This study examines the geometric nonlinearity of large-amplitude vibrations in a symmetrically suspended beam using a multimode approach. The analysis begins with the derivation of the equation of motion, followed by the application of boundary and continuity conditions to formulate the homogeneous determining system. This system is then solved numerically using the Newton-Raphson method to obtain the natural frequencies and linear displacement profiles. The displacement field is developed in spatial and temporal series and integrated into the expressions for total kinetic and potential energy. By applying Hamilton's principle, the system is transformed into a nonlinear formulation, which is then solved using the Benamar method. The study provides a comprehensive analysis of the nonlinear dynamic response of the beam, illustrating the effects of load distribution, vibration amplitude, and geometric nonlinearity. The obtained results highlight the influence of parameters such as the maximum vibration amplitude, the concentrated load, and the distributed load on the geometric nonlinearity of the suspended beam.
Rjilatte et al. (Sun,) studied this question.