Analysis of Tall Buildings with Damped Outriggers

Abstract

A novel damped outrigger system has been recently proposed for tall buildings, and is quite promising. To gain insight into the conceptual design of such systems, a simple beam-damper system model for a building with such dampers installed is developed and studied. A partial differential equation governing the motion is derived assuming a Bernoulli-Euler beam. A closed-form analytical solution is developed for vibration of the beam by analyzing the regions above and below the damper separately using separation of variables. By applying appropriate boundary conditions at the ends, a transcendental characteristic equation is obtained that governs the system’s complex natural frequencies. An explicit form for the complex mode shape is determined for dynamic analysis. A numerical iteration scheme is adopted to solve the characteristic equation for the complex eigenvalues (i.e., the system modal frequencies and damping ratios). This solution was used to determine design curves for optimal damper position and size. For engineering convenience, empirical equations were provided by fitting numerical results. These equations include one for determining the optimal location of the damper for each mode, and two for determining the optimal damping coefficient of the damper, and for calculating the maximum modal damping ratio of the system while the beam vibrates in its first mode. Furthermore, relatively accurate approximations of the pseudoundamped natural frequency and damping ratio of the first mode were obtained using a Taylor expansion of the characteristic equation. All of the results obtained are nondimensionalized for convenience of analysis and application.