Nondimensional triply coupled free vibration analysis of the axially layered thin-walled beams with nonsymmetric open cross sections and its application to the frequency optimization

This study presents triply coupled free vibration analysis of the axially layered Euler-Bernoulli thin-walled beams with unsymmetrical open cross sections in dimensionless and exact manner. Various parameters are normalized with respect to a reference thin-walled beam called as the baseline beam structure. Characteristic equations for determining natural frequencies of the beams are obtained using the transfer matrix method. Different boundary conditions are considered. As a case study, dimensionless solution procedure is applied to the optimization of fundamental natural frequency of cantilevered beam structures. Square of the characteristic equation of the cantilevered beam is taken as an objective function and design variables are chosen as the segment length and volume fraction of the materials. The total mass and total length of the optimized beam are kept equal to those of the reference beam. Axially, two-, three- and five-segmented beam configurations are considered. Optimization routine developed following verification of the coupled free vibration analyses reveals that the final forms of the optimized beam structures can be regarded as tapered beam-like structures and the maximum dimensionless natural frequency without constraint violations is attained for the five-segmented beam and it is 0.51, which represents 22.2142% gain. Also, some conclusions will be drawn.