On the frequency equation of a combined system consisting of a simply supported beam and in-span helical spring–mass with mass of the helical spring considered

This study is concerned with the establishment of the frequency equation of a combined system consisting of a simply supported beam and an in-span helical spring–mass, considering the mass of the helical spring. After obtaining the “exact” frequency equation of the combined system, a Dunkerley-based approximate formula is given for the fundamental frequency. The frequency equation of a simpler system is obtained as a special case. The frequency equations are then numerically solved for various combinations of the physical parameters. Calculated results are also compared with finite element solutions. Further, comparison of the results with the massless spring case reveals the fact that neglecting the mass of the spring can cause considerable errors for some combinations of the physical parameters.