**1. Introduction**

Seismic pounding refers to the collision of adjacent structures during earthquakes. When closely spaced structures vibrate out of phase, they might pound against each other. Pounding can cause local damage at the contact region. In addition, the impact generates short-duration acceleration pulse that can adversely affect the pounding buildings and their contents (see, for example, Abdel Raheem [1], and references therein). Such effects have been documented during past earthquakes. Miari et al. [2] and Abdel Raheem et al. [3] provide very good reviews of seismic pounding and refer to many studies describing pounding observed during past earthquakes. Various methods of modeling pounding and computation of structural response have been discussed in the literature (see, for example, Dimitrakopoulos, [4]).

Pounding can be avoided if adjacent buildings are adequately separated, but there is large uncertainty in just how much separation is adequate. Architectural and financial factors often dictate separation distance between buildings in metropolitan areas where land is expensive and scarce. Mitigation of seismic pounding is, therefore, an important structural engineering problem. One approach is to reduce the consequences of pounding, while the other is to reduce probabilities of pounding. Collision shear walls have been found to be effective in reducing the consequences of pounding (Anagnostopoulos and Karamaneas, [5]). Elastic gap devices (see, for example, Dicleli, [6]), have been shown to be effective in mitigating damage caused by pounding.

Several solutions for reducing probabilities of pounding between adjacent structures have been proposed in the literature. These solutions range from connecting adjacent buildings with springs and viscous/viscoelastic dashpots (Jankowski and Mahmoud, [7]; Richardson et al., [8,9]; Patel and Jangid, [10,11]; and Tubaldi, [12]). Friction dampers and viscous fluid dampers have also been proposed for mitigation of pounding risk (see, for example, Pratesi et al., [13] and Sorace and Terenzi, [14]).

Active, semi-active, and fuzzy control using magneto-rheological dampers have also been shown to be e ffective for seismic pounding mitigation (see, for example, Abdeddaim et al., [15]; Uz and Hadi, [16]). Pounding risk may be reduced by controlling the vibrations of adjacent structures. Tuned mass dampers (TMDs) for vibration control have been extensively studied in the literature (see, for example, Cao and Li, [17] or the review by Elias and Matsagar, [18]).

The risk of pounding in base-isolated buildings is higher due to large displacements concentrated at the isolation level (Agagnostopoulos and Spiliopoulos, [19]). Controlling displacement demands on base-isolated structures has been investigated using di fferent strategies in the literature. These strategies include the use of tuned liquid column dampers [20], and tuned mass dampers located at di fferent locations of the host buildings [20–23]. Coupling of base isolation system with tuned mass damper inerter systems is reported in De Domenico and Ricciardi [24], Hashimoto et al. [25] and De Domenico et al. [26]. Other strategies for displacement control of base-isolated buildings include the use of gap dampers [27].

While several studies on active and semi-active dampers for pounding mitigation are available in the literature, very few have tested passive TMDs for this purpose. The only work we are aware of in this regard is that of Abdullah et al. [28]. The idea was extended to semi-active TMD by Kim [29]. Abdullah et al. [28] presented the concept of a shared TMD to reduce vibrations and seismic pounding between adjacent buildings. Their optimal solution is to connect the TMD mass with a spring to one of the buildings and with a dashpot to the other building. If e ffective, this device could be advantageous due to its simplicity and low cost compared to active control schemes. One of the main appeals of this scheme is that the mass of the shared TMD is only half the mass of two TMDs installed on individual buildings.

Abdullah et al. [28] reported that the device is very e ffective in controlling vibrations and pounding between adjacent buildings. Upon closer inspection, we find that the solution presented by Abdullah et al. [28] is neither a tuned mass damper, nor optimal in mitigating pounding, and needs revisiting. The main objective of this paper is to investigate the dynamics of adjacent buildings connected by a tuned mass damper. We use single degree of freedom (SDOF) and multiple degree of freedom (MDOF) representations of buildings and several earthquake ground motions for numerical simulations to investigate the parameters of TMDs for optimal control of structural displacements and pounding.

#### **2. Conceptual and Mathematical Model of Shared TMDs (STMD)**

For conceptual convenience, let us consider two single degree of freedom (SDOF) systems connected by a shared TMD as shown in Figure 1. The SDOFs are simplified representations of buildings A and B, and their masses, sti ffnesses, and damping coe fficients are denoted as *mA*, *mB*; *kA*, *kB*; and *cA*, *cB*, respectively. The mass, sti ffness and damping coe fficient of the tuned mass damper are denoted by *m*, *k*, and *c*, respectively. An alternate scheme is to connect the TMD mass to building A with a dashpot and to building B with a spring.

**Figure 1.** Schematic representation of a tuned mass damper (TMD) shared by two buildings modeled as single degree of freedom (SDOF) systems.

The system mass, stiffness, and damping matrices are given below.

$$\mathbf{K} = \begin{bmatrix} k\_A + k & 0 & -k \\ 0 & k\_B & 0 \\ -k & 0 & k \end{bmatrix} \tag{1}$$

$$\mathbf{M} = \begin{bmatrix} m\_A & 0 & 0 \\ 0 & m\_B & 0 \\ 0 & 0 & m \end{bmatrix} \tag{2}$$

$$\mathbf{C} = \begin{bmatrix} c\_A & 0 & 0 \\ 0 & c\_B + c & -c \\ 0 & -c & c \end{bmatrix} \tag{3}$$

and the state matrix is defined as

$$\mathbf{S} = \begin{bmatrix} 0 & \mathbf{I} \\ -\mathbf{M}^{-1}\mathbf{K} & -\mathbf{M}^{-1}\mathbf{C} \end{bmatrix} \tag{4}$$

where **I** is an identity matrix. The performance (or cost) function, as defined by Abdullah et al. (2001) is the trace of a matrix **L**, which satisfies the following Lyapunov equation.

$$\mathbf{L}\mathbf{S} + \mathbf{S}^{\mathrm{T}}\mathbf{L} + \mathbf{Q} = 0,\tag{5}$$

where **Q** is a weighting matrix that assigns different importance to the displacements (and/or velocities) at the different degrees of freedom of the system. The design problem is then to estimate *k* and *c* that minimize the trace of **L**, denoted hereafter by *J*. The weighting matrix is taken as **Q**11 = 1 and all other elements as 0.
