Substructure elimination and binding method for bending vibration analysis of beams

This study proposed a vibration analysis employing the substructure elimination and binding method to evaluate the bending vibration of beams. Modal analysis is commonly used for vibration analysis. In modal analysis, it is imperative to obtain highly precise results with fewer degrees of freedom. Thus, to reduce the required degrees of freedom, we proposed the substructure change and elimination methods. Because these methods obtain the desired structure by changing or eliminating certain regions of the original structure, these methods necessitate only a superposition of the eigenmodes of the original structure. Despite no major problems in the substructure elimination method, the precision of the substructure change method decreased owing to the non-smoothness of the shear force etc. at the interface. In addition, a recent study revealed the possibility of analyzing the coupled vibration of a beam using the substructure elimination method with fewer degrees of freedom. In this method, both ends of a beam are eliminated and the coupling with the other structures at the new boundaries were formulated. Owing to the eigenfunctions exhibiting phase variations at the new boundaries, arbitrary amplitudes and phases can be expressed with fewer degrees of freedom at the new boundaries. However, beams are often coupled to other structures other than that at their ends. Despite the long-standing establishment of the analysis method for this case, the precision of this conventional method decreases because of the discontinuity or nonsmoothness of the shear force etc., similar to the substructure change method. Thus, this study proposed a method to set a virtual elimination region at the interface and bind the two ends of the virtual elimination region. The virtual elimination region smoothly connected the discontinuities and non-smooth points. Further, we proposed an analytical model for this method, and an equation of motion for a minute fraction was formulated. In addition, modal analysis was applied to the derived equation of motion. Furthermore, simulations revealed that the length of the virtual elimination region must be set to three to four times the wavelength of the highest eigenmode of the original beam. Moreover, to investigate the advantage of precision, the simulation results obtained using the proposed method were compared with those obtained without installing the virtual elimination regions based on the exact solutions.