EQUILIBRIUM FORMULATION OF THE TRANSVERSE VIBRATION EQUATIONS OF EULER-BERNOULLI BEAMS ON TWO-PARAMETER FOUNDATIONS: RESONANT FREQUENCY SOLUTIONS BY STODOLA-VIANELLO ITERATION METHOD

Abstract

In this study, the partial differential equation (PDE) for the transverse vibration analysis of Euler-Bernoulli beam resting on two-parameter elastic foundation (EBBo2PEF) is derived from a rigorous use of equilibrium method. The equation is nonhomogeneous for forced vibrations, and homogeneous for free vibrations. Free vibration analysis is very important because the natural vibration frequencies are the resonant frequencies responsible for failure if forcing frequency is equal to the natural frequency. The Stodola-Vianello successive iteration method (SVSIM) was adopted in the work to derive the iteration equations for the PDE, thus simplifying the integration problem to an iterative algebraic one. Sinusoidal shape functions satisfying restraint conditions at the simple supports are used to develop characteristic eigenfrequency equations at convergence when the (n + 1)th vibration mode function iteration is made equal to the nth vibration mode function iteration. The resulting eigenvalues are used to determine closed form eigenfrequency solutions for simply supported EBBo2PEF. The closed form eigenfrequency solutions for EBBo2PEF reduce to eigenfrequency solutions for Euler-Bernoulli beams on Winkler foundations (EBBoWFs) if the second modulus of the foundation vanishes. The eigenfrequencies obtained for beam foundation parameters    and    for the first five transverse vibration modes are exact and identical with previously obtained values using the Rayleigh-Ritz method and Legendre polynomial shape functions. Similarly, the dimensionless transverse vibration frequencies for the first five modes for non-vanishing   when   is zero are exact within the framework of thin beam theory and closely agree with previous results.