The application of the Wittrick-Williams algorithm for free vibration analysis of cracked skeletal structures

Abstract

In this study, the dynamic stiffness method is applied to a cracked beam that incorporates Rayleigh bar theory and Timoshenko beam theory, and the Wittrick-Williams algorithm is used to pinpoint natural frequencies. The influence of a crack on the two components of the Wittrick-Williams algorithm, namely, the number of negative leading diagonal terms in the upper triangular form of the global dynamic stiffness matrix at a trial frequency, and the sum of frequencies of each clamped-clamped element of the structure exceeded by the trial frequency, is explored. For the first component, the leading diagonal terms relevant to the additional degrees of freedom introduced by the crack should be considered. While for the second component, it is shown that a crack element itself does not contribute to the second component. The influence of a crack element on the second component only lies in that the crack alters the discretization thus more beam elements are involved. The same philosophy applies to the introduction of an affiliated mass or spring-mass system. To verify this statement, the free vibration of a cracked single-storey frame carrying a roving mass with rotary inertia is studied. The frequency results obtained using the dynamic stiffness method incorporating the Wittrick-Williams algorithm are compared with those exported from a finite element model in ANSYS. Some discussions about the results are made. The effects of roving translational inertia and roving rotary inertia are studied.