Free vibration analysis for tube-in-tube tall buildings

An approximate solution procedure is formulated for free vibration analysis of tube-in-tube tall buildings in this paper. The governing partial differential equation of motion is reduced to an ordinary differential equation with variable coefficients on the assumption that the transverse displacement is a harmonic vibration. A power-series solution which represents the mode shape function of tube-in-tube tall buildings is derived. Applying the boundary conditions yields the boundary value problem, from which the frequency equation is established and solved through a numerical process to determine the natural frequencies. Two numerical examples are performed and compared with results available in the published literature to show the accuracy of the proposed method. Care has been exercised to retain sufficient terms in power series in evaluating natural frequencies of accepted accuracy. The influences of the factors including flexural rigidity, mass per unit length and building height to the natural frequency are discussed. The method proposed herein enables one to calculate as an alternative the natural frequency of tube-in-tube tall buildings with good accuracy associated by calculators and hand, prior to use of the complicated computer programs.