Class of analytical closed-form polynomial solutions for guided–pinned inhomogeneous beams

In this study, new closed-form solutions for the natural frequency are derived for a beam guided at one end and pinned at the other end. As in Elishakoff and Guede (cf. Elishakoff I, Guede Z. Family of analytical polynomial solutions for simply supported inhomogeneous beams, part 1: vibration. Journal of Sound and Vibration 2000, to appear.), we preselected a function that is postulated to be the mode shape of the beam. Then, the stiffness and the material density are taken as a polynomial function of specific order. Finally, from the governing equation of a vibrating beam, we express new closed-form solutions for the natural frequency.