Free vibration analysis of a twisted beam using the dynamic stiffness method

An exact dynamics stiffness matrix is developed and subsequently used for free vibration analysis of a twisted beam whose flexural displacements are coupled in two planes. First the governing differential equations of motion of the twisted beam undergoing free natural vibration are derived using Hamiltonʹs principle. Next the general solutions of these equations are obtained when the oscillatory motion of the beam is harmonic. This is followed by application of boundary conditions for displacements and forces, which essentially leads to the formation of the dynamics stiffness matrix of the twisted beam relating harmonically varying forces with harmonically varying displacements at its ends. The resulting dynamic stiffness matrix is used in connection with the Wittrick–Williams algorithm to compute natural frequencies and mode shapes of a twisted beam with cantilever end condition. These are compared with previously published results to confirm the accuracy of the method, and some conclusions are drawn.