Frequency dependent mass and stiffness matrices of bar and beam elements and their equivalency with the dynamic stiffness matrix

Abstract

Starting from the solutions of the governing differential equations of motion in free vibration, the frequency dependent mass and stiffness matrices of bar and beam elements have been derived in this paper, but importantly, their equivalency with the corresponding dynamic stiffness matrix is established. In sharp contrast to series solutions, reported in the literature, explicit expressions for each term of the frequency dependent mass and stiffness matrices of bar and beam elements are generated in concise form through the application of symbolic computation and their relationship with the single dynamic stiffness matrix (which contains both the mass and stiffness properties) for each of the two element types is highlighted. The theory is demonstrated by numerical results. By splitting the dynamic stiffness matrix into frequency dependent mass and stiffness matrices and at the same time retaining the exactness of results, the investigation paves the way for future research to overcome the difficulty to include damping in the dynamic stiffness research which has not been possible earlier. Furthermore, the frequency dependent mass and stiffness matrices derived in this paper permit the application of the Wittrick-Williams algorithm to compute with certainty the exact natural frequencies of structures comprising bar and beam elements.