Nonlinear dynamics of an axially preloaded hinged–hinged planar beam — Experimental insights, finite element modal analysis and analytical modeling

The dynamic response of a composite beam with hinged boundary conditions and an adjustable span is investigated. Prestress applied via uniform axial stretching alters the natural frequencies across all vibration modes and governs the nonlinear response, including both hardening and softening effects. In certain cases, these effects influence transmissibility (mixed hardening/softening behavior). Uniform axial shortening leads to a pitchfork bifurcation and the emergence of two post-critical configurations. In the experimental part of the study, controlled axial shortening with a prescribed half-wave amplitude ratio was applied to address the extreme sensitivity of longitudinal displacement. The system was then subjected to base excitation, accounting for both in-well and cross-well dynamics. For the first buckling mode, significant changes of the first natural frequency within the linear range were observed, while the other frequencies remained largely unaffected. The experimental findings are followed by analytical studies. In the mathematical formulation, a coupled longitudinal–transverse nonlinear model of an unshearable beam is proposed, retaining nonlinearities up to the fifth order. Additionally, effects related to mid-line extensibility and hinge inertia are included. A system of two partial differential equations governing axial and transverse displacements is derived and subsequently discretized using the Galerkin procedure. The simplified case without the axial load is solved numerically, and frequency response diagrams for individual modal coordinates are plotted accounting for various base excitation amplitudes.