An Improved Approach for Bending Vibrations of Cantilever Beam with Tip Mass at Different End Conditions

Single dof lumped model is frequently utilized due to its simlicity, when studying bending vibration of cantilever beam with a tip mass. For this purpose, an equivalent single dof model is obtained with some assumptions. These assumptions effect the accuracy of the result. This study proposes a new formulation of proportion parameter η which is defined as a function of ratio of the tip mass to the beam mass and used to obtain the value of the first natural frequency. Depending on proposed formula, calculation of the first natural frequency with respect to the mass ratio was simplified and the results were improved. The new formulation of the proportion parameter was applied to the reference model studied in literature to verify its effectiveness. After it is verified, three complex new cases were considered and the corresponding equivalent spring constant and natural frequency parameter were obtained. Lastly, the first non-dimensional natural frequencies were obtained for three different tapered cases with and without tip mass consisting of only height taper, only width taper and double taper. One can see that the results are close to theoretical frequencies presented in the literature.