This study aims to analyze the natural frequency characteristics of multi-cracked extensible beams. The model and governing equations of the multi-cracked beam were derived using Hamilton's principle while considering crack energy. The eigenmode functions were obtained through eigenvalue analysis by applying the patching conditions of the cracks, and the equations for the discretized cracked beam were formulated and solved. The displacement responses from nonlinear system analysis were used to calculate frequencies via Fast Fourier Transform (FFT), and the frequency characteristics were systematically analyzed with respect to the number of cracks, crack depth, and cross-sectional loss. Additionally, the natural frequencies and orthonormal bases of the linear system were derived by exploring the solutions of the characteristic equation reflecting the cracks. Numerical analyses showed that the natural frequency of a cracked extensible beam was higher than that of a cracked EB beam. However, as the number or depth of cracks increased, the natural frequency gradually decreased. Notably, in extensible beams with large deflections, the dynamic changes caused by cracks demonstrated results that could not be obtained through the EB beam model.

This paper aims to advance our understanding of extensible beams with multiple cracks by presenting a crack energy and motion equation, and mathematically justifying the energy functions of axial and bending deformations caused by cracks. Utilizing an extended form of Hamilton's principle, we derive a normalized governing equation for the motion of the extensible beam, taking into account crack energy. To achieve a closed-form solution of the beam equation, we employ a simple approach that incorporates the crack's patching condition into the eigenvalue problem associated with the linear part of the governing equation. This methodology not only yields a valuable eigenmode function but also significantly enhances our understanding of the dynamics of cracked extensible beams. Furthermore, we derive a governing equation that is an ordinary differential equation concerning time, based on orthogonal eigenmodes. This research lays the foundation for further studies, including experimental validations, applications, and the study of damage estimation and detection in the presence of cracks.