Stodola-Vianelo Iteration Method for Solving Transverse Harmonic Natural Vibration Problems of Euler-Bernoulli Beams on Winkler Foundations

Free transverse vibration frequency analysis of Euler-Bernoulli beams on Winkler foundation (EBBoWF) is a significant part of their analysis for averting failures by resonance. Resonant failure of EBBoWF occurs when the loading frequency exciting the vibration coincides with the least natural frequency. This study aims at using the Stodola-Vianello iteration method (SVIM) for the natural transverse vibration analysis of EBBoWF. Generally, the problem is governed by a non-homogenous partial differential equation (PDE) for forced vibrations, but simplifies to a homogeneous PDE for free vibrations where excitation forces are absent. For harmonic vibrations, and harmonic displacement response u(x, t), the equations are decoupled in terms of the independent spatial and time variables, resulting in a fourth order ordinary differential equation (ODE) in the displacement modal function for u(x, t). The study’s focus is on homogenous, prismatic, isotropic thin beams leading to ODEs with constant parameters. SVIM was used to express the ODE as Stodola-Vianello iteration equations with four constants of integration, determinable via the boundary conditions. Specific application of SVIM to the EBBoWF with simple end supports used exact sinusoidal shape functions and boundary conditions to determine the integration constants. Convergence criterion at the nth iteration was used to find the eigenequation which was solved for the eigenvalues. The natural transverse vibration frequencies at the nth modes were found in terms of frequency parameters . Values of calculated for the first five modes n = 1, 2, 3, 4, 5, and for values of showed that the present SVIM gave exact results compared to other previous results. The exact solutions were obtained because exact shape functions were used in the SVIM equations resulting in satisfaction of the governing equations at the domain and the boundaries.