An analytical method for free vibration characterization of continuous multi-span beams with arbitrary boundary conditions and its experimental verification

For the vibration characteristics of continuous multi-span beams under arbitrary boundary conditions, the differential vibration equation for the beam segment between any two supports is derived based on the Timoshenko beam element as the stress analysis object. Then, according to the coordination conditions of elastic support force and displacement deformation between spans, the transverse vibration mode transfer relationship for multi-span continuous beams is derived using the mode shape coefficients as the transfer vectors. Finally, analytical expressions for the frequencies and mode shapes of elastic support multiple-span continuous beams under axial static load for arbitrary boundary conditions are provided. To validate the effectiveness of the proposed method for solving the natural frequencies and mode shapes of elastic support multiple-span continuous Timoshenko beams under arbitrary boundary conditions, vibration characteristic tests of the propeller shaft system were conducted on an experimental platform. The results show that the mode shapes of the propeller shaft system calculated using the proposed method are in good agreement with the experimental results, and the mode frequencies match the experimental measurements well, with a maximum error of 7.62%. This validation confirms the effectiveness of the proposed method for solving natural frequencies and mode shapes, meeting the accuracy requirements of engineering calculations, and particularly suitable for the rapid verification of vibration characteristics during the design phase of propeller shaft systems.