A modified polynomial-based approach to obtaining the eigenvalues of a uniform Euler–Bernoulli beam carrying any number of attachments

Abstract

Free vibration characteristics in uniform beams with several lumped attachments are an important problem in engineering applications that have to deal with mounting different equipment (e.g. motors, oscillators or engines) on a structural beam. In order to solve the lack of a generalized automatic procedure, this investigation presents a simple solving approach based on analytical means applied to a secular frequency equation for obtaining the natural frequencies of an arbitrarily supported single-span, or multi-span Euler-Bernoulli beam carrying any combination of miscellaneous attachments. The approach is obtained by solving a characteristic polynomial equation using a classical method for computing the roots of a polynomial. Interestingly, if the number of elements is greater than one a pole-zero cancellation is needed, but it does not require manual interventions such as initial values and iteration. The mathematical approach is validated with bibliographic references and evaluated for accuracy and computational effectiveness. A good agreement is observed with relative error values practically negligible mostly ranging between 10-3 and 10-9 in the first five natural frequencies, which confirms the validity of the presented approach in this paper. The MatLab code that has been developed with the solving approach is freely available as a supplementary material to this paper. Additionally, a MatLab graphical user interface (GUI) has also been developed in this work which allows to obtain the eigenvalues of a simply supported Euler-Bernoulli beam carrying an undetermined number of lumped elements. The GUI is also available for download, along with help facilities to be run in a Windows operating system and detailed instructions to reproduce the case studies presented here. The proposed scheme (and also the MatLab GUI) is very easy to code, and can be slightly modified to accommodate beams with arbitrary supports.