## 1 Introduction  
Pipelines play a crucial role in various fields, including marine engineering, mechanical engineering, civil engineering, aerospace, and petrochemicals [1, 2]. They are frequently utilized to transport fluids such as lubricating oil, high-pressure steam, or seawater. The coupled effects of external excitations and internal fluid flow induce complex, low-frequency vibration characteristics in pipelines. Excessive pipeline vibrations may lead to machine shutdown, fatigue failure, fluid leakage, and even catastrophic explosion [3, 4]. Traditional approaches such as elastic supports [5], dampers [6, 7, 8], and constrained damping materials [9] demonstrate satisfactory high-frequency vibration suppression, but their effectiveness diminishes significantly at lower frequencies. Active control is a common low-frequency vibration reduction technology. [PERSON] et al. [10] installed two electromagnetic active absorbers on a pipeline that use inertial mass reciprocating motion to obtain control force and can control the multi-directional vibration line spectrum of the pipeline. [PERSON] et al. [11] explored the application of macro-fiber composites (MFCs) in the active control of pipelines. The design methods of Linear-Quadratic Regulator(LQR) controller and Proportional Integral Derivative (PID) controller were introduced, respectively, and the experiments were carried out to verify the effectiveness of the active control. [PERSON] et al. [12] proposed a PI-LQR active controller by integrating the PID and LQR active control methods. Compared with the classical proportional control method, the PI-LQR active controller could reduce the pipeline vibration by 0.29 dB on average and 2.00 dB at maximum. Compared with active control, passive control offers the advantages of low cost and simple structure. A dynamic vibration absorber (DVA) can control the vibration line spectrum at a specific frequency, but it struggles to adapt to frequency shifts. By installing multiple absorbers on the structure to form multiple dynamic vibration absorbers (MDVAs), the vibration reduction frequency band can be effectively broadened. [PERSON] et al. [13] used an optimization program to calculate the optimal design variables of each vibration absorber in the MDVA, achieving satisfactory control of the multi-order vibration line spectrum of the pipeline. [PERSON] et al. [14] presented a method for the optimal design of the MDVA based on frequency response function (FRF) analysis, which can better reflect the modal coupling effect between the absorber and the pipeline when designing the absorber parameters. Compared with commonly used optimization methods, the newly proposed method can further enhance the vibration reduction effect of the MDVA. However, existing MDVAs typically feature large-sized and heavy absorbers, which restricts the installation of sufficient absorbers in the limited space of the pipeline. Meanwhile, mutual interference between absorbers may also lead to overall vibration reduction performance being lower than expected [15]. Therefore, it is necessary to design MDVAs with higher integration and optimize the parameters of each vibration absorber, placing high demands on the structural design and parameter control of MDVAs. Metamaterials are artificial periodic composite structures capable of modifying the propagation properties of elastic waves in a medium and controlling elastic wave transmission [16; 17; 18; 19; 20]. It is widely acknowledged that there are two fundamental bandgap formation mechanisms for metamaterials, i.e., Bragg scattering and localized resonance. Among them, metamaterials based on the local resonance mechanism are more likely to generate low-frequency bandgaps [21]. [PERSON] et al. [22] periodically arranged the resonant units in both the axial and circumferential directions of the pipeline, resulting in a significant bandgap effect and a substantial reduction in pipeline vibration within the bandgap. To broaden the bandgap of the local resonance metamaterial, [PERSON] et al.