Flexural free vibration of cantilevered structures of variable stiffness and mass

Using appropriate transformations, the differential equation for flexural f ree vibration of a cantilever bar with variably distributed mass and stiffn ess is reduced to a Bessel's equation or an ordinary differential equation with constant coefficients by selecting suitable expressions, such as power functions and exponential functions, for the distributions of stiffness an d mass. The general solutions for flexural free vibration of one-step bar w ith variable cross-section are derived and used to obtain the frequency equ ation of multi-step cantilever bars. The new exact approach is presented wh ich combines the transfer matrix method and closed form solutions of one st ep bars. Two numerical examples demonstrate that the calculated natural fre quencies and mode shapes of a 27-storey building and a television transmiss ion tower are in good agreement with the corresponding experimental data. I t is also shown through the numerical examples that the selected expression s are suitable for describing the distributions of stiffness and mass of ty pical tall buildings and high-rise structures.