Arch Comput Methods Eng (2013) 20:419–431 DOI 10.1007/s11831-013-9091-7 Tuned Mass Dampers Mariantonieta Gutierrez Soto · Hojjat Adeli Received: 29 August 2013 / Accepted: 29 August 2013 / Published online: 19 October 2013 © CIMNE, Barcelona, Spain 2013 Abstract A review of representative research on tuned massed dampers (TMD) reported in journals in recent years is presented. TMDs are divided into four categories: con- ventional TMDs, pendulum TMDs (PTMDs), bi-directional TMDs (BTMDs), and tuned liquid column dampers (TL- CDs). 1 Introduction Vibration control of structures can be divided into pas- sive, active, semi-active, hybrid systems. Passive control is the most established technology dating back to 1909 when Frahm received a U.S. patent for Dynamic Vibration Ab- sorber [18]. A large number of articles have been published on passive control of structures subjected to earthquake loading in recent years [6]. Also, there has been a significant amount of activities in the areas of active and semi-active control of structures [11, 34, 43]. Fisco and Adeli [16] pre- sented a state-of-the-art review of journal articles on active control of structures including active tuned mass dampers (ATMD) up to 2010. Fisco and Adeli [17] presented a review of journal articles on hybrid vibration control of structures and improved or new control strategies developed for civil structures including. Sirca and Adeli [53] present a review of journal articles in the area of structural system identifi- cation. More recently El-Khoury and Adeli [14] presented recent advances on vibration control of structures under dy- namic loading. The purpose of this article is to present a M. Gutierrez Soto · H. Adeli (B ) Department of Civil, Environmental, and Geodetic Engineering, The Ohio State University, 470 Hitchcock Hall, 2070 Neil Avenue, Columbus, OH 43210 USA e-mail: adeli.1@osu.edu review of tuned mass dampers (TMDs). TMDs are divided into four categories: conventional TMDs, pendulum TMDs (PTMDs), bi-directional TMDs (BTMDs), and tuned liquid column dampers (TLCDs). 2 Conventional Tuned Mass Dampers The equations representing a single-degree-of-freedom (SDOF) system equipped with a tuned mass damper (Fig. 1) are [13] (1 + μ) ¨ x + 2ξ s ω s ˙ x + ω 2 s x = p m d − μ ¨ u (Primary system) (1) ¨ u + 2ξ d ω d ˙ u + ω 2 d u =−¨ x (TMD system) (2) μ = m d m s (3) ω 2 s = k s m s (4) ω 2 d = k d m d (5) ξ d = c d 2ω d m d (6) ξ s = c s 2ω s m s (7) γ = ω d ω s (8) where μ represents the ratio of the TMD mass (m d ) to struc- tural mass (m s ), k s and c s are the stiffness and damping coefficient of the structure, k d and c d are the stiffness and damping coefficient of the damper, ξ s is the damping ratio