Theoretical Study of Crack‐Induced Eigenfrequency Changes on Beam Structures

The theoretical relationships between eigenfrequency changes and magnitudes and the locations of crack‐induced damage are developed for beam structures with either simply supported or cantilever boundary conditions. For uniform beams, a physical model of a massless rotational spring is used to represent the local flexibility introduced by the crack. A characteristic equation is derived as a base for the development of the relationship. A symbolic computational package, MACSYMA, is used to facilitate the computations of the higher‐order determinant and the corresponding derivatives involved in the characteristic equations. For nonuniform beams, the concept of receptance is used for a system linked with two coordinates (axial and rotational coordinates). Numerical experiments involving the use of a finite‐element program, SAP, to determine the eigenfrequencies of both uniform and nonuniform beams with a variety of damage scenarios are used to validate the derived theoretical relationships. The comparisons between the predicted and simulated damage conditions are satisfactory. Two important assumptions are involved in the derived relationship: one is that the structure is considered to behave linearly, and the other one is that the elastic properties of the structure member are time‐invariant. The application of the theoretical model is discussed at the end of the paper.