An Analysis of Nonlinear Axisymmetric Structural Vibrations of Circular Plates with the Extended Rayleigh–Ritz Method

The nonlinear deformation and vibrations of elastic plates represent a fundamental problem in structural vibration analysis, frequently encountered in engineering applications and classical mathematical studies. In the field of studying the nonlinear phenomena of elastic plates, numerous methods and techniques have emerged to obtain approximate and exact solutions for nonlinear differential equations. A particularly powerful and flexible method, known as the extended Rayleigh–Ritz method (ERRM), has been proposed. In this approach, the temporal variable is introduced as an additional dimension in the formulation. Through expanded integration across both the physical domain and a vibration period, the temporal variable is eliminated. The ERRM builds on the traditional RRM that offers a straightforward, sophisticated, and highly effective way to approximate solutions for nonlinear vibration and deformation issues in the realm of structural dynamics and vibration. In the case of circular plates, the method incorporates the linear displacement function along with high-frequency terms. As a result, it can accurately determine the nonlinear axisymmetric vibration frequencies of circular plates. For scenarios involving smaller deformations, its accuracy is on par with other approximate solution methods. This approach provides a valuable and novel procedure for the nonlinear analysis of circular structural vibrations.