Nonlinear geometrically free vibration analysis of two beams coupled by double spring‐mass

Abstract This article examines the analysis of geometrically nonlinear free vibrations in complex structures. Based on the Euler–Bernoulli beam theory and Von Kármán geometrical nonlinearity theory, a semi‐analytical solution is developed to study the nonlinear vibrations of two homogeneous and isotropic beams coupled by a double spring‐mass system. Research on elastically coupled beams is very limited in the literature, mainly due to the complexity of the problem as well as the absence of efficient, user‐friendly methods for analyzing such structures. Unlike traditional methods, which are difficult to implement and face convergence issues, this study proposes an approximate multimodal approach that offers both rapid and accurate analysis. This method stands out for its ease of application to complex structures, as illustrated in this case, as well as its fast convergence. After applying the boundary conditions and continuity conditions, the eigenvalue problem was solved iteratively using the Newton–Raphson algorithm. By employing Hamilton's principle and an innovative multimodal approach, the nonlinear equations governing the vibrations of this structure were formulated and subsequently solved using an approximate method so‐called the second formulation. The results obtained from the proposed model are compared and validated with previous research presented in the literature. The agreement observed between our results and existing data demonstrates the effectiveness and accuracy of the adopted approach. Furthermore, this work includes a parametric study that provides essential insights into the effects of various parameters influencing structural dynamics, thus opening perspectives for future research on the vibrations of complex structures. By making significant contributions in this domain, this work is a valuable reference for future studies.