2 An Investigation of Vibrational Characteristics of Lap Joints Using Experimental and Analytical Methods 13 Bar sections Solid bolt models Fig. 2.4 Geometry for the bolted structure FE model consisting of two bar sections and four solid bolt models Contact regions Fig. 2.5 Contact regions used to simulate pretension in the bolted structure FE model In order to allow for the approximation of a loosened connection in the modal study, the bars were bonded to the bolts using a circular region around each bolt hole, as shown in Fig. 2.5. A similar method was successfully employed by Liao et al. [11]. These circular regions had shared topology with their respective bar sections, and contact was assigned between the bars and between the bars and the bolt head/nuts. Different levels of pretension were approximated by changing the size of the circular contact regions around the bolt heads. Multiple trials of the bolted structure FE model were solved. First, the sizing of the contact region around each bolt was adjusted to model the effects of a loosened connection in the model. In general, a smaller contact region around each bolt resulted in a model that was less stiff, thus simulating the effects of loosening a bolt in the model while mass properties remain constant. Next, one bolt was completely removed from the model along with its contact parameters. In this type of model, the mass properties and the component contact are different from the undamaged structure. SHM methods were also used to compare the differences between complete (healthy) structures and structures with missing components or differing size contact regions (damaged). 2.3.4 Fractional Strain Energy Method Changes in natural frequencies and damping ratios do not often provide sufficient information to determine the presence of damage in a structure. Even if natural frequencies and damping ratios are sensitive to damage, they cannot be used to locate the damage. In this work, a fractional strain energy method was implemented to compare experimental data and analytical results. The fractional strain energy method (SEM) was implemented for generic planar structures, such as a beam or plate, that have only a single degree-of-freedom of modal response. The fractional strain energy for a beam element for a particular mode shape, ψi(x), is Ui = 1 2 l 0 EI ∂ 2 ψi ∂x2 2 dx. (2.1)
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