## 1 Introduction  
Periodic metastructures are well-known for their wave-filtering properties within a specific frequency range, called the band gap. In phononic crystals and structural materials, Bragg scattering and local resonance are two primary mechanisms of band gap formation [1]. However, achieving low-frequency band gaps through Bragg scattering presents challenges, prompting [PERSON] et al. [2] to propose locally resonant metamaterials as an alternative. Metamaterials are periodic [3] or non-periodic [4] structures composed of unit cells, where the unit cell size is typically smaller than the wavelength of the waves they interact with. These metamaterials can be acoustic [5, 6], elastic [7] or electromagnetic [8, 9] depending on the type of waves they are designed to manipulate. Acoustic metamaterials manipulate sound waves, elastic metamaterials control elastic or mechanical waves, and electromagnetic metamaterials interact with electromagnetic waves. Each class is engineered to exhibit unique properties, such as negative refraction, negative density, or negative stiffness for a tunable response. The integration of local resonators has become a pivotal development in controlling the propagation of waves. Locally resonant metamaterials manipulate wave behavior by tuning the system frequency to match that of the local resonators, thereby enabling selective wave transmission or attenuation [10]. Those metamaterials find applications in vibration control [11], energy harvesting [12], sound manipulation [13], electromagnetic waves control [14] and in the development of tunable devices [15]. Moreover, locally resonant metamaterials are attracting attention in civil engineering due to their unique properties for stress wave isolation, which are useful for structural protection [16]. The dispersion relations in locally resonant metamaterials have been studied for more than a decade [17; 18]. The dispersion of locally resonant metamaterials with a single local resonator is characterized by a narrow fixed band gap, which limits wave control to a small frequency range, making it unsuitable for applications requiring broadband attenuation. The wave propagation properties are dictated by the configuration and physical properties of the respective unit cells of the metamaterials. To address the challenge of narrow and fixed band gaps, numerous approaches have been proposed. The most prominent approaches include the integration of multiple local resonators [19], gradient geometry [20], grading [21; 22], and electromechanical coupling [23; 24]. A locally resonant metamaterial that integrates multiple local resonators at different scales is called a hierarchical metamaterial [25], and is well-established in the literature. The dispersion profiles of a multi-resonator metamaterial were investigated, and band structure tuning was demonstrated by changing the stiffness constants and masses of the resonators [26; 27]. [PERSON] et al. [28] studied coupled multi-resonator metamaterials to broaden an elastic band gap, and [PERSON] et al. [29] introduced a multi-resonator concept to 3D acoustic metamaterials. Hierarchical metamaterials provide a flexible band structure and dynamically stable periodic configurations. Hierarchically configured resonators have been shown to achieve multiple broad local resonance band gaps in beams [30] and mass-in-mass chains [31]. These multiple band gaps emerge as a result of the presence of various tuning frequencies within the hierarchy. Some researchers proposed nonlinear metamaterials with hierarchical unit cells for amplitude-dependent broadband wave filtering [32; 33]. Moreover, hierarchical metamaterials exhibit nonreciprocal vibrational energy transfer in nonlinear systems. The nonreciprocal nature of wave propagation in hierarchical metamaterials makes them suitable for the design of mechanical diodes [34]. A comprehensive understanding of the dispersion dynamics in different hierarchical configurations is essential for effective energy flow control [35; 36].