Vibration analysis of beams with tip mass using differential transform method

Dynamic analysis of beam-tip mass systems is very important for the successful design of mechanical systems 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 coincided with the attachment point of the beam and the tip mass is assumed to be a point mass. In this study, beam with a three dimensional (3D) rigid mass whose center of gravity is not coincided with the attachment point of the beam and subject to both torsional and flexural deformations in two orthogonal planes is considered. Euler-Bernoulli and Timoshenko beam models have been used for the mathematical modelling of the system. In addition, it is assumed that the material and geometry properties of the beam is variable along the beam and the left side of the beam is taken account to be clamped or free. Using Hamilton's principle based on energy approach, the equations of motion of the system with the possible boundary conditions are derived. First, equations of motion were solved analytically for the uniform beam. The natural frequencies and mode shapes of the uniform beam-tip mass systems were obtained and compared with those by finite element method (FEM) and experimental modal analysis. The analytical solution of the beam-tip mass systems is not easy, especially for the non-uniform systems, because the governing equations consist of high-order differential equations. For this reason, the Differential Transform Method (DTM) has been applied. DTM results obtained according to the Timoshenko beam theory were compared with both FEM and experimantal results. Finally, another new and semi-numerical method called Multi-Step Differential Transform Method (MDTM) is used for free vibration analysis of non-uniform beam-tip mass system due to the deficiency of DTM for this system. The MDTM results of Timoshenko beam theory were verified by the FEM and experimental results. However, the results obtained accordingly EB beam theory were only compared with the FEM results because of the difficulty of preparing the non-uniform test sample at the appropriate length for this theory. Furthermore, the effects of the tip mass dimensions, beam length and taper ratio on the natural frequencies are examined for the EB beam theory. Consequently, it is observed that the results obtained by DTM and MDTM are compatible with FEM and experimental results.