On the eigenfrequencies of a cantilever beam carrying a tip spring–mass system with mass of the helical spring considered

In the investigation of the vibrations of elastic arms and their suppression in the field of mechanical engineering and robotics, various systems may be modeled as a clamped-free Bernoulli-Euler beam to which one or several helical spring-mass systems are attached. Some of the numerous publications on this subject are given in Refs. [1-4]. The common aspect of all these works is that the mass of the helical springs is not taken into account. It was observed that the effects of this assumption on the numerical values of the eigenfrequencies of these combined systems had not been investigated in the literature. As a first step to cover this gap, the frequency equation of the combined system described in Ref. [1] is derived, but with the helical spring having mass and without the added tip mass. To this end, the helical spring, as in literature [5], is modeled by an appropriate elastic rod that vibrates longitudinally. The frequency equation obtained is solved numerically for various non-dimensional mass and spring parameters. Comparison with the massless spring case reveals that neglecting the mass can lead to serious errors for some parameter combinations.