Grouped inerter-resonator arrays for broadband reduction of plate mobility and radiated sound power using modal receptance synthesis

A plate-type vibroacoustic treatment is developed in which a simply supported Kirchhoff plate is coupled to a grouped array of parallel spring-damper-mass-inerter attachments. The emphasis is intentionally computational. The plate equation is reduced to a modal receptance form using 36 orthogonal sine modes, and the local attachments are eliminated analytically so that each attachment contributes a rank-one dynamic stiffness term. The resulting frequency-domain matrix remains explicit and can therefore be evaluated over large parameter grids without rederiving the governing equations. For the present plate, the bending rigidity is , the areal density is , the planform area is , and the total plate mass is . The sixteen attachments are arranged on a  grid and grouped columnwise at target frequencies, which are  Hz. With the reference internal mass , inertance ratio , tuning scale 0.9, spread coefficient 0.18, and damping ratio , the grouped inerter-resonator array yields average reductions of  dB in point mobility and  dB in both mean-square velocity and radiated sound-power proxy over 80–600 Hz. The spring-mass reference under identical internal mass produces corresponding lower averages of  and  dB. As per this study, the peak-by-peak reductions exceed 20 dB at several dominant modal peaks, while tuning scatter of 5% root-mean-square leaves the average velocity reduction essentially unchanged.