Free vibration analysis of Euler-Bernoulli beams with non-ideal clamped boundary conditions by using Padé approximation

Using the non-ideal boundary condition model, which is a linear combination of ideal simply supported and ideal clamped boundary conditions, the equation governing the free vibration of Euler-Bernoulli beams produces nonlinear rational functions that relate natural frequencies with the weighting factors of the non-ideal boundary condition model. The natural frequencies in practice are numerically computed by using a standard root-finding method with suitable initial guesses. In the present study, the nonlinear rational functions are approximated using Padé approximants to get analytical formulations of natural frequencies as functions of the weighting factors. Numerical examples are provided for cantilever and beams clamped at both ends with non-ideal boundary conditions. The formulas in most cases are accurate enough to get the natural frequencies up to two-digit accuracy. Those approximations can be easily utilized as starting values in the root finding method to avoid ambiguities in selecting initial guesses.