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

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 sti€ness 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 di€erential equation of the second-order without the ®rst-order derivative by means of functional transformation. Then, this kind of di€erential 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 ®eld measured data, suggesting that the proposed methods are applicable to engineering application and practice