In the industrial field, many factors can cause structural vibrations, including external forces such as periodic loads, blast waves, and gust loads; the mass eccentricity of rotating machinery; and the inertia of moving parts. Aircraft in flight, ships on voyage, and machine tools in processing also contribute, among many others. The additional dynamic stress generated by vibration causes discomfort, numerous noises, fatigue failure of structures, and impacts the operational accuracy and precision of machines, potentially shortening their operating life. Passive isolation is a common method of vibration isolation. By applying materials with high damping coefficients or elastic components, such as rubber sleeves, to block vibration transmission, it effectively controls high-frequency vibrations, though it is less effective on low-frequency vibrations. However, isolating low-frequency vibrations remains complex because the external excitation frequency must exceed the natural frequency of the host structure.

The phononic crystals offer a novel approach for isolating low-frequency vibrations [1] if they feature local resonance. There are two mechanisms through which locally resonant acoustic metamaterial structures can achieve vibration isolation: the locally resonant mechanism and the Bragg scattering mechanism [2,3]. Achieving a low-frequency bandgap via the locally resonant mechanism is simpler, as it does not require attention to the wavelength and lattice constant, which poses challenges for the Bragg scattering mechanism. Recently, locally resonant metamaterials, which are periodically enhanced with low-frequency resonators at a sub-wavelength scale, have been utilized for low-frequency vibration isolation [4].

As flexible structures, beams have been widely utilized in various industries, including suspended cables, flexible manipulators, spacecraft, helicopter fan blades, suspension systems, and conveyor systems. Vibrations of flexible beams occur when they are excited by external forces. For this reason, the vibration and control of beam structures have always concerned researchers and engineers. Wan [5] researched the forced vibration of the Timoshenko beam in relation to various boundary conditions using an improved Timoshenko beam theory. Utilizing the finite element method, Kee [6] investigated the geometrical nonlinear dynamics of cantilever composite beams and shell-type blades. Sahoo [7] explored the nonlinear resonant response and vibration reduction of cantilever beams subjected to multi-harmonic excitations that combined low-frequency and high-frequency components. Abdelhafez [8] studied the nonlinear self-excited vibration control of a cantilever beam under harmonic support motion by applying a time-delayed feedback controller based on the favorable position. Considering viscous damping effects and geometric nonlinearities, El-Bassiouny [9] derived a kinetic equation for the active control of a cantilever beam vibrating in its first mode and proposed a nonlinear control law. Sayed [10] researched the bifurcation and stability of buckling utilizing the active control method. Yang [11] examined the dynamic stability of an axial Timoshenko beam under accelerated movement, employing a combination of the Galerkin method and the averaging method. Bera [12] analyzed the vibration reduction of a rotating Euler–Bernoulli torsion beam by using periodic Duffing resonators and Lagrange’s approach to derive the system’s equations of motion. Huang [13] utilized the Euler buckling beam as a nonlinear stiffness corrector and analyzed the dynamics of the primary resonance of the system by applying the harmonic balance method.

Recently, researchers have devoted considerable attention to LRMBs and their band gaps. Dominguez [14] presented a study of a beam’s flexural–torsional coupling vibrations with periodically distributed spring-mass resonators based on an improved Vlasov theory. Bao [15] used FEM to study the frequency response and bandgaps of a locally resonant meta-beam in which all the resonators were connected by horizontal springs. El-Borgi [16] opened multiple bandgaps by attaching double cantilever beam resonators of two different lengths to the host beam and discussed the numerous bandgap mechanisms by applying the modal analysis approach. Failla [17] employed the transfer matrix approach, an exact analytical method, and the generalized function approach to analyze the frequency response of an LRMB attached by a resonator with frequency-dependent stiffness. Lu [18] presented an LRMB with two degrees of freedom for the resonators, including linear and angular displacements. Meanwhile, the dispersion equations were obtained with the help of the transfer matrix method, resulting in two band gaps. Russillo [19] studied the vibration of a sandwich LRB where the multi-degree-of-freedom resonator has viscoelastic damping.

A class of locally resonant beams with a multi-resonator composite was presented to realize their potential applications in vibration control. Zuo [20] proposed a multi-resonator for LRB to achieve a broadened bandgap. Wu [21] designed a non-periodic mass distribution meta-beam to extend the bandgap. Pai [22] researched an LRMB with a dual-mass spring local resonator to create a wider frequency band. Chen [23] designed a sandwich beam with an internal resonator and conducted theoretical and experimental research on the propagation of flexural waves in the base beam. It was found that adjusting the resonator frequency could alter the range and position of the bandgap for the LRMB. Yu [24] studied the bandgap and vibrations of the Timoshenko host beam. Liu [25] analyzed the influence of periodicity, including material properties, geometric shape, and resonator mass, on the bandgap of Timoshenko beams as the matrix.

Further, Cai [26] acquired low-frequency bandgaps by utilizing the new quasi-zero-stiffness resonator as the core for sandwich beams. In the lever resonator with eddy current damping proposed by Yan [27], the band gaps were adjusted by altering the lever ratio. Moreover, by using eddy current damping, the vibration isolation effect was enhanced. Li [28] employed a linkage mechanism to mass amplify the unit cell of the LRM. Pernas-Salomón [29] developed a composite beam with spring-mass local resonators. Liu [30] considered a negative mass density LRMB, designed by Huang [31], and discussed its band gaps. Wang [32] examined the bandgaps of the meta-beam using Euler-Bernoulli theory, in which the inertia of resonators could be amplified. Other methods exist for adjusting the natural frequency. Sousa [33] treated shape memory alloy as a resonator for the LRB and investigated the effect of temperature on band structures both theoretically and experimentally. Additional adjustable LRMB include helical structures [34], adjustable stiffness springs [35], rubber cylinders, and various soft materials and structures acting as rigid springs [[36], [37], [38], [39], [40]].

The existing research methods for activating low-frequency bandgaps can be summarized as reducing the stiffness of elastic components and increasing mass. However, there remains a gap in the literature regarding meta-beams with adjustable zero-stiffness resonators. This paper proposes a novel meta-beam featuring adjustable stiffness BSM resonators. By employing the Euler-Bernoulli buckled beam, low-frequency bandgaps can be achieved by adjusting the axis displacement of the buckling beam. The stiffness properties of the BSM are derived with the aid of static analysis. Concerning the kinetic equations of the meta-beam, Bloch’s theorem, and the plane wave expansion method, the band gaps are computed. Additionally, the transmissibility of the bending wave is provided to verify the theoretical band gap of the meta-beam. Finally, the effects of the mass block, resonator damping, and lattice constant on the band gap structure are discussed.