Figure 1(a) shows a conventional system with small mass-ratio TMD on the roof which is effective only for wind loading. On the other hand, Fig. 1(b) presents a high-rise building with large mass-ratio TMD on the roof which is believed to be effective for long-period ground motion and to cause significant vertical load on the building. Consider next a base-isolated building system, as shown in Fig. 1(c), with large mass-ratio TMD on the roof which lengthens the fundamental natural period of the high-rise building and also causes large vertical load on the building. The models in Fig. 1(b) and (c) are thought to be unrealistic because of their excessive vertical load. Figure 2(a) indicates the proposed base-isolated building system with large mass-ratio TMD at basement using sliders and rails. This model shown in Fig. 2(a) is called the *Proposed-1 model*. In Fig. 2(a) the large mass-ratio TMD is located on the sliders and rails and in Fig. 2(b) the large mass-ratio TMD is set on the floor just above the base-isolation system.

### Base-isolated building without TMD

Consider a base-isolated building without TMD. This model is called a *BI model* (see Ariga et al. 2006). Let *k*
_{
I
}, *c*
_{
I
}, *m*
_{
I
} denote the stiffness, damping coefficient and mass of the base-isolation story in the BI model. Furthermore let *k*
_{1}, *c*
_{1}, *m*
_{1} denote the stiffness, damping coefficient and mass of the superstructure. The displacements of masses *m*
_{1} and *m*
_{
I
} relative ground are denoted by *u*
_{1} and *u*
_{
I
}, respectively. This model is subjected to the base ground acceleration *ü*
_{
g
}. The equations of motion for this model can be expressed by

$$ \left(\begin{array}{cc}\hfill {m}_I\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill {m}_1\hfill \end{array}\right)\left(\begin{array}{c}\hfill {\ddot{u}}_I\hfill \\ {}\hfill {\ddot{u}}_1\hfill \end{array}\right)+\left(\begin{array}{cc}\hfill {c}_I+{c}_1\hfill & \hfill -{c}_1\hfill \\ {}\hfill -{c}_1\hfill & \hfill {c}_1\hfill \end{array}\right)\left(\begin{array}{c}\hfill {\dot{u}}_I\hfill \\ {}\hfill {\dot{u}}_1\hfill \end{array}\right)+\left(\begin{array}{cc}\hfill {k}_I+{k}_1\hfill & \hfill -{k}_1\hfill \\ {}\hfill -{k}_1\hfill & \hfill {k}_1\hfill \end{array}\right)\left(\begin{array}{c}\hfill {u}_I\hfill \\ {}\hfill {u}_1\hfill \end{array}\right)=\left(\begin{array}{c}\hfill -{m}_I{\ddot{u}}_g\hfill \\ {}\hfill -{m}_1{\ddot{u}}_g\hfill \end{array}\right) $$

(1)

### Conventional base-isolated building with large-mass ratio TMD

Recently some systems of a base-isolated building with large-mass ratio TMD have been proposed. Mukai et al. (2005) proposed a new-type active response control system to improve the effectiveness of base-isolated buildings. In this system, the TMD mass is connected both to a superstructure and the basement (ground). A negative stiffness mechanism is used to amplify the response of the TMD mass which enables the avoidance of introduction of large mass-ratio TMD. Nishii et al. (2013) revised the system due to Mukai et al. (2005) by replacing the active damper with negative stiffness with a passive inertial mass damper system. This model is called the *Imass TMD model* in this paper. Although their system is demonstrated to be effective for the reduction of superstructure response, the performance check on the reaction of the TMD system is not conducted. Xiang and Nishitani (2014) presented a system for a base-isolated building with a TMD mass which is located on the base-isolation story level and connected directly to the ground. This model is called the *NewTMD model* in this paper. They demonstrated that their system is effective for a broad range of excitation frequency and proposed an optimization method for determining the system parameters.

Consider an Imass TMD model and a NewTMD model as shown in Fig. 3. Let *k*
_{2}, *c*
_{2}, *m*
_{2} denote the stiffness, damping coefficient and mass of the TMD system. *z*
_{2} indicates the inertial mass capacity of the inertial mass damper installed between TMD mass and ground in the Imass TMD model.

For later comparison, the Imass TMD model and the NewTMD model are explained in the following. The equations of motion for Imass TMD model may be expressed by

$$ \begin{array}{l}\left(\begin{array}{ccc}\hfill {m}_I\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill {m}_1\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill {m}_2+{z}_2\hfill \end{array}\right)\left(\begin{array}{c}\hfill {\ddot{u}}_I\hfill \\ {}\hfill {\ddot{u}}_1\hfill \\ {}\hfill {\ddot{u}}_2\hfill \end{array}\right)+\left(\begin{array}{ccc}\hfill {c}_I+{c}_1+{c}_2\hfill & \hfill -{c}_1\hfill & \hfill -{c}_2\hfill \\ {}\hfill -{c}_1\hfill & \hfill {c}_1\hfill & \hfill 0\hfill \\ {}\hfill -{c}_2\hfill & \hfill 0\hfill & \hfill {c}_2\hfill \end{array}\right)\left(\begin{array}{c}\hfill {\dot{u}}_I\hfill \\ {}\hfill {\dot{u}}_1\hfill \\ {}\hfill {\dot{u}}_2\hfill \end{array}\right)\\ {}\kern11.75em +\left(\begin{array}{ccc}\hfill {k}_I+{k}_1+{k}_2\hfill & \hfill -{k}_1\hfill & \hfill -{k}_2\hfill \\ {}\hfill -{k}_1\hfill & \hfill {k}_1\hfill & \hfill 0\hfill \\ {}\hfill -{k}_2\hfill & \hfill 0\hfill & \hfill {k}_2\hfill \end{array}\right)\left(\begin{array}{c}\hfill {u}_I\hfill \\ {}\hfill {u}_1\hfill \\ {}\hfill {u}_2\hfill \end{array}\right)=\left(\begin{array}{c}\hfill -{m}_I{\ddot{u}}_g\hfill \\ {}\hfill -{m}_1{\ddot{u}}_g\hfill \\ {}\hfill -{m}_2{\ddot{u}}_g\hfill \end{array}\right)\end{array} $$

(2)

On the other hand, the equations of motion for NewTMD model may be presented by

$$ \left(\begin{array}{ccc}\hfill {m}_I\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill {m}_1\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill {m}_2\hfill \end{array}\right)\left(\begin{array}{c}\hfill {\ddot{u}}_I\hfill \\ {}\hfill {\ddot{u}}_1\hfill \\ {}\hfill {\ddot{u}}_2\hfill \end{array}\right)+\left(\begin{array}{ccc}\hfill {c}_I+{c}_1\hfill & \hfill -{c}_1\hfill & \hfill 0\hfill \\ {}\hfill -{c}_1\hfill & \hfill {c}_1\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill {c}_2\hfill \end{array}\right)\left(\begin{array}{c}\hfill {\dot{u}}_I\hfill \\ {}\hfill {\dot{u}}_1\hfill \\ {}\hfill {\dot{u}}_2\hfill \end{array}\right)+\left(\begin{array}{ccc}\hfill {k}_I+{k}_1+{k}_2\hfill & \hfill -{k}_1\hfill & \hfill -{k}_2\hfill \\ {}\hfill -{k}_1\hfill & \hfill {k}_1\hfill & \hfill 0\hfill \\ {}\hfill -{k}_2\hfill & \hfill 0\hfill & \hfill {k}_2\hfill \end{array}\right)\left(\begin{array}{c}\hfill {u}_I\hfill \\ {}\hfill {u}_1\hfill \\ {}\hfill {u}_2\hfill \end{array}\right)=\left(\begin{array}{c}\hfill -{m}_I{\ddot{u}}_g\hfill \\ {}\hfill -{m}_1{\ddot{u}}_g\hfill \\ {}\hfill -{m}_2{\ddot{u}}_g\hfill \end{array}\right) $$

(3)

### Base-isolated building with large-mass ratio TMD at basement using inertial mass damper for stroke reduction

The equations of motion for a base-isolated building with large-mass ratio TMD at basement may be expressed by

$$ \begin{array}{l}\left(\begin{array}{ccc}\hfill {m}_I\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill {m}_1\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill {m}_2\hfill \end{array}\right)\left(\begin{array}{c}\hfill {\ddot{u}}_I\hfill \\ {}\hfill {\ddot{u}}_1\hfill \\ {}\hfill {\ddot{u}}_2\hfill \end{array}\right)+\left(\begin{array}{ccc}\hfill {c}_I+{c}_1+{c}_2\hfill & \hfill -{c}_1\hfill & \hfill -{c}_2\hfill \\ {}\hfill -{c}_1\hfill & \hfill {c}_1\hfill & \hfill 0\hfill \\ {}\hfill -{c}_2\hfill & \hfill 0\hfill & \hfill {c}_2\hfill \end{array}\right)\left(\begin{array}{c}\hfill {\dot{u}}_I\hfill \\ {}\hfill {\dot{u}}_1\hfill \\ {}\hfill {\dot{u}}_2\hfill \end{array}\right)\\ {}\kern10.25em +\left(\begin{array}{ccc}\hfill {k}_I+{k}_1+{k}_2\hfill & \hfill -{k}_1\hfill & \hfill -{k}_2\hfill \\ {}\hfill -{k}_1\hfill & \hfill {k}_1\hfill & \hfill 0\hfill \\ {}\hfill -{k}_2\hfill & \hfill 0\hfill & \hfill {k}_2\hfill \end{array}\right)\left(\begin{array}{c}\hfill {u}_I\hfill \\ {}\hfill {u}_1\hfill \\ {}\hfill {u}_2\hfill \end{array}\right)=\left(\begin{array}{c}\hfill -{m}_I{\ddot{u}}_g\hfill \\ {}\hfill -{m}_1{\ddot{u}}_g\hfill \\ {}\hfill -{m}_2{\ddot{u}}_g\hfill \end{array}\right)\end{array} $$

(4)

A base-isolated building, as shown in Fig. 2(c), with large-mass ratio TMD at basement using an inertial mass damper for stroke reduction is called the *Proposed-2 model*. A mechanism example of inertial mass dampers is shown in Fig. 2(d) (Takewaki et al. 2012a). The equations of motion for this model may be expressed by

$$ \begin{array}{l}\left(\begin{array}{ccc}\hfill {m}_I+{z}_2\hfill & \hfill 0\hfill & \hfill -{z}_2\hfill \\ {}\hfill 0\hfill & \hfill {m}_1\hfill & \hfill 0\hfill \\ {}\hfill -{z}_2\hfill & \hfill 0\hfill & \hfill {m}_2+{z}_2\hfill \end{array}\right)\left(\begin{array}{c}\hfill {\ddot{u}}_I\hfill \\ {}\hfill {\ddot{u}}_1\hfill \\ {}\hfill {\ddot{u}}_2\hfill \end{array}\right)+\left(\begin{array}{ccc}\hfill {c}_I+{c}_1+{c}_2\hfill & \hfill -{c}_1\hfill & \hfill -{c}_2\hfill \\ {}\hfill -{c}_1\hfill & \hfill {c}_1\hfill & \hfill 0\hfill \\ {}\hfill -{c}_2\hfill & \hfill 0\hfill & \hfill {c}_2\hfill \end{array}\right)\left(\begin{array}{c}\hfill {\dot{u}}_I\hfill \\ {}\hfill {\dot{u}}_1\hfill \\ {}\hfill {\dot{u}}_2\hfill \end{array}\right)\\ {}\kern12.25em +\left(\begin{array}{ccc}\hfill {k}_I+{k}_1+{k}_2\hfill & \hfill -{k}_1\hfill & \hfill -{k}_2\hfill \\ {}\hfill -{k}_1\hfill & \hfill {k}_1\hfill & \hfill 0\hfill \\ {}\hfill -{k}_2\hfill & \hfill 0\hfill & \hfill {k}_2\hfill \end{array}\right)\left(\begin{array}{c}\hfill {u}_I\hfill \\ {}\hfill {u}_1\hfill \\ {}\hfill {u}_2\hfill \end{array}\right)=\left(\begin{array}{c}\hfill -{m}_I{\ddot{u}}_g\hfill \\ {}\hfill -{m}_1{\ddot{u}}_g\hfill \\ {}\hfill -{m}_2{\ddot{u}}_g\hfill \end{array}\right)\end{array} $$

(5)

The model parameters of BI Model, Proposed-1 Model and Proposed-2 Model as shown in Fig. 3 are specified as follows. The same model parameters are used for Imass TMD Model and NewTMD Model. The influence of the rail friction on the response of the proposed models will be discussed in Section ‘Influence of rail friction on response of proposed system’.

The superstructure is a 20-story or 50-story reinforced concrete building and is modeled into a single-degree-of-freedom (SDOF) model. This modeling into an SDOF model is thought to be appropriate in a base-isolated building. The equal story height of the original building is 3.5 m. The building has a plan of 40 × 40 m and the floor mass is obtained from 1.0 × 10^{3} kg/m^{2}. The floor mass in each floor is 1.6 × 10^{6} kg. The fundamental natural period of the superstructure with fixed base is *T*
_{1} = 1.4 s for a 20-story building and *T*
_{1} = 3.5 s for a 50-story building. The structural damping ratio is assumed to be *h*
_{1} = 0.02. The stiffness and damping coefficient of the SDOF model are computed by \( {k}_1={m}_1{\omega}_1^2 \), *c*
_{1} = 2*h*
_{1}
*k*
_{1}/*ω*
_{1} with the fundamental natural circular frequency *ω*
_{1} = 2*π*/*T*
_{1}.

The mass of the base-isolation story is 4.8 × 10^{6} kg. The fundamental natural period of the BI model with rigid superstructure is *T*
_{
I
} = 5.0 s for the 20-story model and *T*
_{
I
} = 6.0 s for the 50-story model. The damping ratio of the BI model with rigid superstructure is *h*
_{
I
} = 0.1. The stiffness and damping coefficient of the SDOF model are computed by \( {k}_I=\left({m}_I+{m}_1\right){\omega}_I^2 \), *c*
_{
I
} = 2*h*
_{
I
}
*k*
_{
I
}/*ω*
_{
I
} with the fundamental natural circular frequency *ω*
_{
I
} = 2*π*/*T*
_{
I
}. As for TMD, the mass ratio *m*
_{2}/*m*
_{1} is set to *μ* = 0.1 and the inertial mass damper ratio *z*
_{2}/*m*
_{1} is set to *η*
_{
s
} = 0.06. The damping ratio is assumed to be *h*
_{2} = 0.3. The stiffness and damping coefficient of TMD are given by \( {k}_2=\left({m}_2+{z}_2\right){\omega}_2^2 \), *c*
_{2} = 2*h*
_{2}
*k*
_{2}/*ω*
_{2} in terms of the natural circular frequency *ω*
_{2} of TMD . The process of determining *ω*
_{2} is explained in Section ‘Determination of stiffness and damping coefficient of TMD’.

### Determination of stiffness and damping coefficient of TMD

In this section, the procedure of determination of stiffness and damping coefficient of TMD for the proposed model, Imass TMD model and NewTMD model is explained. The tuning of TMD is performed by minimizing the response ratio *D* of the deformation of the base-isolation story to the base input (displacement amplitude) as shown in Fig. 4.

Let us assume the input ground acceleration as

$$ {\ddot{u}}_g=A{e}^{i\omega t} $$

(6)

The harmonic response of the systems may be expressed by

$$ \left(\begin{array}{ccc}\hfill {u}_I\hfill & \hfill {u}_1\hfill & \hfill {u}_2\hfill \end{array}\right)=\left(\begin{array}{ccc}\hfill {U}_I\hfill & \hfill {U}_1\hfill & \hfill {U}_2\hfill \end{array}\right){e}^{i\omega t} $$

(7)

By solving the equations of motion, the response amplitude may be obtained as

$$ {\left(\begin{array}{ccc}\hfill {U}_I\hfill & \hfill {U}_1\hfill & \hfill {U}_2\hfill \end{array}\right)}^T={\left(-{\omega}^2\mathbf{M}+i\omega \mathbf{C}+\mathbf{K}\right)}^{-1}{\left(\begin{array}{ccc}\hfill -{m}_IA\hfill & \hfill -{m}_1A\hfill & \hfill -{m}_2A\hfill \end{array}\right)}^T $$

(8)

where ()^{T} indicates the matrix transpose. The displacement response ratio *D* can then be defined by

$$ D=\left|\frac{U_I}{A/{\omega}_{I1}^2}\right| $$

(9)

where *ω*
_{
I1} is the undamped natural circular frequency of the BI model.