Dynamic analysis of beam-tip mass systems is a significant issue in terms of providing successful design of mechanisms such as robot arms and manipulators. In the relevant literature, uniform beams have been mostly considered for the free vibration analysis of these systems whereas the number of studies using nonuniform beam is limited. Furthermore, center of tip mas is, in general, coincident with the attachment point of the beam, and the tip mass is assumed to be a point mass. This research concerns a non-uniform beam with a three dimensional rigid tip mass whose center of gravity is not coincident with the attachment point of the beam and subjected to both torsional and flexural deformations in two orthogonal planes. A semi-numerical method called Multi-Step Differential Transform Method (MDTM) is employed for the solution of governing equations of the system. Natural frequencies and mode shapes of the system are obtained for two different boundary conditions, i.e. the left end clamped or free. The influence of tip mass dimensions, beam length and taper ratio on the natural frequencies are studied, and the well-known finite element software (ANSYS) is employed to compare the results and confirm the accuracy of the model. It is proved that results obtained by MDTM and ANSYS are sufficiently close to each other.