Development of an exact dynamic stiffness matrix for free vibration analysis of a twisted Timoshenko beam

Abstract

An exact dynamic stiffness matrix for a twisted Timoshenko beam is developed in this paper in order to investigate its free vibration characteristics. First the governing differential equations of motion and the associated natural boundary conditions of a twisted Timoshenko beam undergoing free natural vibration are derived using Hamilton's principle. The inclusion of a given pretwist together with the effects of shear deformation and rotatory inertia, gives rise in free vibration to four coupled second order partial differential equations of motion involving bending displacements and bending rotations in two planes. For harmonic oscillation these four partial differential equations are combined into an eighth order ordinary differential equation, which is identically satisfied by all components of bending displacements and bending rotations. This difficult task has become possible only with the help of symbolic computation. Next the exact solution of the differential equation is obtained in completely general form in terms of eight arbitrary constants. This is followed by application of boundary conditions for displacements and forces. The procedure leads to the formation of the dynamics stiffness matrix of the twisted Timoshenko beam relating harmonically varying forces with harmonically varying displacements at its ends. The resulting dynamic stiffness matrix is used with particular reference to the Wittrick–Williams algorithm to compute the natural frequencies and mode shapes of a twisted Timoshenko beam with cantilever end condition. The exact results from the present theory are compared with numerically simulated results using simpler theories, and some conclusions are drawn.