## 2 Experimental Model Considering CSI Effect
### CSI Effect Analysis  
This section analyzes the essence of the CSI effect of an AMD control system with a DC motor, shown in Figure 5. It is modeled as a permanent magnet DC motor that is usually used in the control field of civil structures. Because there is a natural feedback path between the structure and the actuator, the CSI effect is unavoidable and needs to be considered [13]. Since the AMD mass is embedded with magnetic coil sets, the input-output relationship of a DC motor that allows for a concise analysis is:  
\[\dot{T}(t)=-\frac{R_{a}}{L_{a}}T(t)-\frac{K_{b}K_{i}K_{g}}{L_{a}r_{m}}\dot{x}_{A }(t)+\frac{K_{i}}{L_{a}}V(t) \tag{5}\]  
where \(T\) is the motor torque, \(V\) is the applied voltage, \(L_{a}=0.00073\) H is the armature inductance, \(R_{a}=0.0099\)\(\Omega\) is the armature resistance, \(K_{b}=1.27\) V/s is the back electromotive force constant, \(K_{i}=1.2\) Nm/A is the motor torque constant, and \(K_{g}=1\) and \(r_{m}=0.15\) m are the gear ratio and the lead of the ball screw, respectively.  
The absolute velocity of the AMD is:  
\[\dot{x}_{A}=\dot{x}_{a}-\dot{x}_{s4} \tag{6}\]  
where \(\dot{x}_{s4}\) and \(\dot{x}_{a}\) are the relative velocities of the fourth floor and the AMD.  
The horizontal control force produced by the rotary motor with its ball-screw can be defined as:  
\[u(t)=\frac{K_{g}}{r_{m}}T(t) \tag{7}\]An equation for the force and its transform can be obtained by substituting Equation (6) into Equation (5).  
\[\dot{u}(t)=-\frac{R_{a}}{L_{a}}u(t)-\frac{K_{b}K_{i}K_{g}^{2}}{L_{a}r_{m}^{2}}\dot {x}_{A}(t)+\frac{K_{i}K_{g}}{L_{a}r_{m}}V(t) \tag{8}\]  
Then:  
\[u(t)=-\frac{L_{a}}{R_{a}}\dot{u}(t)-\frac{K_{b}K_{i}K_{g}^{2}}{R_{a}r_{m}^{2}} \dot{x}_{A}(t)+\frac{K_{i}K_{g}}{R_{a}r_{m}}V(t) \tag{9}\]  
where \(-K_{b}K_{i}K_{g}^{2}\dot{x}_{A}/R_{a}r_{m}^{2}\) and \(-L_{a}\dot{u}/R_{a}\) are the control force loss caused by the reverse induction voltage and the inductance of the motor coils, respectively.  
The AMD is regarded as an analysis object, and the control force can be written as:  
\[u=m_{a}\ddot{x}_{A}+c_{a}\dot{x}_{A}+k_{a}x_{A} \tag{10}\]  
By using the Laplace transform, Equation (10) can be written as:  
\[u=s^{2}m_{a}x_{A}+sc_{a}x_{A}+k_{a}x_{A} \tag{11}\]  
The absolute displacements \(x_{A}\) of the AMD is:  
\[x_{A}=\frac{u}{s^{2}m_{a}+sc_{a}+k_{a}} \tag{12}\]  
As can be seen from Equations (9) and (12), there is an additional variable (relative structural velocity of the fourth floor) in Equation (9), meaning that the dynamical property of the AMD control system is coupled with the structural dynamics. Furthermore, a first-order differential of force also exists in Equation (9). As a result, corresponding to the different accuracy requirements of the solution, three models with different CSI effects can be established. Thus, Figure 6 is a block diagram representation of the different AMD models given in Equations (9) and (12).  
Regarding the DC motor installed on the example frame, three models can be shown in Figure 6: (1) Non-CSI effect only contains the item (a); (2) Considering the reduced-order CSI effect, the items (a) and (b) are contained, and the higher-order item (c) is neglected; (3) Considering the higher-order CSI effect, the items (a-c) are all considered. Different transfer functions from the control voltage to the control force are shown in Figures 7 and 8.  
Figure 5: A photograph of the DC motor.