Displacements and Forces in Structures with Inerters when Subjected to Earthquakes Nicos Makris, M.ASCE 1 ; and Gholamreza Moghimi 2 Abstract: This study investigates the seismic response of a two-degrees-of-freedom structure with supplemental rotational inertia at its first story. The proposed response-modification strategy uses an inerter—a mechanical device the resisting force of which is proportional to the relative acceleration between the first story and the ground. This arrangement complements the traditional supplemental damping strategies, which are also examined in this work. The paper develops a time-domain and a frequency-domain formulation for the response analysis and shows that the seismic protection of structures with supplemental rotational inertia suppresses effectively interstory drifts at the expense of transferring appreciable forces at the support of the inerter. Both a single inerter and a pair of inerters that can only resist the motion of the structure are examined. The paper examines the extent to which a compliant support of the inerter affects the dynamics of the structure and concludes that as the compliance of the support frame increases, a single inerter may lead to a more favorable response. The proposed response-modification strategy is attractive for cases with large relative displacement demands. DOI: 10.1061/(ASCE)ST.1943- 541X.0002267. © 2018 American Society of Civil Engineers. Introduction The concept of modifying structural response with supplemental rotational inertia was apparently first introduced in Japan in the late 1990s by Arakaki et al. (1999a, b), who proposed a ball– screw assembly to modify the seismic response of structures. When the lead angle of the screw is appropriate, a ball–screw can be backdriven, producing a force output that is proportional only to the relative acceleration between its end nodes, while stor- ing kinetic energy. Accordingly, a backdriven ball–screw is the pre- cise mechanical analog of an electrical capacitor in a force-current/ velocity-voltage analogy between mechanical and electrical net- works. This missing analogy was first recognized by Smith (2002), who coined the term inerter for any mechanical arrangement in which the output force is proportional only to the relative acceler- ation between its end nodes. For instance, the driving spinning top in Fig. 1 is a physical realization of the inerter because the driving force is only proportional to the relative acceleration between Nodes 1 and 2. The constant of proportionality of the inerter is called the inertance, M R (Smith 2002) and has a unit of mass M. The unique characteristic of the inerter is that it has an appreciable inertial mass as opposed to a marginal gravitational mass. Accord- ingly, if F 1 and u 1 and F 2 and u 2 are the forces and displacements, respectively, at the end nodes of the inerter with inertance M R ; the constitutive relation is defined as (Saitoh 2007; Makris 2018) ( F 1 ðtÞ F 2 ðtÞ ) ¼ " M R -M R -M R M R #( ¨ u 1 ðtÞ ¨ u 2 ðtÞ ) ð1Þ Smith and coworkers developed and tested both a rack-and- pinion inerter and a ball–screw inerter (Papageorgiou and Smith 2005; Papageorgiou et al. 2009). Upon its conceptual development and experimental validation, the inerter was implemented to control the suspension vibrations of racing cars under the name J-damper (Chen et al. 2009; Kuznetsov et al. 2011). In parallel with the aforementioned developments in vehicle mechanics and dynamics, and following the pioneering work of Arakaki et al. (1999a, b), a growing number of publications have proposed the use of rotational inertia dampers for the wind and seismic protection of civil structures. As an example, Hwang et al. (2007) proposed a rotational inertia damper in association with a toggle bracing for vibration control of building structures. Ikago et al. (2012) studied the dynamic response of a single-degree-of- freedom (SDOF) structure equipped with a rotational damper that is very similar to the rotational inertia damper initially proposed by Hwang et al. (2007). Their configuration contained an additional flywheel to accentuate the rotational inertia effect of the proposed response modification device. About the same time, Takewaki et al. (2012) examined the response of SDOF and multiple-degree-of- freedom (MDOF) structures equipped with supplemental rotational inertia offered from a ball-screw-type device that sets in motion a rotating flywheel. More recent studies of the response of MDOF structures equipped with supplemental rotational inertia were pre- sented by Miranda (2005), Hoang and Warnitchai (2005), Wang et al. (2010), Marian and Giaralis (2014), Lazar et al. (2014), Chen et al. (2014), and Giaralis and Taflanidis (2018). In addition to the traditional design of inerters that involve rotating masses (rack– pinion–flywheel or ball–screw inerters), Swift et al. (2013) pro- posed a fluid inerter. The aforementioned studies focused invariably on the effective- ness of inerters to reduce structural displacements without looking into the resulting forces and the overall demand in base shear when such response modification devices are used. In a recent paper, Makris and Kampas (2016) showed that seismic protection of struc- tures with supplemental rotational inertia is most effective in reduc- ing spectral displacements of long-period SDOF structures at the expense of transferring appreciable forces to the support of the 1 Professor, Dept. of Civil and Environmental Engineering, Southern Methodist Univ., Dallas, TX 75275; formerly, Dept. of Civil, Environmen- tal, and Construction Engineering, Univ. of Central Florida, Orlando, FL 32816 (corresponding author). Email: nmakris@smu.edu 2 Doctoral Candidate, Dept. of Civil, Environmental, and Construction Engineering, Univ. of Central Florida, Orlando, FL 32816. Note. This manuscript was submitted on March 2, 2018; approved on August 17, 2018; published online on December 14, 2018. Discussion period open until May 14, 2019; separate discussions must be submitted for individual papers. This paper is part of the Journal of Structural En- gineering, © ASCE, ISSN 0733-9445. © ASCE 04018260-1 J. Struct. Eng. J. Struct. Eng., 2019, 145(2): 04018260 Downloaded from ascelibrary.org by Nikolaos Makris on 12/16/18. Copyright ASCE. For personal use only; all rights reserved.