Vibration Mode Analysis of Frames by the Method of Reverberation Ray Matrix

Abstract

The method of reverberation ray matrix (MRRM) has been developed by (Pao et al. 1999, “Dynamic Response and Wave Propagation in Plane Trusses and Frames,” AIAA J., 37(5), pp. 594–603) recently based on the theory of wave propagation for transient analysis of truss or frame structures. In this study, the MRRM is employed to obtain the frequency response function (FRF) of displacement of a frame under the action of a unit impulse load. The natural frequencies of the frame are determined from the FRF, since the curve of FRF has peak when a resonant frequency is approached. The vibration mode is retrieved from the adjoint matrix of the coefficient matrix of the governing equations of MRRM. The MRRM has advantage over numerical methods, such as finite element method (FEM), since in MRRM the frame is treated as an assembly of multiconnected beams, and exact solutions to the beam differential equations are employed to yield the system matrix of the frame. The vibration mode obtained is therefore exact. A planar frame made of 17 aluminum bars is analyzed. The vibration modes, as well as natural frequencies obtained from MRRM, coincide accurately with those obtained from FEM of ANSYS for the first a few modes; however, the difference of the frequencies between the two methods becomes a bit obvious when high order modes are examined.