {\mathrm{\Phi }}_{\beta 11}^{+}]+{\mu }_{x22}{a}_{21}{n}^2[-(n^2+2n{\dot{\Delta }}_2+{\dot{\Delta }}_2^2)\sin {\mathrm{\Phi }}_{12}^{-}-{\ddot{\Delta }}_2\cos {\mathrm{\Phi }}_{12}^{-}]\\ +{\displaystyle \frac{1}{2}}{\mu }_{\psi 22}{a}_{21}{A}_{22}{n}^2{l}^2\{-(n^2+2n{\dot{\Delta }}_2+{\dot{\Delta }}_2^2)[\sin {\mathrm{\Phi }}_{\beta 21}^{+}+\sin {\mathrm{\Phi }}_{\beta 12}^{-}]\\ +{\ddot{\Delta }}_2[\cos {\mathrm{\Phi }}_{\beta 21}^{+}-\cos {\mathrm{\Phi }}_{\beta 21}^{-}]\}\}+{\epsilon }^2\{T_1^{(2)}-2F_1^{(2)}(1+{\dot{\Delta }}_1)\}+{\epsilon }^3\dots \\ {\ddot{\Delta }}_2=\epsilon \{T_2^{(1)}-2F_2^{(1)}(n+{\dot{\Delta }}_2)+k_{2}sin(n\tau +{\Delta }_2)-{\mu }_{x21}[(1+2{\dot{\Delta }}_1+{\dot{\Delta }}_1^2)\sin {\mathrm{\Phi }}_{21}^{-}+{\ddot{\Delta }}_1\cos {\mathrm{\Phi }}_{21}^{-}]\\ +{\displaystyle \frac{{\mu }_{\psi 21}{A}_{22}{l}^2}{2}}[-(1+2{\dot{\Delta }}_1+{\dot{\Delta }}_1^2)(\sin {\mathrm{\Phi }}_{\beta 21}^{+}+\sin {\mathrm{\Phi }}_{\beta 21}^{-})+{\ddot{\Delta }}_1\cos {\mathrm{\Phi }}_{\beta 21}^{+}-{\ddot{\Delta }}_1\cos {\mathrm{\Phi }}_{\beta 12}^{-}]-{\mu }_{x22}{a}_{21}{n}^2{\ddot{\Delta }}_2\\ +{\displaystyle \frac{{\mu }_{\psi 22}{a}_{21}{A}_{22}{n}^2{l}^2}{2}}[-(n^2+2n{\dot{\Delta }}_2+{\dot{\Delta }}_2^2)\sin {\mathrm{\Phi }}_{\beta 22}^{+}+{\ddot{\Delta }}_2\cos {\mathrm{\Phi }}_{\beta 22}^{+}-{\ddot{\Delta }}_2]\}\\ +{\epsilon }^2\{T_2^{(2)}-2F_2^{(2)}(n+{\dot{\Delta }}_2)\}+{\epsilon }^3\dots \end{array}$$

(17)

Equation (17) is fundamentally sufficient to represent the synchronization process of EBs, but the correlation term of ε contains ${\ddot{\Delta }}_i$, making it difficult to solve for the phase difference. According to the asymptotic method, Equation (17) can be rearranged into the Bogoliubov standard form as follows:

$$\begin{array}{l}{\dot{\Delta }}_1=\sqrt{\epsilon }{v}_1,{\dot{\Delta }}_2=\sqrt{\epsilon }{v}_2\\ {\dot{v}}_1={\dot{v}}_1^{(1)}+{\dot{v}}_1^{(2)}+{\dot{v}}_1^{(3)}+\dots \dots \\ {\dot{v}}_1^{(1)}=\sqrt{\epsilon }\{T_1^{(1)}-2F_1^{(1)}+k_{1}sin(\tau +{\Delta }_1)-{\displaystyle \frac{{\mu }_{\psi 21}{A}_{22}{l}^2}{2}}\sin {\mathrm{\Phi }}_{\beta 11}^{+}-{\mu }_{x22}{a}_{21}{n}^4\sin {\mathrm{\Phi }}_{12}^{-}\\ -{\displaystyle \frac{1}{2}}{\mu }_{\psi 22}{a}_{21}{A}_{22}{n}^4{l}^2[\sin {\mathrm{\Phi }}_{\beta 21}^{+}+\sin {\mathrm{\Phi }}_{\beta 12}^{-}]\};\\ {\dot{v}}_1^{(2)}=\epsilon \{-2F_1^{(1)}{v}_1-{\mu }_{\psi 21}{A}_{22}{l}^2{v}_1\sin {\mathrm{\Phi }}_{\beta 11}^{+}-2{\mu }_{x22}{a}_{21}{n}^3{v}_2\sin {\mathrm{\Phi }}_{12}^{-}\\ -{\mu }_{\psi 22}{a}_{21}{A}_{22}{n}^3{l}^2{v}_2[\sin {\mathrm{\Phi }}_{\beta 21}^{+}+\sin {\mathrm{\Phi }}_{\beta 12}^{-}]\}\\ {\dot{v}}_1^{(3)}=\sqrt{{\epsilon }^3}\{T_1^{(2)}-2F_1^{(2)}-{\displaystyle \frac{{\mu }_{\psi 21}{A}_{22}{l}^2}{2}}{v}_1^2\sin {\mathrm{\Phi }}_{\beta 11}^{+}-{\mu }_{x22}{a}_{21}{n}^2{v}_2^2\sin {\mathrm{\Phi }}_{12}^{-}-{\displaystyle \frac{1}{2}}{\mu }_{\psi 22}{a}_{21}{A}_{22}{n}^2{l}^2{v}_2^2\\ \times [\sin {\mathrm{\Phi }}_{\beta 21}^{+}+\sin {\mathrm{\Phi }}_{\beta 12}^{-}]+[T_1^{(1)}-2F_1^{(1)}+k_{1}sin(\tau +{\Delta }_1)-{\displaystyle \frac{{\mu }_{\psi 21}{A}_{22}{l}^2}{2}}\sin {\mathrm{\Phi }}_{\beta 11}^{+}\\ -{\mu }_{x22}{a}_{21}{n}^4\sin {\mathrm{\Phi }}_{12}^{-}-{\displaystyle \frac{1}{2}}{\mu }_{\psi 22}{a}_{21}{A}_{22}{n}^4{l}^2(\sin {\mathrm{\Phi }}_{\beta 21}^{+}+\sin {\mathrm{\Phi }}_{\beta 12}^{-})][{\displaystyle \frac{{\mu }_{\psi 21}{A}_{22}{l}^2}{2}}(\cos {\mathrm{\Phi }}_{\beta 11}^{+}-1)\\ -{\mu }_{x21}]+[T_2^{(1)}-2F_2^{(1)}n+k_{2}sin(n\tau +{\Delta }_2)-{\mu }_{x21}\sin {\mathrm{\Phi }}_{21}^{-}-{\displaystyle \frac{{\mu }_{\psi 21}{A}_{22}{l}^2}{2}}(\sin {\mathrm{\Phi }}_{\beta 21}^{+}+\sin {\mathrm{\Phi }}_{\beta 21}^{-})\\ -{\displaystyle \frac{{\mu }_{\psi 22}{a}_{21}{A}_{22}{n}^4{l}^2}{2}}\sin {\mathrm{\Phi }}_{\beta 22}^{+}][{\displaystyle \frac{1}{2}}{\mu }_{\psi 22}{a}_{21}{A}_{22}{n}^2{l}^2(\cos {\mathrm{\Phi }}_{\beta 21}^{+}-\cos {\mathrm{\Phi }}_{\beta 21}^{-})-{\mu }_{x22}{a}_{21}{n}^2\cos {\mathrm{\Phi }}_{12}^{-}]\}\end{array}$$

(18)

$$\begin{array}{l}{\dot{v}}_2={\dot{v}}_2^{(1)}+{\dot{v}}_2^{(2)}+{\dot{v}}_2^{(3)}+\dots \dots \\ {\dot{v}}_2^{(1)}=\sqrt{\epsilon }\{T_2^{(1)}-2F_2^{(1)}n+k_{2}sin(n\tau +{\Delta }_2)-{\displaystyle \frac{{\mu }_{\psi 21}{A}_{22}{l}^2}{2}}(\sin {\mathrm{\Phi }}_{\beta 21}^{+}+\sin {\mathrm{\Phi }}_{\beta 21}^{-})\\ -{\mu }_{x21}\sin {\mathrm{\Phi }}_{21}^{-}-{\displaystyle \frac{{\mu }_{\psi 22}{a}_{21}{A}_{22}{n}^4{l}^2}{2}}\sin {\mathrm{\Phi }}_{\beta 22}^{+}\}\\ {\dot{v}}_2^{(2)}=\epsilon \{-2F_2^{(1)}{v}_2-2{\mu }_{x21}{v}_1\sin {\mathrm{\Phi }}_{21}^{-}-{\mu }_{\psi 21}{A}_{22}{l}^2{v}_1(\sin {\mathrm{\Phi }}_{\beta 21}^{+}+\sin {\mathrm{\Phi }}_{\beta 21}^{-})\\ -{\mu }_{\psi 22}{a}_{21}{A}_{22}{n}^3{l}^2{v}_2\sin {\mathrm{\Phi }}_{\beta 22}^{+}\}\\ {\dot{v}}_2^{(3)}=\sqrt{{\epsilon }^3}\{T_2^{(2)}-2n{F}_2^{(2)}-{\mu }_{x21}{v}_1^2\sin {\mathrm{\Phi }}_{21}^{-}-{\displaystyle \frac{{\mu }_{\psi 21}{A}_{22}{l}^2}{2}}{v}_1^2(\sin {\mathrm{\Phi }}_{\beta 21}^{+}+\sin {\mathrm{\Phi }}_{\beta 21}^{-})\\ -{\displaystyle \frac{{\mu }_{\psi 22}{a}_{21}{A}_{22}{n}^2{l}^2}{2}}{v}_2^2\sin {\mathrm{\Phi }}_{\beta 22}^{+}+[T_1^{(1)}-2F_1^{(1)}+k_{1}sin(\tau +{\Delta }_1)-{\displaystyle \frac{{\mu }_{\psi 21}{A}_{22}{l}^2}{2}}\sin {\mathrm{\Phi }}_{\beta 11}^{+}\\ -{\mu }_{x22}{a}_{21}{n}^4\sin {\mathrm{\Phi }}_{12}^{-}-{\displaystyle \frac{1}{2}}{\mu }_{\psi 22}{a}_{21}{A}_{22}{n}^4{l}^2(\sin {\mathrm{\Phi }}_{\beta 21}^{+}+\sin {\mathrm{\Phi }}_{\beta 12}^{-})]\\ \times [{\displaystyle \frac{{\mu }_{\psi 21}{A}_{22}{l}^2}{2}}(\cos {\mathrm{\Phi }}_{\beta 21}^{+}-\cos {\mathrm{\Phi }}_{\beta 12}^{-})-{\mu }_{x21}\cos {\mathrm{\Phi }}_{21}^{-}]+[T_2^{(1)}-2F_2^{(1)}n+k_{2}sin(n\tau +{\Delta }_2)\\ -{\mu }_{x21}\sin {\mathrm{\Phi }}_{21}^{-}-{\displaystyle \frac{{\mu }_{\psi 21}{A}_{22}{l}^2}{2}}(\sin {\mathrm{\Phi }}_{\beta 21}^{+}+\sin {\mathrm{\Phi }}_{\beta 21}^{-})-{\displaystyle \frac{{\mu }_{\psi 22}{a}_{21}{A}_{22}{n}^4{l}^2}{2}}\sin {\mathrm{\Phi }}_{\beta 22}^{+}]\\ \times [{\displaystyle \frac{{\mu }_{\psi 22}{a}_{21}{A}_{22}{n}^2{l}^2}{2}}(\cos {\mathrm{\Phi }}_{\beta 22}^{+}-1)-{\mu }_{x22}{a}_{21}{n}^2]\}\end{array}$$

From Equation (18), ${\dot{v}}_i$ is numerically related to the small parameter $\sqrt{\epsilon }$ in the process of realizing synchronization, thus v_i (i = 1, 2) is treated as a slowly varying quantity. According to the nonlinear vibration theory, v_i is equal to the sum of the smooth term and integrates the small oscillation term in ${\dot{v}}_i$ with respect to τ. By ignoring the higher-order small parameter terms with lower weights, the approximate solutions for v_i can be expressed as follows:

$$\begin{array}{l}{v}_1={\mathrm{\Omega }}_1+\sqrt{\epsilon }\{-k_1\cos (\tau +{\Delta }_1)+{\displaystyle \frac{{\mu }_{\psi 21}{A}_{22}{l}^2}{4}}\cos {\mathrm{\Phi }}_{\beta 11}^{+}+{\displaystyle \frac{{\mu }_{x22}{a}_{21}{n}^4}{1-n}}\cos {\mathrm{\Phi }}_{12}^{-}\\ +{\displaystyle \frac{1}{2}}{\mu }_{\psi 22}{a}_{21}{A}_{22}{n}^4{l}^2[{\displaystyle \frac{1}{1+n}}\cos {\mathrm{\Phi }}_{\beta 21}^{+}+{\displaystyle \frac{1}{1-n}}\cos {\mathrm{\Phi }}_{\beta 12}^{-}]\}+\epsilon \{{\displaystyle \frac{1}{2}}{\mu }_{\psi 21}{A}_{22}{l}^2{\mathrm{\Omega }}_1\cos {\mathrm{\Phi }}_{\beta 11}^{+}\\ +{\displaystyle \frac{2}{1-n}}{\mu }_{x22}{a}_{21}{n}^3{\mathrm{\Omega }}_2\cos {\mathrm{\Phi }}_{12}^{-}+{\mu }_{\psi 22}{a}_{21}{A}_{22}{n}^3{l}^2{\mathrm{\Omega }}_2[{\displaystyle \frac{1}{1+n}}\cos {\mathrm{\Phi }}_{\beta 21}^{+}+{\displaystyle \frac{1}{1-n}}\cos {\mathrm{\Phi }}_{\beta 12}^{-}]\}\\ v_2={\mathrm{\Omega }}_2+\sqrt{\epsilon }\{-{\displaystyle \frac{1}{n}}{k}_2\cos (n\tau +{\Delta }_2)+{\displaystyle \frac{1}{n-1}}{\mu }_{x21}\cos {\mathrm{\Phi }}_{21}^{-}+{\displaystyle \frac{{\mu }_{\psi 21}{A}_{22}{l}^2}{2}}({\displaystyle \frac{1}{n+1}}\cos {\mathrm{\Phi }}_{\beta 21}^{+}\\ +{\displaystyle \frac{1}{n-1}}\cos {\mathrm{\Phi }}_{\beta 21}^{-})+{\displaystyle \frac{{\mu }_{\psi 22}{a}_{21}{A}_{22}{n}^4{l}^2}{4n}}\cos {\mathrm{\Phi }}_{\beta 22}^{+}\}+\epsilon \{{\displaystyle \frac{2}{n-1}}{\mu }_{x21}{\mathrm{\Omega }}_1\cos {\mathrm{\Phi }}_{21}^{-}\\ +{\mu }_{\psi 21}{A}_{22}{l}^2{\mathrm{\Omega }}_1({\displaystyle \frac{1}{n+1}}\cos {\mathrm{\Phi }}_{\beta 21}^{+}+{\displaystyle \frac{1}{n-1}}\cos {\mathrm{\Phi }}_{\beta 21}^{-})+{\displaystyle \frac{1}{2n}}{\mu }_{\psi 22}{a}_{21}{A}_{22}{n}^3{l}^2{\mathrm{\Omega }}_2\cos {\mathrm{\Phi }}_{\beta 22}^{+}\}\end{array}$$

(19)

where Ω_i (i = 1, 2) represents the smoothing term, and the remaining are the periodic oscillatory term. Since the value of the oscillatory term is very small, v_i (i = 1,2) can be replaced by the integral mean. The duration of a single cycle is negligible with respect to the system operation process, and ${\dot{v}}_i$ is continuous as a function of time, whereupon the instantaneous value of the periodic oscillatory term can be expressed as the integral average over a single cycle. Combining Equations (18) and (19) and then integrating them over τ = 0~2π, finally the mean value is considered. Treating Ω_i and Δ_i as constants during this process, the equilibrium equations for EBs are derived as follows:

$$\begin{array}{l}{\dot{\mathrm{\Omega }}}_1=\sqrt{\epsilon }\{T_1^{(1)}-2F_1^{(1)}\}-2\epsilon F_1^{(1)}{\mathrm{\Omega }}_1+\sqrt{{\epsilon }^3}\{T_1^{(2)}-2F_1^{(2)}+[T_1^{(1)}-2F_1^{(1)}][-{\displaystyle \frac{{\mu }_{\psi 21}{A}_{22}{l}^2}{2}}-{\mu }_{x21}]\}\\ {\dot{\mathrm{\Omega }}}_2=\sqrt{\epsilon }\{T_2^{(1)}-6F_2^{(1)}\}-2\epsilon F_2^{(1)}{\mathrm{\Omega }}_2+\sqrt{{\epsilon }^3}\{T_2^{(2)}-6F_2^{(2)}+[T_2^{(1)}-6F_2^{(1)}][-{\displaystyle \frac{{\mu }_{\psi 22}{a}_{21}{A}_{22}{n}^2{l}^2}{2}}-{\mu }_{x22}{a}_{21}{n}^2]\\ +{\displaystyle \frac{{\mu }_{\psi 21}{A}_{22}{l}^2{\mu }_{\psi 21}{A}_{22}{l}^2}{4}}\sin [3{\Delta }_1-{\Delta }_2-(3{\beta }_1-{\beta }_2)]+{\displaystyle \frac{{\mu }_{\psi 21}{A}_{22}{l}^2{\mu }_{x21}}{2}}\sin (3{\Delta }_1-{\Delta }_2-2{\beta }_1)\}\end{array}$$

(20)

where, 3Δ₁ − Δ₂ is the phase difference between the two EBs. When the system is synchronized, the value of the phase difference should be stabilized at a fixed value, and the parameter Ω_i should be equal to zero. Meanwhile, the parameter ${\dot{\mathrm{\Omega }}}_i$ as the derivative of Ω_i, should be also zero; thus, the phase difference can be solved. However, the phase difference of Equation (20) contains two solutions. Which one is the correct solution? This needs filtering by a stability determination. Therefore, the formula for the stability criterion of synchronization is derived in the following section.

5.2. Stability Criterion

To obtain the solutions of the relative phase, Equation (20) can be simplified to the following form:

$$\begin{array}{l}{\dot{\mathrm{\Omega }}}_1=\sqrt{\epsilon }\{T_1^{(1)}-2F_1^{(1)}\}-2\epsilon F_1^{(1)}{\mathrm{\Omega }}_1+\sqrt{{\epsilon }^3}\{T_1^{(2)}-2F_1^{(2)}+[T_1^{(1)}-2F_1^{(1)}][-{\displaystyle \frac{{\mu }_{\psi 21}{A}_{22}{l}^2}{2}}-{\mu }_{x21}]\}\\ {\dot{\mathrm{\Omega }}}_2=\sqrt{\epsilon }\{T_2^{(1)}-6F_2^{(1)}\}-2\epsilon F_2^{(1)}{\mathrm{\Omega }}_2+\sqrt{{\epsilon }^3}\{T_2^{(2)}-6F_2^{(2)}+[T_2^{(1)}-6F_2^{(1)}][-{\displaystyle \frac{{\mu }_{\psi 22}{a}_{21}{A}_{22}{n}^2{l}^2}{2}}-{\mu }_{x22}{a}_{21}{n}^2]\\ +{\mu }_{\psi 21}{A}_{22}{l}^2\sqrt{A^2+B^2}\sin [3{\Delta }_1-{\Delta }_2+\arctan ({\displaystyle \frac{B}{A}})]\}\end{array}$$

(21)

with

$$\begin{array}{l}\mathrm{A}={\displaystyle \frac{{\mu }_{\psi 21}{A}_{22}{l}^2}{4}}\cos (3{\beta }_1-{\beta }_2)+{\displaystyle \frac{{\mu }_{x21}}{2}}\cos (2{\beta }_1)\\ \mathrm{B}=-[{\displaystyle \frac{{\mu }_{\psi 21}{A}_{22}{l}^2}{4}}\sin (3{\beta }_1-{\beta }_2)+{\displaystyle \frac{{\mu }_{x21}}{2}}\sin (2{\beta }_1)]\end{array}$$

Assuming that δ_i and ξ_i are small steady-state fluctuations, the following settings are given as follows:

$${\Delta }_i={\Delta }_{i0}+{\delta }_i,{\mathrm{\Omega }}_i={\mathrm{\Omega }}_{i0}+{\xi }_i,i=1,2$$

(22)

where Δ_i0 represents the value of Δ_i when the vibrating system operates in the synchronous state. By substituting the above equation into Equation (21), the perturbation equations can be obtained, i.e., as follows:

$$\begin{array}{l}\begin{array}{c}{\dot{\delta }}_i=\sqrt{\epsilon }{\xi }_i\end{array}\\ {\dot{\xi }}_1+2\epsilon F_1^{(1)}{\xi }_1=0\\ {\dot{\xi }}_2+2\epsilon F_2^{(1)}{\xi }_2+\sqrt{{\epsilon }^3}{\mu }_{\psi 21}{A}_{22}{l}^2{\delta }_2\sqrt{A^2+B^2}\cos [3{\Delta }_{10}-{\Delta }_{20}+\arctan ({\displaystyle \frac{B}{A}})]\\ =\sqrt{{\epsilon }^3}{\mu }_{\psi 21}{A}_{22}{l}^2\sqrt{A^2+B^2}\sin [3{\Delta }_{10}-{\Delta }_{20}+\arctan ({\displaystyle \frac{B}{A}})]\\ -3\sqrt{{\epsilon }^3}{\mu }_{\psi 21}{A}_{22}{l}^2{\delta }_1\sqrt{A^2+B^2}\cos [3{\Delta }_{10}-{\Delta }_{20}+\arctan ({\displaystyle \frac{B}{A}})]\end{array}$$

(23)

Through replacing ξ_i by δ_i and rearranging Equation (23), the equation with respect to δ_i is written as follows:

$$\begin{array}{l}{\ddot{\delta }}_1+2\epsilon F_1^{(1)}{\dot{\delta }}_1=0\\ {\ddot{\delta }}_2+{\dot{\delta }}_{2}2\epsilon F_2^{(1)}+{\delta }_2{\epsilon }^2{\mu }_{\psi 21}{A}_{22}{l}^2\sqrt{A^2+B^2}\cos [3{\Delta }_{10}-{\Delta }_{20}+\arctan ({\displaystyle \frac{B}{A}})]\\ ={\epsilon }^2{\mu }_{\psi 21}{A}_{22}{l}^2\sqrt{A^2+B^2}\sin [3{\Delta }_{10}-{\Delta }_{20}+\arctan ({\displaystyle \frac{B}{A}})]\\ -3{\epsilon }^2{\mu }_{\psi 21}{A}_{22}{l}^2{\delta }_1\sqrt{A^2+B^2}\cos [3{\Delta }_{10}-{\Delta }_{20}+\arctan ({\displaystyle \frac{B}{A}})]\end{array}$$

(24)

The characteristic equation of Equation (24) can be expressed as follows:

$$\begin{array}{l}{\lambda }_1^2+2{\lambda }_1\epsilon F_1^{(1)}=0\\ {\lambda }_2^2+2{\lambda }_2\epsilon F_2^{(1)}+{\epsilon }^2{\mu }_{\psi 21}{A}_{22}{l}^2\sqrt{A^2+B^2}\cos [3{\Delta }_{10}-{\Delta }_{20}+\arctan ({\displaystyle \frac{B}{A}})]=0\end{array}$$

(25)

According to the Routh–Hurwitz criterion, all elements in the first column of the Routh table should be identical in their positivity and negativity. A system that satisfies the criterion is called a closed-loop stable system. Otherwise, it would be an unstable duo to a shift in the sign of the elements in the first column. The solution of the system is therefore restricted as follows:

$$\begin{array}{l}2\epsilon F_1^{(1)}>0;2\epsilon F_2^{(1)}>0\\ H_1\cos [3{\Delta }_{10}-{\Delta }_{20}+\arctan ({\displaystyle \frac{B}{A}})]>0\end{array}$$

(26)

with

$$H_1={\epsilon }^2{\mu }_{\psi 21}{A}_{22}{l}^2\sqrt{A^2+B^2}$$

that is

$$\begin{array}{l}[3{\Delta }_{10}-{\Delta }_{20}+\arctan ({\displaystyle \frac{B}{A}})]\in (-{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{\pi }{2}}), \mathrm{when} H_1>0\\ [3{\Delta }_{10}-{\Delta }_{20}+\arctan ({\displaystyle \frac{B}{A}})]\in ({\displaystyle \frac{\pi }{2}},{\displaystyle \frac{3\pi }{2}}), \mathrm{when} H_1<0\end{array}$$

(27)

6. Numerical Analysis

The dynamical equations, steady-state solutions, synchronization conditions, and stability criterion are deduced in Section 2, Section 3, Section 4 and Section 5. It is concluded that the characteristics of the system mainly depend on parameters l, β₁, a₂₁, k_y1, and k_w (w = ψ₁, ψ₂). Combined with the selected mechanical model, the basic parameters of the system during numerical analysis are shown in

Table 1. The standard parameters of the system.

VB	IB	Vibrator 1/Vibrator 2
M1 = 60 (kg)	M2 = 60~120 (kg)	m1 = 3 (kg)
kx1 = 0~3 × 106 (N/m)	kx2 = 47,300 (N/m)	m2 = 1.2~3 (kg)
ky1 = 0~3 × 106 (N/m)	ky2 = 47,300 (N/m)	β1 = π/6 or π/4 or π/3 (rad)
kψ1 = 0~11 × 104 (N∙m/rad)	kψ2 = 0~1 × 105 (N∙m/rad)	β2 = π − β1
J1 = 30 (kg∙m2)	J2 = 60 (kg∙m2)	rl = 0~6
\	\	r = 0.05 (m)

.

6.1. Vibration Absorption Ability

Based on the characteristics of two-degree-of-freedom forced vibration dynamics, in a triple-frequency forced vibration system, the excellent absorption points where the VB amplitude is larger than the IB amplitude can be obtained by appropriately selecting system parameters. In order to ascertain the vibration absorption characteristics of the system, the absorption capacity factor ζ is defined in Equation (15), and it can be seen that the larger the ζ, the greater the IB absorption capacity. The magnitude of ζ in the y-direction is significantly affected by the system parameters k_y1, k_y2, a₂₁, and ω_m; thus, the parameters are relevantly discussed in this section. According to the actual situation of the anti-resonance system with dual-body, the stiffness coefficient of the isolation spring is relatively small compared to the main vibration spring, and the spring stiffness is required to satisfy the relationship k_y1 >> k_y2, whereupon only k_y1 is discussed. The variation of ζ with k_y1 and ω_m is revealed in Figure 2, where two brighter stripes representing a higher vibration absorption capacity are observed. The left striped area is defined as the low-frequency absorption zone, and the right is named the high-frequency absorption zone. With the continuous increase of a₂₁, the low-frequency vibration absorption area is increasingly brightened up, while the high-frequency vibration absorption area is gradually darkened. Furthermore, according to the trend of the absorption zone, it is noticed that k_y1 is increased with the growth of ω_m. The above findings indicate that with the increase of parameter a₂₁, the absorption capacity of the low-frequency absorption zone is continuously increased, while the high-frequency absorption zone is gradually weakened. In addition, the higher the stiffness coefficient of the main vibration spring, the greater the disturbance frequency requirement that makes the system in the absorption zone.

It must be noted that the vibration system is driven by triple-frequency excitation force; considering the actual situation, the fundamental frequency ω_m should be limited; thus, the low-frequency vibration absorption zone is chosen as the research object. According to the trend of the low-frequency vibration absorption zone, the stiffness coefficient of the main vibration spring is exponentially increased with the increase of the excitation frequency. Thus, the stiffness of the main vibration spring should also be limited. Therefore, as shown in Figure 2c, k_y1 = 1.235 × 10⁶ (N/m) and ω_m = 49.23 (rad/s) are selected to analyze the vibration characteristics of the anti-resonance system. The amplitude–frequency characteristic curve of the vibrating system in the y-direction is shown in Figure 3, where Y₁ and Y₂ represent the maximum displacement of the VB and IB in y-direction, respectively. It is evident that as the disturbance frequency increases, four points where the amplitude tends to infinity emerge in the amplitude–frequency response curve. Measurements show that the excitation frequencies corresponding to the four points of infinite amplitude, arranged in ascending order, are 5.9 rad/s, 17.7 rad/s, 61.9 rad/s, and 185.8 rad/s. Calculated by Equation (14), the first-order natural frequency of the system in the y-direction is 17.7030 rad/s, and the second-order natural frequency is 185.7894 rad/s. Thus, ω_n1 and ω_n2 in Figure 3 are separately the first-order natural frequency and the second-order natural frequency. Since EB2 is excited by a triple-frequency force, the corresponding frequencies of identical rotational speed, ω_h1 and ω_h2, are one-third of ω_n1 and ω_n2, respectively. This reflects the unique amplitude–frequency response characteristics of the triple-frequency excitation system. The phenomenon shows that resonance has occurred when the frequency of EBs is equal to either the first- or second-order natural frequency of the system. However, the position of the resonance points in three figures is unchanged with the change of dimensionless parameter a₂₁, which demonstrates that the mass of the EBs is negligible in affecting the natural frequency (ω_n1, ω_n2, ω_h1, and ω_h2).

6.2. Vibration Property

To investigate the vibration property of the anti-resonance system driven by triple-frequency excitation, the values of the system parameters are given, namely k_y1 = 1.235 × 10⁶ (N/m) and ω_m = 49.23 (rad/s). Calculated by Equation (10) without considering the phase of the EBs, Figure 4 and Figure 5 separately reflect the displacement in the y-direction and the trajectory of the VB centroid at different a₂₁. From Figure 4, when the mass of EB1 is constant, the maximum amplitude of the VB in the y-direction is increased with the increase of a₂₁, and the effect of parameter a₂₁ on the amplitude can be visually observed in Figure 5. In order to reveal the triple-frequency vibration regularity, Figure 4 is partially enlarged. From the enlarged image of Figure 4, three wave crests are contained in a vibration cycle. It can be shown that the number of crests in a period is identical with the multiple of the system excitation frequency. Furthermore, by comparing the trajectory plots, it can be seen that the vibrational trajectory can be accurately described as a superposition of three closed roundabout trajectories due to the presence of three wave crests within one period. Among them, the largest trajectory is named main-trajectory, and the two relatively small trajectories are called sub-trajectories. Therefore, the trajectory size ratio (TSR) is employed as the ratio of the sub-trajectory size to the main-trajectory size. From Figure 5a, one can easily notice that the TSR is quite small, while the TSR is continuously increased with the increase of a₂₁. How is this formed? By cross-referencing Figure 4, It is easy to notice that the ratio of the two lower wave heights to the higher wave height also increases with the increase of a₂₁ during a vibration cycle. Thus, the magnitude of TSR and the wave height ratio are inextricably linked. It follows that with the increase of a₂₁, the TSR is constantly increased with the wave height ratio.

6.3. Stability Criterion

According to Equation (20), when solving the synchronous phase difference, two solutions are obtained under a set of structural parameters. Therefore, a stability assessment of the system is of necessity. From Equation (27), the stability condition can be divided into two cases for analysis: on the one hand, when H₁ > 0, and on the other hand, when H₁ < 0. The stability of the phase difference is subject to different constraints in these two cases. At the same time, the magnitude of H₁ is influenced by multiple system parameters, where the only one that affects the properties of positive and negative is the parameter μ_ψ21. Furthermore, the magnitude of μ_ψ21 is mainly determined by the system parameters k_w (w = ψ₁, ψ₂) and the fundamental frequency ω_m. The latter is decisive for the effectiveness of the anti-resonance system and was determined in the previous analysis as ω_m = 49.23 (rad/s). Therefore, the black–blue graph of k_w (w = ψ₁, ψ₂) with respect to the sign of the stability coefficient H₁ is given in this section. As illustrated in Figure 6, notice that area with H₁ < 0 is represented in the black region. Meanwhile, blue means that H₁ > 0. Analysis shows that when k_ψ1 is small, the properties of the positive and negative of H₁ are determined by both k_ψ1 and k_ψ2, while when k_ψ1 > 7.2, the value of H₁ is consistently negative. In addition, with the increase of M₂, the size of the blue area is reduced steadily.

6.4. Stable Phase Difference

In Section 6.3, the stable synchronization conditions are discussed; thus, the stable phase difference (SPD) can be determined. In this section, the system parameters M₂ = 90 (kg) and k_ψ2 = 5000 (N∙m/rad) are selected as invariant parameters. According to Equation (20), the calculation results are significantly influenced by the structural parameters l. Further, considering the overall size of the IB, the parameter r_l (r_l = l/l_e2) was introduced (where the parameter l_e2 is explained behind Equation (7)). As shown in Figure 7, the solutions of SPD are in different intervals when k_ψ1 = 4 × 10⁴ (N∙m/rad) and k_ψ1 = 5 × 10⁴ (N∙m/rad). Based on an analysis of the data, the stable phase difference is rapidly changed when r_l is quite small. Nevertheless, when r_l exceeds the value of 2, the SPD remains nearly constant and the trend converges towards a horizontal line. In addition, there are some sudden jumps in the SPD at some points with the magnitude of π, which indicates that the reverse synchronization phenomenon (i.e., the phase difference is abruptly decreased or suddenly increased by π) has occurred in the EBs.

7. Computer Simulation

To further demonstrate the reliability of the theoretical analysis and enhance the accuracy of the simulation results, the Runge–Kutta algorithm is employed to establish the system simulation model based on Equation (7). The Runge–Kutta method is chosen primarily for its outstanding advantages. It offers high accuracy by providing a more precise approximation of the solution to differential equations through a weighted average of function evaluations at multiple points within each integration step, effectively reducing the local truncation error compared to simpler methods like Euler’s method. In terms of stability, it can handle a wide range of problems without an excessive growth of errors over time, making it suitable for long-term simulations. Moreover, it exhibits excellent performance in dealing with nonlinear differential equations, which are common in complex systems. By leveraging the Runge–Kutta algorithm, the model can accurately capture the dynamic behavior of the system described by Equation (7), ensuring that the simulation results are both reliable and reflective of the real-world system characteristics. It should be noted that the system parameters used in the simulation are the same as the actual project, while the theoretical part is forced to make numerical methods such as neglecting small values and substituting average values in order to simplify derivation. Therefore, the results obtained from the simulation and the theoretical analysis can only be highly consistent and cannot be totally identical. That is, the error is acceptable within a small range. A total of two sets of computer simulations are performed, and the selection of stiffness and perturbation frequencies is marked in Figure 2c. The difference is that one set is simulated when H₁ > 0, while the other is when H₁ < 0. Since the mass of the EBs in the theoretical calculation is 3 kg and the fundamental frequency rotational speed is 49.23 rad/s (approximately 470 r/min), the parameters set in the simulation are consistent with the theory. In engineering, many high-frequency excitation motors choose bipolar logarithmic motors, such as YZS-1.5-2. During use, they can stably drive the high-speed rotation of 3 kg EBs. According to the formula f = pn₀/60 (p represents the number of pole pairs, and n₀ represents the rotational speed of the asynchronous motor, r/min), the power supply frequencies are 15.67 Hz and 47 Hz, respectively. To overcome the load torque under complex working conditions and ensure the ideal speed is achieved, the power supply frequency is slightly adjusted upwards to ensure consistency with the theory. The parameters of the vibrator are shown in

Table 2. The parameters of vibrators during simulation.

Parameters of Vibrator	Vibrator 1	Vibrator 2
Nominal power	3.7 × 104 (VA)	3.7 × 104 (VA)
Power frequency	15.8323 (Hz)	47.5006 (Hz)
Rated velocity	470 (r/min)	1410 (r/min)
Pole pairs	2	2
Mass	m1 = 3 (kg)	m2 = 3 (kg)

.

7.1. Simulation Results Under H₁ > 0

When H₁ > 0, the parameters k_ψ1 = 5 × 10⁴ (N∙m/rad), k_ψ2 = 5 × 10³ (N∙m/rad), r_l = 2, and β₁ = π/3 are selected to conduct the simulation, and the simulation results are shown in Figure 8. It is of notice that a perturbation with the value of π/2 is added to the EB2 at the simulation time of 20 (s). After a period of time, the system is stabilized again, which proves the high stability of triple-frequency synchronization. Figure 8a,b reflect the curve of displacement versus time in x- and y-directions, respectively; the reliability of the vibration absorption capacity is verified. From the locally enlarged waveform diagram, within one period, the amplitude has three peaks, and the peaks show two different heights, indicating that the system has maintained stable operation under vibrations of two different intensities and frequencies. To reduce the influence of resonance, the selection of system parameters is symmetric in the x- and y-directions, so that the natural frequencies of the system in the x- and y-directions are consistent, and the width of the non-resonant region in the frequency domain has been significantly increased. Therefore, the waveform laws in Figure 8a,b are basically the same. Meanwhile, the value of SPD in Figure 8c is stabilized around 0.76 (rad) and the theoretical calculation result shown in Figure 6 is 0.8946 (rad). Thus, the difference between simulation and numerical analysis is about 0.13 (rad), within the error tolerance, indicating that the system stability determination condition is valid when H₁ > 0. Figure 8d visualizes the working trajectory of the VB, where the scattered points are formed when the disturbance is applied, which is highly consistent with Figure 5c. At the same time, the trajectory route of the IB is reflected in Figure 8e, in which the size of the sub-trajectory is close to zero, resulting in an elliptical shape for the overall trajectory. In the stable state, there are differences in the TSR of the upper and lower plastids, indicating that the vibration isolation system has different vibration isolation effects on the vibrations caused by the two excitation forces. Furthermore, from Figure 8f, obviously the rotational speed of vibrator 1 is stabilized around 49.23 (rad/s), while vibrator 2 is around 147.69 (rad/s), which indicates that the results are obtained when the system is driven by triple-frequency excitation, which also proves the feasibleness of triple-frequency synchronization.

7.2. Simulation Results Under H₁ < 0

When H₁ < 0, the parameters k_ψ1 = 4 × 10⁴ (N∙m/rad), k_ψ2 = 5 × 10³ (N∙m/rad), r_l = 1, and β₁ = π/4 are chosen to conduct the simulation. As shown in Figure 9, operating as Section 7.1 did, a perturbation with the value of π/2 is added to the EB2 at the simulation time of 20 (s). After a period, the system can also be stabilized. Figure 9a,b reflect the curve of displacement versus time in x- and y-directions, respectively. Among them, the amplitude of the vibration-isolating body is much smaller than that of the vibrating body, indicating a good vibration isolation effect. Like the simulation study in Section 7.1, the partial enlarged view reflects that two types of vibrations with different frequencies and different amplitudes occur alternately. The value of the SPD in Figure 9c is stabilized around 3.402 (rad) (3.402 = 9.685 − 2π), while the value of the corresponding theoretical value is 3.334 (rad). The difference between simulation and numerical analysis is 0.068 (rad), satisfying the error tolerance. This means that the stability determination condition of the system is available when H₁ < 0. The motion trajectory of the VB is visualized in Figure 9d, where the scatter points are left when the system is unstable or perturbed, and the stable trajectory is highly consistent with Figure 5c. Meanwhile, Figure 9e reflects the motion trajectory of IB; the amplitude of the secondary trajectory is close to zero, and thus the overall trajectory is close to an ellipse. Additionally, Figure 9f shows that the average speed of vibrator 1 is about one third of vibrator 2, which indicates that triple-frequency synchronization can be satisfied under selected structural parameters. When a disturbance is applied to the system, the rotational speed of the eccentric block is also affected, experiencing a recovery period before gradually stabilizing. The reason is that the disturbance changes the system’s amplitude, thereby altering the load torque exerted by the system in the reverse direction on the EBs rotating shaft, which indirectly influences the rotational speed of the rotor.

7.3. Result Analysis

The simulation results are highly consistent with the theoretical analysis. However, the absolute error only reflects the absolute difference between the measured value and the true value, without considering the value range of the measured value itself. In this paper, the relative error is introduced and measured by the ratio of the absolute error to the value range, which can eliminate the influence of the dimension and the magnitude of the value, and more objectively reflect the accuracy of the measurement.

$$\Delta =\left|V_T-V_S\right|,\delta ={\displaystyle \frac{\Delta }{L}}\times 100\%$$

(28)

Among them, V_T represents the theoretical analysis result, and V_S represents the simulation result under the same parameters as the theoretical analysis. Δ is the absolute error; δ is the relative error; L is the length of the value range. Since the phase difference is randomly taken in the range of 0 to 2π, L = 2π.

The error analysis is shown in

Table 3. Error analysis.

Project	Theory Analysis	Simulation	Absolute Error	Relative Error
Simulation under H1 > 0	0.8946 rad	0.76 rad	0.1346	2.1422%
Simulation under H1 < 0	3.334 rad	3.402 rad	0.068	1.0823%

. When H₁ > 0 and H₁ < 0, the absolute errors between the simulation results and the theory are 0.1346 rad and 0.068 rad, respectively, and the relative errors are 2.1422% and 1.0823%, respectively. Such tiny values indicate a high degree of consistency between the theoretical analysis and the simulation results. The negligible differences show that the simulation model accurately reproduces the theoretical framework, verifying the accuracy and reliability of the theoretical method and the adopted simulation method.

8. Conclusions

In antecedent sections of this paper, the vibration isolation mechanism, vibration trajectory, and self-synchronization condition of the triple-frequency anti-resonance system were discussed quantitatively. Ultimately, the reliability of the theory was fortified by simulation results. It is imperative to highlight the following conclusions:

(1)

In the near-resonance region of the triple-frequency anti-resonance system, two absorption zones are observed. When the mass of the low-frequency EB is constant, the vibration absorption capacity in the low-frequency absorption zone is proportional to a₂₁ (the mass ratio of the EBs). Conversely, the vibration absorption capacity in the high-frequency absorption zone is inversely proportional to a₂₁.

(2)

During a single vibrational cycle, the number of peaks presented in the curve of the displacement is consistent with the frequency multiple, which indicates that the number of closed roundabout trajectories in the VB trajectory is equivalent to the multiple of the disturbance frequency. In this case, the working trajectory of the triple-frequency excitation vibration system can be characterized as the superposition of three closed roundabout trajectories.

(3)

When the mass of low-frequency EB remains constant, the influence of a₂₁ on the magnitude of the natural frequency is considered negligible; however, the amplitude of the system is significantly impacted, i.e., the amplitude is increased with an increasing a₂₁. In addition, the magnitude of TSR is also affected by a₂₁, and larger values of a₂₁ result in greater values of TSR.

(4)

Two solutions of phase difference exist in triple-frequency synchronization: one is stable and the other is unstable. The stability range is greatly affected by the rotational stiffness of bodies. In addition, when r_l < 2, SPD is significantly influenced by r_l; however, when r_l > 2, the value of the SPD remains almost constant.

(5)

The validity of the adopted theoretical method and the feasibility of triple-frequency anti-resonance and synchronization characteristics are verified through theoretical comparative analysis, numerical qualitative analysis, and dynamical simulation.