Investigation of free vibration analysis of axially accelerating beams by using differential transform method

Dynamic analysis of axially moving beams have became very important for engineering systems with developments in technology. Vibrations of engineering structures that arised from external loads have been investigated by many researchers. In these studies about free vibration analysis, the mass which is one of the most important parameter of the dynamic analysis, is assumed as lumped at some points on the beam axis or distributed along the beam. Even if the solution of the lumped mass system is easy, it is a fact that the real behaviour can not be demonstrated especially using limited number of lumped masses. In contrast, the models with distrubuted mass reflect the real behaviour of the moving beam. Thus, in this study, the model is formed according to distributed mass approach by using Timoshenko beam theory and Reddy-Bickford beam theory, which is one of the high order shear deformation theories. In this thesis, the free vibration analysis of accelerating beams, which can be defined as special cases of moving beams, are investigated. First of all, the governing equations of motion of Timoshenko beam and Reddy-Bickford beam are derived using Hamilton's principle. Due to acceleration, the axial speed and axial tensile force are assumed as harmonically varying about a constant speed and tensile force. Parameters for the nondimensionalised multiplication factors for the velocity and axial tensile force are used to investigate their effects on the natural frequencies. The natural frequencies are calculated by solving the differential equations for both Timoshenko and Reddy-Bickford beam theories. After the analytical solution, governing equations of motion of both Timoshenko and Reddy-Bickford beam theories are solved by using Differential Transform Method (DTM), which is based on finite Taylor series. For different boundary conditions, the calculated first three natural frequencies from DTM and analytic solution are tabulated. For different boundary conditions and different parameters, the mode shapes are figured for both Timoshenko and Reddy-Bickford beam theories and the normalized moment diagrams are presented for Reddy-Bickford beam theory. All of the numerical analysis are performed by using computer programs prepared in Matlab.