A new exact approach for determining natural frequencies and mode shapes of non-uniform shear beams with arbitrary distribution of mass or stiffness

Abstract

A new exact approach which combines the basic solutions with unit matrix property and recurrence formula for determining natural frequencies and mode shapes of non-uniform shear beams is presented in this paper. The function for describing the distribution of mass is arbitrary, and the distribution of shear stiffness is expressed as a functional relation with the mass distribution and vice versa. The governing equation for free vibration of a non-uniform shear beam is reduced to a differential equation of the second-order without the first-order derivative by means of functional transformation. Then, this kind of differential equation is reduced to Bessel equations and other solvable equations for six cases. The exact solutions of mode shape functions are thus found. The basic solutions, which have a unit matrix property, are derived and used to obtain the frequency equations and mode shapes of multi-step shear beams with varying cross-sections. Numerical examples show that the calculated natural frequencies and mode shapes of two symmetric buildings are very close to the corresponding field measured data, suggesting that the proposed methods are applicable to engineering application and practice.