Despite the development of more and more locally resonant (LR) structures of various resonant attachments, the application of LR structures in vibration attenuation is still limited by the widths of bandgaps. To achieve broadband vibration suppression at low frequencies, a LR plate with periodically attached stepped-frequency resonators (SFR) is proposed in this paper, where SFR refers to a group of local resonators with step-variable resonant frequencies. The extended plane wave expansion method is used to analyze the bandgap properties of the LR plate with SFR, and an exact series solution is used to analyze its vibration response. The merging of bandgaps based on SFR is revealed, and the parametric analysis shows that increasing the mass ratio or the damping of resonators is conducive to the merging effects, while increasing the frequency spacing makes the bandwidth of the merged bandgap to first increase and then decrease, as the bandgaps are gradually separated. Then, the optimal resonant frequencies of the SFR are obtained by maximizing the dominant vibration attenuation region. The numerical validation shows that the LR plate with SFR consisting of six resonators can provide broadband attenuation performance covering a frequency range from 190 to 450 Hz, which makes the LR structures more practical in vibration control.

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An improved plane wave expansion (PWE) method is developed based on the governing equations of a lattice unit cell, providing support for calculating the band structures of a locally resonant (LR) plate with multiple arrays of multiple degree-of-freedom resonators. The extended plane wave expansion (EPWE) method is correspondingly presented to obtain the complex band structures of the LR plate. Then the bandgap properties of LR plate systems with single/multiple arrays of single/multiple degree-of-freedom resonators are analyzed, which present a good agreement with the results calculated by the classical theory or finite element method. Compared with the classical theory, the improved PWE and EPWE methods have excellent efficiency and broad applicability for LR plate systems with complex attachments. Furthermore, when the damping of resonators is taken into account, the merging of the multiple bandgaps can be observed using the EPWE method adopted in this paper, which gives a useful tool to analyze the band structures of an LR plate with merged bandgaps for broadband vibration suppression.

Active vibration control (AVC) effect of the classical strategy is mainly restricted to a limited region around the error sensor, while the structural vibration at other locations may be dramatically enlarged. The enlargement is non-negligible when the global vibration suppression is considered. Aiming at this enlargement, an indicator is presented to reveal the effect of the classical strategy on the whole controlled structure. By establishing a cost function related with the indicator, an improved AVC strategy is proposed. Utilizing the proposed strategy, an optimized secondary force is obtained to achieve the global vibration suppression. Numerical simulations of AVC are undertaken on a typical plate to verify the superiority of the proposed strategy. Results demonstrate that the proposed strategy is still effective to achieve the global vibration suppression at frequencies where the classical strategy fails. This performance indicates that the proposed strategy is advantageous for global vibration suppression by attenuating the total input power, which provides more attraction for the actual application of AVC.

Ultralow-frequency band gaps are realized to suppress the ultralow-frequency flexural wave propagation in the beams with periodically attached quasi-zero-stiffness (QZS) resonators, which are designed by inserting quasi-zero-stiffness systems into the mass-in-mass structures. The band structure and the transfer matrix of the QZS locally resonant beam are derived by the transfer matrix method to quantify the wave attenuation performance of band gaps. Then, the effect of the stiffness ratio on the bandgap characteristic is studied. It is shown that, thanks to the introduction of the QZS system, the band gaps can be easily transferred to lower frequency or even ultralow frequency without weakening the static stiffness of the resonators. Finally, the flexural wave propagation in locally resonant beam consisting of multiple periodic arrays of QZS resonators is investigated. The result shows that differential design of the bandgap frequencies can be easily realized by adjusting the negative stiffness coefficient of the QZS resonators, so as to obtain broadband flexural wave suppression performance.