Derivation of the Mode Shape Functions for a Three-span Cantilever Beam

Based on the Euler-Bernoulli beam theory, the mode shape functions and the relationship matrix of undetermined coefficients for each beam segment were constructed through piecewise solution and separation of variables. The natural frequencies and mode shape functions of a three-span cantilever beam were calculated. The results of ANSYS software simulation for the three-span cantilever beam model were almost consistent with the theoretical calculation frequencies, with very small errors. This verified the effectiveness of the proposed method for mode shape function calculation in complex structural analysis. This method lays a solid theoretical foundation for further research on the flow-induced vibration of beams under axial flow with multi-support constraints.