A novel unified dynamical modeling of perforated plates based on negative mass equivalence

Plate-shaped structures are extensively utilized in aerospace, automotive, marine, and civil engineering, often featuring holes or cutouts for weight reduction, material savings, and design flexibility. However, the irregular shapes of their structures pose challenges in accurately extracting modal shapes using traditional analytical dynamical modeling methods. To address this challenge, this paper proposes a novel semi-analytical method for modal extraction in flat plates, particularly suitable for those with holes or cutouts, based on negative mass equivalence. Taking the perforated flat plates as examples, their dynamical models are developed using polynomial interpolation with Chebyshev polynomials and the Rayleigh-Ritz method. The concepts of “negative mass” and “positive mass” are employed to represent the features of openings, with Lagrange multipliers ensuring compatibility at connection nodes. Model validations are performed through comparisons of results regarding the natural frequencies, modal shapes, and dynamical responses obtained from this method, the finite element method, and the experimental method. The results demonstrate that the RMS error of the natural frequency is within 3.89 % and the accuracy of the model improves as the number of polynomial terms and connection nodes increases. Furthermore, the proposed method employs analytical modal shapes with minimal degrees of freedom to develop models, achieving exceptional computational efficiency with over 98 % time savings compared to the finite element method. This work provides a new pathway for advancing dynamical modeling methods of plates and enhancing simulation efficiency, with the potential to contribute to the development of active control and aeroelastic analysis techniques in plate-shaped structures.