Calculation of vertical dynamic characteristics of tall buildings with viscous damping

The magnitude of the vertical component of earthquake ground motion is often about one-third of the horizontal component. Thus, it is necessary to calculate vertical dynamic characteristics of tall buildings and high-rise structures, it is possible to regard such structures as a cantilever bar with variable cross-section. In this paper, the differential equations of free longitudinal vibrations (in vertical direction) of bars with variably distributed mass and stiffness considering damping effect are established. The damping coeffient of a bar is assumed to be proportional to its mass, and the general solutions of mode shapes of damped distributed parameter systems are reduced to bessel's equations by selecting suitable expressions, such as power functions and exponential functions, for the distributions of stiffness and mass. An approach to determine the natural frequencies and mode shapes in vertical direction for tall buildings with variably distributed stiffness and variably distributed mass is proposed. The presneted method is also applicable to the free longitudinal vibration analysis without considering damping effect (damping coefficient in vibration equations is equal to zero). A numerical example shows that the computed values of the fundamental longitudinal natural frequency and mode shape by the proposed method are close to the full scale measured data. It is shown through the numerical example that the selected expressions are suitable for describing the distributions of stiffness and mass of typical tall buildings. A comparison between undamped strucural dynamic characteristics and damped natural frequencies, mode shapes is made in this paper.