A new method for free vibration of beams using theory of elasticity

A refined theory which accounts for both longitudinal and transverse displacements as well as for the Poisson’s effect is developed for free vibration analysis of beams. This is achieved by using the theory of elasticity and symbolic computation. The formulation is based on a two dimensional (time dependent) stress field in which both normal and shear stresses along the length and depth/thickness of the beam are considered but the stresses in the lateral/width direction are ignored. Both Newton’s law and Hamilton’s principle are used to derive the governing differential equations of motion in free vibration. The derivation led to two coupled partial differential equations from which the time dependent terms are eliminated by assuming harmonic oscillation. In order to secure an approximate, but accurate formulation, the ordinary differential equations obtained in this way (still coupled in the space variables) are then solved by expanding the solution in series and then selecting appropriate terms from the series. An important feature of this work is the application of symbolic computation when obtaining the differential equations and their solution in explicit analytical form. The theory developed provides general solution for longitudinal and transverse displacements of the beam in free vibration in terms of arbitrary constants. Results for specific cases such as cantilever, simply-supported and built-in beams can be obtained by imposing the boundary conditions and then eliminating the arbitrary constants. Numerical evaluation of the theory has not been possible at present because the volume of the work needed has proved to be enormous and would take the paper further than it is intended. However, a numerical study will be undertaken at a later stage which will constitute a separate investigation of the problem. The theory presented can be extended further to develop the frequency dependent dynamic stiffness matrix of the beam for its free vibration analysis using the Wittrick-Williams algorithm.