Separating the Location and Severity Effects in Frequency-Based Crack Detection Using the Dynamic Stiffness Matrix

The Dynamic Stiffness Matrix (DSM) of a structure is a frequency-dependent stiffness matrix relating the actions (forces and moments) and displacements (translations and rotations) when the structure vibrates at a given frequency. The DSM may be used to find the natural frequencies, modes, and structural response. For many structures, including skeletal frames of prismatic members, exact transcendental expressions for the DSM are readily available. This paper presents a mathematical proof of a linear determinantal relationship between the DSM of a skeletal frame when it is undamaged, cracked, and hinged at the crack location. The rotational stiffness or flexibility of the crack also appears as a linear term. This relationship gives, for the first time, an explicit equation to directly calculate the stiffness of the rotational spring representing a crack from measured natural frequencies for any potential crack location. Numerical examples demonstrate that computing the DSM of the intact and hinged structures gives an efficient solution method for the inverse problem of identifying crack location and severity. This paper also shows that an approximate DSM based on a finite element model can be used in the same way, making this procedure more versatile. Furthermore, new approximate expressions for the natural frequencies of structures with very small or very severe cracks are derived. An interesting relationship between the square of the bending moment in an undamaged beam and the determinant of the DSM of a hinged beam is also derived. This relationship, which can also be inferred from previous work, leads to a better understanding of the effect of crack location in specific vibration modes.