This study investigates the low-frequency vibro-acoustic characteristics of a finite locally resonant (LR) plate. A dynamic model of the finite LR plate consisting of periodic arrays of beam-like resonators attached to a thin aluminum plate with simply supported boundary conditions is established, and the average vibration response and radiated efficiency are theoretically determined by using modal-superposition and harmonic-balance methods. In addition, the study investigates the influence of the parameters and number of additional resonators on the vibro-acoustic performance of the finite LR plate. Finally, a vibration experiment of a finite plate with 8 × 10 uniformly distributed beam-like resonators validates the theoretical analysis results. The numerical and experimental results show that the finite LR plate has a low-frequency bandgap that can suppress the vibration and radiated noise of the structure, and the bandgap position is close to the resonance frequency of resonators. The position and performance of bandgaps can be influenced by changing the parameters and number of resonators. The experimental results show a bandgap ranging from 370 to 425 Hz, which is consistent with the theoretical prediction. The finite LR plates proposed in this study can find potential applications in the attenuation of low-frequency vibration and noise.

An analytical, numerical, and experimental investigation on the transverse vibrations of a finite beam with periodically arrayed beam-like resonators was carried out. A continuous-discrete model of the finite locally resonant beam was established by employing the “mass-spring- mass” subsystem. The analytical solution of the coupling vibration equations was derived based on the modal superposition method, and the analytical expression of average velocity response and vibration transmissibility were given. Then, the minimum periodic number of different units which could result in a bandgap was determined. Finally, the bandgap of a finite locally resonant beam was confirmed by a vibration experiment on a simply supported beam with twelve uniformly distributed beam-like resonators. The numerical and experimental results show that finite locally resonant beams have low-frequency bandgaps like infinite locally resonant beams, and the bandgap position is close to the resonance frequency of resonators. In addition, for a beam with a different type of locally resonant units, the minimum number of units that can generate the bandgap is nearly the same. Within considered frequency ranges, the experimental results are consistent with the theoretical results, meaning that the transverse vibration in locally resonant beams could be substantially attenuated. The conclusions may be supported to the application of locally resonant theory to control low-frequency vibration and radiation noise.

The approximate analytical solution of the limit cycle and the acoustic radiation characteristics of a nonlinear vibroacoustic coupling system consisting of a non-autonomous van der Pol oscillator, a plate, and the surrounding acoustic medium are investigated. The nonlinear coupling equations of the vibro-acoustic coupling system under external excitation are derived using variational theory. The modal expression method and the incoherent superposition assumption are employed to discretize the displacements and the surface acoustic pressure of the plate, and then the incremental harmonic balance (IHB) method is used to drive the approximate analytical expressions for the limit cycles of the coupling system. The acoustic radiation power level and the surface average velocity under multi-frequency excitation are defined to describe the response of the plate. The results are that there are limit cycle motions in the vibration of the vibro-acoustic coupling system due to the van der Pol oscillator's excitation, and there are more harmonic components in the system when the parameters are increased. The vibro-acoustic characteristics of the plate present nonlinear variation because of the oscillator's nonlinear damping force. The method combining the IHB method and the modal expression method can be used to investigate the nonlinear vibro-acoustic coupling characteristics of a structure under external and self-excited excitation.

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