Free longitudinal vibration analysis of multi-step non-uniform bars based on piecewise analytical solutions

Using appropriate transformations, the differential equations of free longitudinal vibrations of bars with variably distributed mass and stiffness are reduced to Besselʹs equations or ordinary differential equations with constant coefficients by selecting suitable expressions, such as power functions and exponential functions, for the distributions of stiffness and mass. Exact analytical solutions to determine the longitudinal natural frequencies and mode shapes for a one step non-uniform bar are derived and used to obtain the frequency equation of a multi-step non-uniform bar with several boundary conditions. This approach which combines the transfer matrix method and closed-form solutions of one step non-uniform bars leads to a single frequency equation for any number of steps. Numerical example shows that the computed values of the longitudinal fundamental natural frequency and mode shape of a tall building by the proposed method are close to the field measured data. It is also demonstrated through the numerical example that the selected expressions are suitable for describing the distributions of stiffness and mass of typical tall buildings.