Forced vibrations of double beams discretely connected by multiple springs and dashpots

Abstract

This paper investigates the forced vibrations of double beams, consisting of upper and lower beams, which are discretely connected by N sets of springs and dashpots, when the upper beam is subjected to harmonic excitation. In the theoretical analysis, the orthogonality conditions of the vibrational modes of the system are derived and allow ones to obtain the modal equations of motion. Then, the solutions of the forced vibrations for the two beams can be theoretically obtained by summing up the results for the vibrational modes. In the numerical calculation, two cases, Cases A and B, are examined. In Case A, the two beams are connected by a single set of a spring and a dashpot, while in Case B they are connected by two sets of them. In Case A, when the two beams have identical materials and dimensions, the resonant peaks for the odd-order vibrational modes are independent of the connecting spring and dashpot because they are not stretched. However, the resonant peaks for the evenorder vibrational modes are influenced by the spring and dashpot. In Case A, when the two beams have different dimensions, the lower beam may vibrate at amplitudes lower than those of the upper beam due to the changes of the vibrational mode shapes. In Case B, the amplitudes of the even-order vibrational modes for the lower beam may be increased because increasing the number of connecting springs and dashpots results in the changes of the vibrational mode shapes and modal forces. The validity of the theoretical analysis was confirmed by comparing the theoretical results with the results obtained by the FEM analysis when the damping coefficients of the dashpots are comparatively small.

Keywords

• double-beam system
• discretely connecting spring and dashpot
• harmonic excitation
• forced vibration
• modal analysis
• orthogonality of vibrational modes