In this stury, control performance of tuned mass damper (TMD) with torsional rigidity for an eccentric structure showing torsional responses is investigated. To this end, an eccentric structure subjected to earthquake excitation is used to evaluate the control performance of torsional TMD by varying installed location and torsional rigidity of TMD, To reduce computational time required for repetitive time history analysis of an example structure having non-proportional damping system due to TMD, an equivalent analytical model is used in this study. Torsional properties of TMD usually neglected in typical TMD are verified to be effective in reduction of torsional responses of the eccentric structure. In the case of eccentric structures, it has been seen that the center of a plane of a structure may not be optimal location of TMD.