Vibration analysis of a multi-span Timoshenko beam with flexible constraints subjected to a two degrees-of-freedom moving system

An analytical method is developed to determine the effects of the frame angle on the behavior of multi span Timoshenko beams with flexible constraints subjected to a two degrees-of-freedom moving system. In a multi-span beam, there are discontinuities in each frame angle in addition to discontinuities in flexible constraints. The correlation among every segment can be obtained considering the compatibility equations in frame angles and cracks. Eigensolutions of the serial-frame system can be calculated explicitly, using the analytical transfer matrix method. The forced responses can be determined by the modal expansion theory using the eigenfunctions. The orthogonality of the mode shapes is applied to calculate the forced response. A new formulation is introduced and confirmed for the orthogonality of the mode shapes for the case of a beam with the frame angle. It is observed that the natural frequencies will be increased by increasing the frame angles, while the maximum deflection of the beam will be decreased. The crack modeling leads to lower natural frequencies and higher maximum deflection for the beam in compared with the no crack model. The validity of the developed technique is probed by comparing the results with the analytical results reported in other articles and those from numerical solutions.