An innovative approach for evaluating mode shapes and natural frequencies of tubular frame and damped outriggers

A tall building comprised of tubular frame, central shear core, and outrigger-belt truss systems is considered as an equivalent flexural hollow box beam with lumped masses at each floor level. Natural frequencies and mode shapes of the un-damped hybrid system are obtained by forming mass and stiffness matrices and solving the eigenvalue problem. Previous researches did not deal with the overall intrinsic damping of the structure or the damping added by external instruments, nor they considered the hybrid structure comprised of a tubular frame with geometrical discontinuous and damped outrigger systems in free vibration studies. By utilizing the achieved frequencies, first to fifth mode shape vectors, mass matrix, and modal participant mass and considering 5% modal damping ratio for each mode, the classically damping matrix of the hybrid system is achieved to fill these gaps. Then the non-classical damping matrix of the hybrid system with damped outriggers is obtained by adding the viscosity of passive linear viscous dampers to the classical damping matrix at the level of damped outrigger-belt truss systems. The eigenvalue problem is solved, and natural frequencies and mode shapes are obtained. 3-D finite element models of two 60- and 70-story buildings with conventional and damped outrigger systems are analyzed, and comparing the results of the proposed method and those of SAP2000 models reveals the correctness, reliability, and robustness of the method. Moreover, calculating natural frequencies by the proposed method is much less time consuming in comparison with finite element models.