Bandgap Control of a Locally Resonant Beam Through Multiple Sub-Divided Resonators

Purpose: Locally resonant metamaterials demonstrate excellent performance in low-frequency vibration suppression. However, metamaterials with a single resonator type typically produce narrow bandgaps. Although using multiple resonators is a common strategy to widen bandgaps, existing studies often assume that the total bandgap width results from the simple superposition of individual resonator responses. This paper examines the bandgap distribution patterns in a locally resonant beam with multiple resonators, aiming to uncover their coupling effects on bandgap characteristics. Methods: The transfer matrix method is employed to develop a theoretical model for a locally resonant beam with multiple resonators. The dispersion relations of the beams with single and multiple resonators are analyzed. The bandgap formation mechanism is investigated by analyzing the mode shapes at the bandgap boundaries. The concept of negative effective mass density is introduced to explore resonator coupling effects. The parametric analysis is performed for obtaining the bandgap behaviors. Results: The bandgaps align with the natural frequencies of the resonators. When the natural frequency spacing between adjacent resonators is small, the bandgap widths at low frequencies are significantly reduced. A uniform and smooth bandgap distribution can be achieved by increasing the mass ratio of the resonator at low frequencies and introducing an appropriate damping value. Conclusion: The coupling effects of the multiple resonators strongly influence the bandgap distributions of the locally resonant beam. The resonators at high frequencies suppress the bandgap formation of the resonators at low frequencies. The effect intensifies as the frequency spacing reduces. These results offer valuable guidance for bandgap tuning in locally resonant metamaterials.