Vibration Analysis of Beams with and without Cracks Using the Composite Element Model

Abstract

Beams are fundamental models for the structural elements of many engineering applications and have been studied extensively. There are many examples of structures that may be modeled with beam-like elements, for instance, long span bridges, tall buildings, and robot arms. The vibration of Euler–Bernoulli beams with one step change in cross-section has been well studied. Jang and Bert (1989) derived the frequency equations for combinations of classical end supports as fourth order determinants equated to zero. Balasubramanian and Subramanian (1985) investigated the performance of a four-degree-of-freedom per node element in the vibration analysis of a stepped cantilever. De Rosa (1994) studied the vibration of a stepped beam with elastic end supports. Recently, Koplow et al. (2006) presented closed form solutions for the dynamic response of Euler–Bernoulli beams with step changes in cross section. There are also some works on the vibration of beams with more than one step change in cross-section. Bapat and Bapat (1987) proposed the transfer matrix approach for beams with n-steps but provided no numerical results. Lee and Bergman (1994) used the dynamic flexibility method to derive the frequency equation of a beam with n-step changes in crosssection. Jaworski and Dowell (2008) carried out a study for the free vibration of a cantilevered beam with multiple steps and compared the results of several theoretical methods with experiment. A new method is presented to analyze the free and forced vibrations of beams with either a single step change or multiple step changes using the composite element method (CEM) (Zeng, 1998; Lu & Law, 2009). The correctness and accuracy of the proposed method are verified by some examples in the existing literatures. The presence of cracks in the structural components, for instance, beams can have a significant influence on the dynamic responses of the whole structure; it can lead to the catastrophic failure of the structure. To predict the failure, vibration monitoring can be used to detect changes in the dynamic responses and/or dynamic characteristics of the structure. Knowledge of the effects of cracks on the vibration of the structure is of importance. Efficient techniques for the forward analysis of cracked beams are required. To this end, the composite element method is then extended for free and forced vibration analysis of cracked beams. The principal advantage of the proposed method is that it does not need to partition the stepped beam into uniform beam segments between any two successive discontinuity points