A multimode approach to nonlinear geometrically free and forced vibration analysis of two beams interconnected by several double spring‐mass

Abstract

Abstract This study aims to provide a semi‐analytical solution for the geometrically nonlinear vibrations of two identical, homogeneous, isotropic beams, clamped at their ends, coupled by several double spring‐mass systems, and subjected to harmonic excitation forces. The analysis is conducted within the framework of Euler‐Bernoulli beam theory and Von Kármán's geometric nonlinearity assumption. After solving the linear formulation using the iterative Newton‐Raphson method, the nonlinear formulation is obtained by applying an approach based on Hamilton's principle combined with a multimodal method for nonlinear vibrations with large displacements. This approach has been successfully used in previous studies for continuous beams, plates, and nanostructures. Although nonlinear vibrations of coupled beams have received limited attention in the literature due to their complexity, this work represents the first application of this methodology to such coupled systems. New findings were obtained by applying this approach to analyze the nonlinear behavior of coupled beams. The results indicate that, at a dimensionless amplitude of 1, the frequency ratio increases by 4.23% for a system with a single double spring‐mass, and by 3.96% for a system with three double spring‐mass. Finally, an in‐depth parametric study was performed to evaluate the influence of the number of coupling systems, the spring stiffness constants, and the mass connecting the double spring in the free vibration case, as well as the influence of the intensity of concentrated or distributed harmonic forces in the forced vibration case. This research provides new insights for the design of coupled beam systems and also contributes novel quantitative results to the literature, which can serve as a solid reference for future studies.