An exact method for free vibration of beams and frameworks using frequency-dependent mass, elastic and geometric stiffness matrices

Abstract

The frequency-dependent mass, elastic and geometric stiffness matrices of an axially loaded Bernoulli-Euler beam are developed through rigorous application of symbolic computation. These three matrices are related to the corresponding dynamic stiffness matrix so that free vibration analysis of axially loaded beams and frameworks can be carried out in an exact manner by applying the Wittrick-Williams algorithm as solution technique. Representative results from the proposed theory are presented for different boundary conditions of beams and frameworks, carrying tensile and compressive loads. Comparative results from finite element method are also presented. The duality between the free vibration and buckling problems is captured in that when the compressive load in a beam or frame approaches its critical buckling load, the fundamental natural frequency tends to zero and thus, buckling can be thoughtfully interpreted as free vibration at zero frequency. The investigation has opened the possibility of including damping in free vibration analysis of beams and frameworks when applying the dynamic stiffness method.