Quantitative validation of the analytical mode shapes of a beam-like structure with a Z-shaped configuration

Beam-like structures are widely used in engineering. The accuracy of the analytical mode shapes of these structures is important for studying their dynamic characteristics. A method is presented in this study to theoretically obtain and quantitatively validate the analytical mode shapes of a beam-like structure with a Z-shaped configuration. The governing equations and boundary conditions of the structure's planar motion are derived using Hamilton's principle. Frequencies and analytical mode shapes of the structure are theoretically obtained and compared with the numerical results from finite element method. The comparison of frequencies and modal assurance criterion is used to quantitatively validate the analytical mode shapes. Examples are presented to show the analytical mode shapes of Zshaped beams with different fold angles. The proposed method is useful for improving the accuracy of analytical mode shapes, which is beneficial to the design and control of beam-like structures in engineering fields.