| Summary: | This paper investigates the free vibrations of double beams, consisting of upper and lower beams, which are discretely connected by N springs. In the theoretical analysis, the modal analysis approach is employed to determine the expressions for the natural frequencies and vibrational modes of the system. In the numerical calculation, two cases, Cases A and B, are examined. In Case A, the two beams are connected by a single spring, while in Case B they are connected by two springs. In Case A, when the two beams have identical materials and dimensions, as the spring constant K1 increases, the natural frequencies of the odd-order vibrational modes are constant because the two beams vibrate in phase in their vibrational modes and the connecting spring is not stretched. The natural frequencies of the even-order vibrational modes are increased with the increase of K1 because the two beams vibrate out of phase. When the spring is attached at the middle of the beams, the natural frequencies p4n-1 (n=1, 2, …) equal p4n. In addition, when K1 reaches the specific values K1,n in this case, a set of three natural frequencies satisfies p4n-2=p4n-1=p4n, and a magnitude relationship of the natural frequencies is switched when K1 crosses the value K1,n. In Case B, when two springs are attached in symmetry with respect to the midpoints of the beams, a set of two natural frequencies satisfies p2n=p2n+1 for the specific values of the spring constants even if the springs are not attached to the positions of the nodes of the independent beams. The validity of the theoretical analysis was confirmed by comparing the theoretical results with the results obtained by the FEM analysis for Case A.
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