Exact Analytical Solutions for Free Single-Mode Nonlinear Cantilever Beam Dynamics: Experimental Validation Using High-Speed Vision

This work investigates the nonlinear flexural dynamics of a macroscale cantilever beam by combining analytical modeling, symbolic solution techniques, numerical simulation, and vision-based experiments. Starting from the Euler–Bernoulli equation with geometric and inertial nonlinearities, a reduced-order model is derived via a single-mode Galerkin projection, justified by the experimentally confirmed dominance of the fundamental bending mode. The resulting nonlinear ordinary differential equation is solved analytically using two symbolic methods rarely applied in structural vibration studies: the Extended Direct Algebraic Method (EDAM) and the Sardar Sub-Equation Method (SSEM). Comparison with high-accuracy numerical integration shows that EDAM reproduces the nonlinear waveform with high fidelity, including the characteristic non-sinusoidal distortion induced by mid-plane stretching. High-speed vision-based measurements provide displacement data for a physical cantilever beam undergoing free vibration. After calibrating the linear stiffness, analytical and experimental responses are compared in terms of the dominant oscillation frequency. The analytical model predicts the classical hardening-type amplitude–frequency dependence of an ideal Euler–Bernoulli cantilever, whereas the experiment exhibits a clear softening trend. This contrast reveals the influence of real-world effects, such as initial curvature, boundary compliance, or micro-slip at the clamp, which are absent from the idealized formulation. The combined analytical–experimental framework thus acts as a diagnostic tool for identifying competing nonlinear mechanisms in flexible structures and provides a compact physics-based reference for reduced-order modeling and structural health monitoring.