A transfer matrix method for free vibration analysis of Euler-Bernoulli beams with variable cross section

To perform vibration analysis, many structures could be modeled entirely or partially as beams with variable or constant cross section. In this study, the differential equations for free bending vibrations of straight beams with variable cross section are solved using Bessel’s functions. Then, the general equations for one-step conical beams are used together with corresponding equations of cylindrical beams to model multi-step beams with various boundary conditions. The natural frequencies of a cantilever conical beam, calculated using its derived field matrix, were shown to be close to those obtained experimentally. In addition, the influence of geometrical factors upon the value of fundamental frequencies is investigated for both cantilever and clamped-clamped conical beams. It was shown numerically that changing a cantilever cylindrical beam into a conical beam by reducing the diameter of the free end increases its fundamental natural frequency even though its rigidity decreases. This property was not observed in a clamped-clamped conical beam. Finally, the effectiveness of a proposed transfer matrix on the calculation of natural frequencies for a multi-step beam was verified. It is believed that the current method could help in the design of various structures, including beams with variable cross section.